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+arXiv:1001.0029v2  [hep-th]  3 Aug 2010Gravity assisted solution
+of the mass gap problem
+for pureYang-Mills fields
+Arkady L.Kholodenko
+375 H.L.Hunter Laboratories, Clemson University, Clemson,
+SC 29634-0973, USA. e-mail: string@clemson.edu
+In 1979 Louis Witten demonstrated that stationary axially symmetr ic Ein-
+stein field equationsand those for static axiallysymmetric self-dual SU(2) gauge
+fields can both be reduced to the same (Ernst) equation. In this pa per we
+use this result as point of departure to prove the existence of the mass gap
+for quantum source-free Yang-Mills (Y-M) fields. The proof is facilit ated by
+results of our recently published paper, JGP 59 (2009) 600-619. S ince both
+pure gravity, the Einstein-Maxwell and pure Y-M fields are describe d for axi-
+ally symmetric configurations by the Ernst equation classically, their quantum
+descriptions are likely to be interrelated. Correctness of this conj ecture is suc-
+cessfully checked by reproducing (by different methods) results o f Korotkin and
+Nicolai, Nucl.Phys.B475 (1996) 397-439, on dimensionally reduced qua ntum
+gravity. Consequently, numerous new results supporting the Fad deev-Skyrme
+(F-S) -type models are obtained. We found that the F-S-like model is best
+suited for description of electroweak interactions while strong inte ractions re-
+quire extension of Witten’s results to the SU(3) gauge group. Such an extension
+is nontrivial. It is linked with the symmetry group SU(3) ×SU(2)×U(1) of the
+Standard Model. This result is quite rigid and should be taken into acco unt in
+development of all grand unified theories. Also, the alternative (to the F-S-like)
+model emerges as by-product of such an extension. Both models a re related to
+each other via known symmetry transformation. Both models poss ess gap in
+their excitation spectrum and are capable of producing knotted/lin ked config-
+urations of gauge/gravity fields. In addition, the paper discusses relevance of
+the obtained results to heterotic strings and to scattering proce sses involving
+topology change. It ends with discussion about usefulness of this in formation
+for searches of Higgs boson.
+Keywords : ExtendedRicciflow; Bose-Einsteincondensation; Ernst,Landa u-
+Lifshitz, Gross-Pitaevski, Richardson-Gaudin equations; Einstein ’s vacuum and
+electrovacuumequations; Floer’stheory; instantons,monopoles ,calorons;knots,
+links and hyperbolic 3-manifolds; Standard Model; Higgs boson.
+Mathematics Subject Classifications 2010 . Primary: 83E99, 53Z05, 53C21,
+83E30,81T13,82B27;Secondary :82B23
+11 Introduction
+1.1 General remarks
+History of physics is full of situations when experimental observat ions lead to
+deep mathematical results. Discoveryof Yang-Mills (Y-M) fields in 19 54[1] falls
+out of this trend. Furthermore, if one believes that theory of the se fields makes
+sense, they should never be directly observed. To make sure that these fields
+do exist, it is necessary to resort to all kinds of indirect methods to probe them.
+Physically, the rationale for the Y-M fields is explained already in the or iginal
+Yang and Mills paper [1]. Mathematically, such a field is easy to understa nd.
+It is a non Abelian extension of Maxwell’s theory of electromagnetism. In
+1956 Utiyama [2] demonstrated that gravity, Y-M and electromagn etism can
+be obtained from general principle of local gauge invariance of the u nderlying
+Lagrangian. The explicit form of the Lagrangian is fixed then by assu mptions
+about its symmetry. For instance, by requiring invariance of such a Lagrangian
+with respect to the Abelian U(1) group, the functional for the Max well field
+is obtained, while doing the same operations but using the Lorentz gr oup the
+Einstein-Hilbert functional for gravitational field is recovered. By employing
+the SU(2) non Abelian gauge group the original Y-M result [1] is recov ered.
+Only Maxwell’s electromagnetic field is reasonably well understood bot h at
+the classical and quantum level. Due to their nonlinearity, the Y-M fie lds are
+much harder to study even at the semi/classical level. In particular , no classical
+solutions e.g. solitons (or lumps) with finite action are known in Minkows ki
+space-time. This result was proven by many authors, e.g. see [3-4] and refr-
+erences therein. The situation changes dramatically in Euclidean spa ce where
+the self-duality constraint allows to obtain meaningful classical solu tions [5,6].
+These are helpful for development of the theory of quantum Y-M fi elds. Such
+solutions are useful in the fields other than quantum chromodynam ics (QCD)
+since the self-duality equations are believed to be at the heart of all exactly
+integrable systems [7]. Although the self-duality equations originate from study
+of the Y-M functional, not all solutions [6] of these equations are re levant to
+QCD. In this paper we discuss the rationale behind the selection proc edure.
+In QCD solutions of self-duality equations, known as instantons , are describ-
+ing tunneling between different QCD vacua [8]. It should be noted thou gh
+that treatment of instantons in mathematics [9-11] and physics lite rature [8]
+is different. This fact is important. It is important since one of the ma jor
+tasks of nonperturbative QCD lies in developing mathematically corre ct and
+physically meanigful description of these vacua. According to a poin t of view
+existing in physics literature the QCD has a countable infinity of topolo gically
+different vacua. Supposedly, the Faddeev-Skyrme (F-S) model is designed for
+description of these vacua. If this model can be used for this purp ose, then
+2each vacuum state is expected to be associated with a particular kn ot (or link)
+configuration. Under these conditions the instantons are believed to be well
+localized objects interpolating between different knotted/linked va cuum config-
+urations [12-16]. These configurations upon quantization are expe cted to posses
+a tower of excited states. Whether or not such a tower has a gap in its spec-
+trum or the spectrum is gaplessis the essence ofthe millennium prize p roblem1.
+Originally, the above results were obtained and discussed only for SU (2) gauge
+fields [17]. They were extended to SU(N) case, N≥2,only quite recently [18]2.
+Although such a description of QCD vacua is in accord with general pr inciples
+of instanton calculations [8], it is in formaldisagreement with results known in
+mathematics [9-11]. Indeed, it is well known that complement of a par ticular
+knot inS3is 3-manifold. Since instantons ”live” in R4(or any Riemannian
+4-manifold allowing an anti self -dual decomposition of the Y-M field (e .g. see
+Ref.[9], pages 38-393), this means that all knots in R4(orS4) are trivial and
+one should talk about knotted spheres instead of knotted rings [20 ]. This known
+topological fact is in apparent contradiction with results of [13-15]. In this work
+we shall provide evidence that such a contradiction is only apparent and that,
+indeed, knotted configurations in S3are consistent with the notion of instan-
+tons as formulated in mathematics. This is achieved by using results b y Floer
+[21]. It should be noted, though, that known to us ”proofs” [22-24 ] of the
+existence of the mass gap in pure Y-M theory done at the physical le vel of rigor
+ignore instanton effects altogether. Among these papers only Ref .[22] uses the
+F-S SU(2) model for such mass gap calculations. It also should be no ted that
+results of such calculation sensitively depend upon the way the F-S m odel is
+quantized. For instance, in the work by Faddeev and Niemi [25], done for the
+SU(2) gauge group, the results of quantization produce gaplesss pectrum. To fix
+the problem the same authors suggested to extend the original mo del in ad hock
+fashion. Other authors, e.g. see Ref.[26], proposed different ad ho c solution of
+the same problem.
+The above results are formally destroyed by the effects of gravity . Indeed, in
+1988 Bartnik and McKinnon numerically demonstrated [27] that the c ombined
+Y-M and gravity fields lead to a stable particle-like (solitonic) solutions while
+neither source-freegravitynor pure Y-M fields are capable of pro ducing such so-
+lutions4. Such situation has interesting cosmological ramifications5[28] causing
+disappearance of singularities in spacetime as shown by Smoller et al [2 9]. In
+this work we do not discuss implications of these remarkable results. Instead, in
+the spirit of Floer’s ideas [21], we argue that even without taking thes e results
+into account, the effects of gravity on processes of high energy p hysics are quite
+substantial.
+1E.g. see
+http://www.claymath.org/millennium/Yang-Mills Theory/
+2In this work, in accord with experimental evidence, we demon strate that N≤3.
+3In physics literature, both anti and self dual instantons ar e allowed to exist, e.g. see
+Ref.[19], page 481.
+4More accurately, neither pure Y-M fields nor pure gravity hav e nontrivial static globally
+regular (i.e.nonsingular, asymptotically flat) solitons.
+5E.g. Einstein-Y-M hairy black holes
+31.2 Statements of the problems to be solved
+Inthispaperseveralproblemsareposedandsolved. Inparticular ,wewouldlike
+to investigate the physics and mathematics behind gravity-Y-M cor respondence
+discovered by Louis Witten [30] for SU(2) gauge fields. Is this corre spondence
+accidental? If it is not accidental, how it should be related to commonly shared
+opinion that the Standard Model (SM) of particle physics does not a ccount
+for gravity? Can this correspondence be extended to other gaug e fields, e.g.
+SU(N), N>2 ? If the answer is ”yes”, will such correspondence be valid for all
+N’s or just for few? In the last case, what such a restriction means physically?
+Howthe noticed correspondenceis helping to solvethe gapproblem? What role
+the F-S model is playing in this solution? Is this model instrumental in s olving
+the gap problem or are there other aspects of this problem which th e F-S model
+is unable to account? How this correspondence affects known strin g-theoretic
+and loop quantum gravity (LQG) results ? What place the topology-c hanging
+(scattering) processes occupy in this correspondence? Is ther e any relevance of
+the results of this work to searches for Higgs boson?
+1.3 Organization of the rest of the paper and summary of
+obtained results
+Sections 2,3 and 6, and Appendix A are devoted to detailed investigat ion of
+gravity-Y-Mcorrespondence. Section4isdevotedtothephysics -styleexposition
+ofworksbyAndreasFloer[11, 21]on Y-Mtheorywith purposeofco nnectinghis
+mathematical formalism for Y-M fields with the F-S model. In the same section
+we also consider the Y-M fields monopole and instanton solutions and t heir
+meaning and place within Floer’s theory. Our exposition is based on res ults of
+Sections 2 and 3. Section 5 is entirely devoted to solution of the gap p roblem for
+pureY-M fields. Although the solution depends onresultsofpreviou ssections,
+numerous additional facts from statistical mechanics and nuclear physics are
+being used. In Section 6 we discuss various implications/corollaries of the
+obtained results, especially for the SM of particle physics. In Sectio n 7 we
+discuss possible directions for further research based on the res ults presented
+in this paper. These include (but not limited to): connections with the LQG,
+the role and place of the Higgs boson, relationship between real spa ce-time
+scattering processes of high energy physics and processes of to pology change
+associated with such scattering. Based on the results of this pape r, we argue
+that this task can be accomplished with help of the formalism develope d by G.
+Perelman for his proof of the Poincare′and geometrization conjectures.
+The major new results of this paper are summarized as follows.
+1. In subsection 5.4.4, while solving the gap problem, we reproduced b y
+employing entirely different methods, the main results of the paper b y Korotkin
+and Nicolai[31]on quantizingdimensionally reducedgravity. From the se results
+it follows that for gravity and Y-M fields possessing the same symmet ry the
+nonperrturbative quantization proceeds essentially in the same wa y.
+2. In subsection 6.3 we demonstrated that gravity-Y-M correspo ndence dis-
+4covered by L.Witten for gauge group SU(2) can be extended onlyto the SU(3)
+gauge group. This group contains SU(2) ×U(1) group as a subgroup. This fact
+allowedustocomeupwiththe anticipated(but neverproven!) conclu sionabout
+symmetry of the SM. It is given by SU(3) ×SU(2)×U(1). The obtained result is
+very rigid. It is deeply rooted into not widely known/appreciated (dis cussed in
+Appendix A) properties of the gravitational field. It is these prope rties which
+ultimately determine the conditions of gravity-Y-M correspondenc e.
+3. The latest papers Refs.[32-34] are aimed at reproduction of the classifi-
+cation scheme of particles and fields in the SM within the framework of LQG
+formalism. These results match perfectly with the results of our pa per be-
+cause of the noticed and developed gravity-Y-M correspondence . In view of this
+correspondence, the results of Refs.[32-34] can be reproduced with help of min-
+imalgravity model described in subsections 3.2, 3.4, and 7.2 . This minimal
+model has differential-geometric /topological meaning in terms of th e dynamics
+of the extended Ricci flow [35,36]. Such a flow is the minimal extension o f the
+Ricci flow now famous because of its relevance in proving the Poincar e′and
+geometrization conjectures.
+4. The formalism developed in this paper explains why using pure gravit y
+onecantalk aboutthe particle/fieldcontentofthe SM. Not onlyit isc ompatible
+with just mentioned LQG results but also with those, coming from non commu-
+tative geometry [37], where it is demonstrated that use of pure gra vity (that is
+”minimal model”) combined with 0- dimensional internal space is sufficie nt for
+description of the SM.
+2 Emergence of the Ernst equation in pure grav-
+ity and Y-M fields
+2.1 Some facts about the Ernst equation
+Study of static vacuum Einstein fields was initiated by Weyl in 1917. Co n-
+siderable progress made in later years is documented in Ref.[38]. To de velop
+formalism of this paper we need to discuss some facts about these s tatic fields.
+Following Wald [39], a spacetime is considered to be stationary if there is a one-
+parameter group of isometries σtwhose orbits are time-like curves e.g. see [40].
+With such group of isometries is associated a time-like Killing vector ξi.Fur-
+thermore, a spacetime is axisymmetric if there exists a one-parameter group of
+isometriesχφwhose orbits are closed spacelikecurves. Thus, a spacelike Killing
+vector field ψihas integral curves which are closed. The spacetime is station-
+ary and axisymmetric if it possesses both of these symmetries, provided that
+σt◦χφ=χφ◦σt.Ifξ= (∂
+∂t) andψ= (∂
+∂φ) so that [ξ,ψ] = 0,one can choose
+coordinates as follows: x0=t,x1=φ,x2=ρ,x3=z.Under such identification,
+the metric tensor gµνbecomes a function of only x2andx3.Explicitly,
+ds2=−V(dt−wdφ)2+V−1[ρ2dφ2+e2��(dρ2+dz2)],(2.1)
+5where functions V,wandγdepend on ρandzonly. In the case when V=
+1,w=γ= 0,the metric can be presented as ds2=−(dt)2+(d˜s)2, where
+(d˜s)2=ρ2dφ2+dρ2+dz2(2.2)
+is the standard flat 3 dimensional metric written in cylindrical coordin ates. The
+four-dimensional set of vacuum Einstein equations Rij= 0 with help of metric
+given by Eq.(2.2) acquires the following form
+∇·{V−1∇V+ρ−2V2w∇w}= 0 (2.3a)
+and
+∇·{ρ−2V2∇w}= 0. (2.3b)
+Intheseequations ∇·and∇arethree-dimensionalflat(thatiswithmetricgiven
+byEq.(2.2)) divergenceand gradientoperatorsrespectively. In a ddition to these
+two equations, there are another two needed for determination o f factorγin the
+metric, Eq.(2.1). They require knowledge of Vandwas an input. Solutions
+of Eq.s(2.3) is described in great detail in the paper by Reina and Trev ers [41]
+with final result:
+(Reǫ)∇2ǫ=∇ǫ·∇ǫ. (2.4)
+This equation is known in literature as the Ernst equation. The comple x po-
+tentialǫis defined in by ǫ=V+iωwithVdefined as above and ωbeing an
+auxiliary potential whose explicit form we do not need in this work. As it was
+recognized by Ernst [42,43] such an equation can be also used for de scription of
+the combined Einstein-Maxwell fields. We shall exploit this fact in Sect ion 6.
+In Appendix A and in Section 6 we provide proofs that knowledge of st atic vac-
+uum solutions of the Ernst equation is necessary and sufficient for restoration of
+static Einstein-Maxwell fields.6Fields other than Y-M should be also restorable
+7. To proceed, we need to list several properties of the Ernst equa tion to be
+used below. First, following [41] and using prolate spheroidal coordin ates, the
+Ernst equation reproduces the Schwarzschild metric, and with ano ther choice
+of coordinates it reproduces the Kerr and Taub-NUT metric. Thus , the Ernst
+equation is the most general equation describing physically interest ing vac-
+uum spacetimes compatible with the Cauchy formulation of general r elativity
+[39,40,44,45]. Such a formulation is convenient staring point for quant ization
+of gravitational field via superspace formalism [39] leading to the Whe eler- De
+Witt equation, etc. Since in this work we advocate different approach to quan-
+tization of gravity, this topic is not being discussed further. Second, following
+Ref.[38], page 283, a stationary solution of Einstein’s field equations is called
+staticif the timelike Killing vector is orthogonal to the Cauchy surface. In s uch
+a case from the Table 18.1. of the same reference it follows that the Ernst
+6Surprisingly, upon changes of variables in these static sol utions, the exact results for
+propagating gravitational waves can be obtained as well.
+7This is so because each of these fields is a source of gravitati onal field which, in turn, can
+be eliminated locally. See Appendix A.
+6potentialǫis real. This observation allows us to simplify Eq.(2.4) considerably.
+For the sake of notational comparison with Ref.[38] we redefine the potential
+ǫ=V+iΦ.In the static case we have ǫ≡ −F≡ −e2u8.Using this result in
+Eq.(2.4) produces
+∆ρ,zu= 0, (2.5)
+where ∆ρ,zis flat Laplacian written in cylindrical coordinates defined by the
+metric, Eq.(2.2).
+2.2 Isomorphism between the SU(2) self-dual gauge and
+vacuum Einstein field equations
+This isomorphism was discovered by Louis Witten in 1979 [30]. His work wa s
+inspired by earlier works of Ernst [42] and Yang [46]. To our knowledge , since
+time when Ref.[30] was published such an isomorphism was left undevelo ped.
+In this paper we correct this omission in order to demonstrate that when both
+fields are mathematically indistinguishable, their quantization should p roceed
+in the same way. The result analogous to that discovered by Witten w as ob-
+tained using different arguments a year later by Forgacs, Horvath and Palla [47]
+and, in a simpler form, by Singleton [48]. These authors used essentia lly the
+paper by Manton, Ref.[49], in which it was cleverly demonstrated that the ’t
+Hooft-Polyakov monopole can be obtained without actual use of the auxiliary
+Higgs field. Both Refs.[47,48] and the original paper by Witten [30] use the
+axial symmetry of either gravitational or Y-M fields essentially. Only in this
+case it can be shown that the axisymmetric version of the self-dualit y equations
+obtained by Manton can be rewritten in the form of the Ernst equat ion. In
+the light of above information, following Ref.[5 ],we shall discuss briefly con-
+tributions of Yang and Witten. For this purpose, we need to conside r first the
+following auxiliary system of linearequations
+Ψx=XΨ;Ψt=TΨ. (2.6)
+HereΨx=∂
+∂xΨandΨt=∂
+∂tΨ.In this system XandTare square matrices
+of the same dimension and such that
+Xt−Tx+[X,T] = 0 (2.7)
+This result easily follows from the compatibility condition: Ψxt=Ψtx. The
+matrices XandTcan be realized as
+X=/parenleftbigg−iζ q(x,t)
+r(x,t)iζ/parenrightbigg
+,T=/parenleftbiggA B
+C−A/parenrightbigg
+(2.8)
+withζbeing a spectral parameter and, A,BandCbeing some Laurent poly-
+nomials in ζ.The above system can be extended to four variables x1,x2,t1,t2
+8The minus sign in front of Fis written in accord with conventions of Chapter 30.2 of the
+1st edition of Ref.[38].
+7in a simple minded fashion as follows
+(∂
+∂x1+ζ∂
+∂x2)Ψ= (X1+iX2)Ψ, (2.9a)
+(∂
+∂t1+ζ∂
+∂t2)Ψ= (T1+iT2)Ψ. (2.9b)
+In the most general case, the matrices X1,X2,T1,T2are made of functions
+which ”live” in C4.They are representatives of the Lie algebra sl(n,C) ofn×n
+trace-free matrices. The compatibility conditions for this case are equivalent to
+the self-duality condition for the Y-M fields associated with algebra sl(n,C).It
+is instructive to illustrate these general statements explicitly.
+InR4the (anti)self-duality condition for the Y-M curvature reads: ∗F=
+(−1)Fso that for the self-dual case we obtain:
+F01=F23,F02=F31,F03=F12. (2.10)
+In the ”light cone” coordinates σ=1√
+2(x1+ix2),τ=1√
+2(x0+ix3) the Y-M
+field one-form can be written as Aµdxµ=Aσdσ+Aτdτ+A¯σd¯σ+A¯τd¯τwith the
+overbarlabeling the complex conjugation. In such notations A0=1√
+2(Aτ+A¯τ),
+A1=1√
+2(Aσ+A¯σ),A2=1√
+2(Aσ−A¯σ),A3=1√
+2(Aτ−A¯τ).In these notations
+Eq.s (2.9) acquire the following form
+Fστ= 0, F¯σ¯τ= 0 andFσ¯σ+Fτ¯τ= 0. (2.11)
+They can be obtained as compatibility condition for the isospectral lin ear prob-
+lem
+(∂σ+ζ∂¯τ)Ψ= (Aσ+ζA¯τ)Ψand (∂τ−ζ∂¯σ)Ψ= (Aτ−ζA¯σ)Ψ,(2.12)
+where the spectral parameter is ζandΨis the local section of the Y-M fiber
+bundle. The compatibility condition reads: ( ∂σ−ζ∂¯τ)(∂σ+ζ∂¯τ)Ψ= (∂σ+
+ζ∂¯τ)(∂σ−ζ∂¯τ)Ψ,thus leading to
+[Fστ−ζ(Fσ¯σ+Fτ¯τ)+ζ2F¯σ¯τ]Ψ= 0. (2.13)
+This equation allows us to recover Eq.s(2.11). The first two equation s of
+Eq.s(2.11) can be used in order to represent the A-fields as follows: Aσ=
+(∂σC)C−1, Aτ=(∂τC)C−1, A¯σ=(∂¯σD)D−1andA¯τ= (∂τD)D−1,where
+bothCandDare some matrices in the Lie group G, e.g.G=SU(2). By
+introducing the matrix M=C−1D∈Gthe last of equations in Eq.(2.11)
+becomes
+∂¯σ(M−1∂σM)+∂¯τ(M−1∂τM) = 0. (2.14a)
+Thus, the self-duality conditions for the Y-M fields are equivalent to Eq.(2.14a).
+For the future use, following Yang [46], we notice that in such formalis m the
+gauge transformations for Y-M fields are expressible through D→DEand
+8C→CEso thatFσ¯σ→E−1Fσ¯σEandFτ¯τ→E−1Fτ¯τEwith the matrix
+E=E(σ,¯σ,τ,¯τ)∈SU(2) leaving self-duality Eq.s(2.10) (or (2.13)) unchanged.
+To connect Eq.(2.14a) with the Ernst equation, following L.Witten [30] it
+is sufficient to assume that the matrix Mis a function of ρ=/radicalbig
+x2
+1+x2
+2and
+z=x3.In such a case it is useful to remember that ρ2= 2σ¯σandz=i√
+2(τ−¯τ).
+With help of these facts Eq.(2.14a) can be rewritten as
+∂ρ(ρM−1∂ρM)+ρ∂z(M−1∂zM) = 0. (2.14b)
+By assuming that the matrix Mis representable by the SL(2,R)-type matrix,
+and writing it in the form
+M=1
+V/parenleftbigg1 Φ
+Φ Φ2+V2/parenrightbigg
+, (2.15)
+Eq.(2.14b) is reduced to the pair of equations
+V∇2V=∇V·∇V−∇Φ·∇Φ andV∇2Φ = 2∇V·∇Φ.
+With help of the Ernst potential ǫ=V+iΦ these two equations can be brought
+to the canonical form of the Ernst equation, Eq.(2.4). Below, we sh all provide
+sufficientevidencethatsuchareductionoftheErnstequationisco mpatiblewith
+analogous reduction in instanton/monopole calculations for the Y-M fields.
+3 From analysis to synthesis
+3.1 General remarks
+The results of previous section demonstrate that for axially symme tric fields
+both pure gravity and pure self-dual Y-M fields are described by th e same
+(Ernst) equation. In this section we reformulate these results in t erms of the
+nonlinear sigma model with purpose of using such a reformulation late r in the
+text. To do so we need to recallsome results from ourrecent work s, Ref.s[50,51].
+In particular, we notice that under conformal transformations ˆ g=e2ugind-
+dimensions the curvature scalar R(g) changes as follows:
+ˆR(ˆg) =e−2u{R(g)−2(d−1)∆gu−(d−1)(d−2)|▽gu|2}.(3,1)
+Since this equation is Eq.(2.11) of our Ref.[50] we shall be interested o nly in
+transformations for which ˆR(ˆg) is a constant. This is possible only if the total
+volume of the system is conserved. Under this constraint we need t o consider
+Eq.(3.1) for d= 3 in more detail. Without loss of generality we can assume that
+initiallyR(g) = 0. For this case we shall write g=g0so that Eq.(3.1) acquires
+the form
+ˆR(ˆg) =−2e−2u[2∆g0u+|▽g0u|2] (3.2)
+in which ∆ g0is the flat space Laplacian. Now we can formally identify it
+with that in Eq.(2.5). Accordingly, we shall be interested in such conf ormal
+9transformations for which ∆ g0u= 0 in Eq.(3.2). If they exist, Eq.(3.2) can be
+rewritten as
+e2uˆR(ˆg) =−2/parenleftBig
+/vector▽g0u/parenrightBig
+·/parenleftBig
+/vector▽g0u/parenrightBig
+. (3.3)
+This allows us to interpret Eq.(3.3) and
+∆g0u= 0 (3.4)
+as interdependent equations: solutions of Eq.(3.4) determine the s calar cur-
+vatureˆR(ˆg) in Eq.(3.3) .Clearly, under conditions at which these results are
+obtained only those solutions of Eq.(3.4) should be used which yield the con-
+stant scalar curvature ˆR(ˆg).Eq.(3.3) contains information about the Ricci
+tensor. To recover this information we notice that ˆ gij=−e2uδij. Therefore we
+obtain:
+ˆRij(ˆg) = 2∇iu∇ju, (3.5)
+inaccordwithEq.(18.55)ofRef.[38]wherethisresultwasobtainedby employing
+entirely different arguments. From the same reference we find tha t Eq.(3.4)
+comes as result of use of the contracted Bianci identities applied to ˆRij(ˆg)9.
+It is instructive to place the obtained results into broader context . This is
+accomplished in the next subsection.
+3.2 Connection with the nonlinear sigma model
+Some time ago Neugebauer and Kramer (N-K), Ref.[38], obtained Eq.s (3.4) and
+(3.5) usingvariationalprinciple. In less generalform this principle wa sused pre-
+viously by Ernst [42] resulting in now famous Ernst equation. Neugeb auer and
+Kramer proposed the Lagrangian and the associated with it action f unctional
+SN−Kproducing upon minimization both Eq.s(3.4)and (3.5). Todescribe the se
+results, we also use some results by Gal’tsov [52].
+The functional SN−Kis given by
+SN−K=1
+2/integraldisplay
+M/radicalbig
+ˆg[ˆR(ˆg)−ˆgijGAB(ϕ)∂iϕA∂jϕB]d3x, (3.6)
+easily recognizable as three-dimensional nonlinear sigma model coup led to 3-d
+Euclidean gravity. The number of components for the auxiliary field ϕas well
+as the metric tensor GAB(ϕ) of the target space is determined by the problem
+in question. In our case upon variation of SN−Kwith respect to ϕA
+iand ˆgijwe
+should be able to re obtain Eq.s(3.4) and (3.5). To do so, following Ref.[5 3], we
+introduce the current
+Ji=M−1∂iM. (3.7)
+In view of results of subsection 2.2, we have to identify the matrix Mwith that
+defined by Eq.(2.15) and, taking into account Eq.(2.14a), the index ishould
+9Ref.[38], page 283, bottom
+10take two values: σandτ.With such definitions we can replace the functional
+SN−Kby
+S=1
+2/integraldisplay
+M/radicalbig
+ˆg[ˆR(ˆg)−ˆgik1
+4tr(JiJk)]d3x. (3.8)
+The actual calculations with such type of functionals can be made us ing
+results of Ref.[53]. Thus, using this reference we obtain,
+ˆRij(ˆg) =1
+4tr(JiJj) (3.9)
+and
+∂iJi= 0. (3.10)
+Evidently, by construction Eq.(3.10) coincides with Eq.(2.14a) and, u ltimately,
+with Eq.(3.4). It is also easy also to check that Eq.(3.9) does coincide w ith
+Eq.(3.5). For this purpose it is sufficient to notice that
+tr(JiJj) =−tr(∂iM∂jM−1). (3.11)
+To check correctness of our calculations the entries of the matrix M, Eq.(2.15),
+can be restricted to V(that is we can put Φ = 0) .SinceV=−F≡ −e2u(e.g.
+see discussion prior to Eq.(2.5)), a simple calculation indeed brings Eq.( 3.9)
+back to Eq.(3.5) as required.
+It is interesting and important to observe at this point that the equa-
+tion of motion, Eq.(3.10), formally is not affected by effects of gravit y. This
+conclusion requires some explanation. From subsection 2.2, especia lly from
+Eq.s(2.14),(2.15), it should be clear that Eq.(3.10) is the Ernst equat ion deter-
+mining gravitational field. Hence, it is physically wrong to expect that it is
+going to be affected by the effects of gravity. Eq.s(3.9) and (3.10) a re the same
+as Eq.s(3.5) and (3.4) whose meaning was explained in the previous sub section.
+Clearly, the functional, Eq.(3.8), can be used for coupling of otherfields to
+gravity. This is indeed demonstrated in Ref.[52]. This is done with purpo se
+of connecting results for the nonlinear sigma models with those for h eterotic
+strings. We would like to discuss this connection now since it will be used later
+in the text.
+3.3 Connection with heterotic string models
+The functional S, Eq.(3.8), is related to that for the heterotic string model,
+e.g. see Ref.[54]. For such a model the sigma model-like functional is ob tainable
+from 10 dimensional supersymmetric string model by means of comp actifica-
+tion scheme (ideologically similar to that used in the Kaluza-Klein theory of
+gravity and electromagnetism) aimed at bringing the effective dimens ionality
+to physically acceptable values (e.g. 2, 3 or 4). For dimensionality D<10 such
+11compactified/reduced action functional reads (e.g. see Ref.[54], E q.(9.1.8)):
+Sheterotic
+D =/integraldisplay
+dDx√
+−detGe−2φ[R+4∂µφ∂µφ−1
+12ˆHµνρˆHµνρ
+−1
+4(M−1)ijFi
+µνFjµν+1
+8tr(∂µM∂µM−1)]. (3.12)
+The compactification procedure is by no means unique. There are ma ny ways
+to make a compactified action to look exactly like that given by Eq.(3.8) (e.g.
+see [55]). Evidently, there should be a way to relate such actions to each ot her
+since they all arehavingthe sameorigin- 10 dimensionalheterotic st ring action.
+Because of this, we would like to make some comments on action given b y
+Eq.(3.12) by specializing to D= 3 for reasons explained in Refs[ 68,69] and
+to be clarified below, in Section 6. Under such conditions if we require t he
+dilatonφ, the antisymmetric H-field (associated with string orientation) and
+the electromagnetic field Fto vanish,the remaining action will coincide with
+that given by Eq.(3.8). Because of this, the following steps can be ma de.
+First, asexplainedinourwork,Ref.[50],forclosed3-manifoldswecan /should
+dropthedilatonfield φ. Second,byproperlyselectingstringmodelwecanignore
+the antisymmetric field H. Third, by taking into account results of Appendix A
+we can also drop the electromagnetic field since it can be always resto red from
+pure gravity. Thus, we end up with the action functional S, Eq.(3.8), which
+we shall call ” minimal”. In Section 6 we shall provide evidence that its mini-
+mality is deeply rooted into gravity-Y-M correspondence which does not leave
+much room for ”improvements” abundant in physics literature. We s hall begin
+explaining this fact immediately below and will end our arguments in Sect ion
+6.
+3.4 The extended Ricci flow
+Thus far use of the variational principle apparently had not brough t us any
+new results (at least at the classical level). Situation changes in the light of
+recent work by List [35]. Following Ref.s[35,36 ], it is convenient to introduce
+Perelman-like entropy functional F(ˆgij,u,f)
+F(ˆgij,u,f) =/integraldisplay
+M(ˆR(ˆg)−2|∇ˆgu|2+|∇ˆgf|2)e−fdv (3.13)
+coinciding with Eq.(7.22b) of our work, Ref.[50], when u= 0.10. Because of
+this observation, if formally we make a replacement R(ˆg;u) =ˆR(ˆg)−2|∇u|2
+in Eq.(3.13), we are able to identify Eq.(3.13) with Perelman’s entropy f unc-
+tional enabling us to follow the same steps as were made in Perelman’s p apers
+aimed at proofof the geometrizationand Poincare′conjectures. Such a program
+10It should be noted that there is an obvious typographical err or in Eq.(7.22b): the term
+|∇hf|2is typed as |∇hf|.
+12was indeed completed in the PhD thesis by List [36]. Minimization of entro py
+functional F(ˆgij,f) produces the following set of equations
+∂
+∂tgij=−2(ˆRij+∇i∇jf) +4∇iu∇ju, (3.14a)
+∂
+∂tu= ∆ˆgu−(∇u)·(∇f), (3.14b)
+and∂
+∂tf=−ˆR−∆ˆgf+2|∇u|2, (3.14c)
+coinciding with Eq.s(7.28a), (7.28b) of our work, Ref.[50], when u= 0.In these
+equations |∇ˆgu|2= ˆgij∇iu∇ju, etc. From the next section and results below it
+follows that physically we should be interested in closed 3 manifolds. Fo r such
+manifolds onecan use Lemma 2.13, provenby List [36], which canbe for mulated
+as follows:
+Let ˆg,u,fbe a solution of Eq.s(3.14) for t∈[0,T) on a closed manifold M.
+Then the evolution of the entropy is given by
+∂tF(ˆgij,u,f) = 2/integraldisplay
+M[|Rij(ˆg;u)+∇i∇jf|2+2(∆ ˆgu−(∇u)·(∇f))2]e−fdv≥0.
+(3.15)
+Thus, the entropy is non decreasing with equality taking place if and o nly if the
+solution of Eq.(3.14) is a gradient soliton. This happens when the follow ing two
+conditions hold
+Rij(ˆg;u)+∇i∇jf= 0 and ∆ ˆgu−(∇u)·(∇f) = 0.(3.16)
+Foru= 0 the result of Perelman, Eq.(7.30) of Ref [50], for steady gradient
+soliton is reobtained, as required. Since for closed compact manifold sf=const
+Eq.s(3.16) coincide with Eq.s(3.4) and (3.5) as anticipated. Thus, existence
+of steady gradient solitons in the present context is equivalent to e xistence of
+solutions of static Einstein’s equations for pure gravity. This fact alone could
+be mathematically interesting but requires some reinforcement to b e of interest
+physically. We initiate this reinforcement process in the following subs ection.
+3.5 Relationship between the F-S and the Ernst function-
+als
+The F-S functional was mentioned in the Itroduction. In this subse ction we
+would like to initiate study of its connection with the Ernst functional. We
+begin with the following observation. In steps leading to Eq.(2.14b) (o r (3.10))
+the Euclidean time coordinate x0was eventually dropped implying that solu-
+tions of selfduality for Y-M equations, when substituted back into Y -M action
+functional, will produce physically meaningless (divergent) results. While in
+subsection 4.4 we discuss a variety of means for removing of such ap parent
+13divergence, in this subsection we notice that already Ernst [42] sug gested the
+action functional whoseminimization produces the Ernst equation. He gavetwo
+equivalent forms for such a functional, now bearing his name. These are either
+SE1[ǫ] =/integraldisplay
+Mdv∇ǫ·∇ǫ∗
+(Reǫ)2(3.17)
+or
+SE2[ξ] =/integraldisplay
+Mdv∇ξ·∇ξ∗
+(ξξ∗−1)2. (3.18)
+Minimization of SE1[ǫ] leads to Eq.(2.4) while functional, Eq.(3.18), is obtained
+fromSE1[ǫ] by means of substitution: ǫ= (ξ−1)/(ξ+1).In both functionals
+dvis 3-dimensional Euclidean volume element so that apparently the man ifold
+Mis justE3(or, with one point compactification, it is S3).Evidently,both
+SE1[ǫ] andSE2[ξ] are functionals for the nonlinear sigma model. If we drop
+the curvature term in Eq.(3.6) such truncated functional can be id entified, for
+example, with SE2[ξ]. This explains why Eq.(3.10) is formally unaffected by
+gravity. In mathematics literature the nonlinear sigma models are kn own as
+harmonic maps. Since Reina [56] demonstrated that the functional SE2[ξ]
+describes the harmonic map from S3toH2,it is not too difficult to write
+analogous functional SE3[ξ] describing the mapping from S3toS2.It is given
+by
+SE3[ξ] =/integraldisplay
+Mdv∇ξ·∇ξ∗
+(ξξ∗+1)2(3.19)
+and is part of the F-S model. If needed, both SE2[ξ] andSE3[ξ] can be supple-
+mented by additional (topological) terms which in the simplest case ar e wind-
+ing numbers. Thus, we shall be dealing either with the truncated F-S model,
+Eq.(3.19), or with its hyperbolic analog, Eq.(3.18). The choice betwee n these
+models is nontrivial and will discussed in detail in Section 6 .To facilitate this
+discussion, we need to observe the following. In the static case, we argued, e.g.
+see Eq.(2.5), that ǫ=−F=−e2u.Substitution of this result back into SE1[ǫ]
+produces (up to a constant) the following result:
+˜SE1[ǫ] =/integraldisplay
+Mdv∇u·∇u (3.20)
+leading to Eq.(2.5) as anticipated. At the same time, consider the follo wing H-E
+action functional
+SH−E[ˆg] =/integraldisplay
+Mdv/radicalbig
+ˆgˆR(ˆg), (3.21)
+and takeinto accountEq.(3.3)and the fact that ˆ gij=−e2uδij. Straightforward
+calculation leads us then to the result (up to a constant):
+SH−E[ˆg] =−/integraldisplay
+Mdv∇u·∇u. (3.22)
+14The minus sign in front of the integral is important and will be explained be-
+low. Before doing so, we notice that the Ernst functional (in whate ver form)
+is essentially equivalent to the H-E functional! Since in the original pap er by
+ErnstMisE3(orS3),apparently, such a functional should be zero. This is
+surely not the case in general but the explanation is nontrivial. Supp ose that
+minimization of the Ernst functional leads to some knotted/linked st ructures11.
+If such knots/links are hyperbolic then, by construction, complem ents of these
+knots/links in S3areH3modulo some discrete group. This conclusion is in
+accord with properties of the Ernst equation discovered by Reina a nd Trevers
+[41]. Following this reference, we introduce the complex space C×C=C2so
+that∀z= (u,v)∈C2the scalar product z∗
+αzαcan be made with the metric
+παβ=diag{1,−1}. Furthermore, the Ernst Eq.(2.4) can be rewritten with help
+of substitution ǫ= (u−v)/(u+v) as the set of two equations
+zαz∗
+α∇2zβ= 2z∗
+α∇zα·∇zβ. (3.23)
+Such a system of equations is invariant with respect to transforma tions from
+unimodular group SU(1,1) which is equivalent to SL(2, C). But SL(2, C) is the
+group of isometries of hyperbolic space H3as was discussed extensively in our
+work,Ref.[57]. Thus, minimization ofboth the F-S andErnstfunction alsshould
+account for knotted/linked structures. This conclusion is streng thened in the
+next subsection.
+3.6 Relationship between the Ernst and Chern-Simons
+functionals
+Even though we need to find this relationship anticipating results of t he next
+section, by doing so, some unexpected connections with previous s ubsection
+are also going to be revealed. For this purpose, we notice that for u= 0 the
+functional F(ˆgij,u,f) introduced earlier is just Perelman’s entropy functional.
+As such, it was discussed in our work, Ref.[50]. Evidently, both Fand Perel-
+man’s functional can be used for study of topology of 3-manifolds. We believe,
+though, that use of Perelman’s functional is more advantageous a s we would
+like to explain now. For this purpose, it is convenient to introduce the Raleigh
+quotientλgvia
+λg= inf
+ϕ/integraltext
+MdV(4|∇ϕ|2+R(g)ϕ2)
+/integraltext
+MdVϕ2, (3.24)
+e.g. see Eq.(7.24)of [50], to be compared against the Yamabe quotien t (p=2d
+d−2
+andα= 4d−1
+d−2) .
+Yg=/integraltextddx√ˆgˆR(ˆg)
+/parenleftbig/integraltext
+ddx√ˆg/parenrightbig2
+p=/parenleftbigg1/integraltext
+Mddx√gϕp/parenrightbigg2
+p/integraldisplay
+Mddx√g{α(∇gϕ)2+R(g)ϕ2} ≡E[ϕ]
+/ba∇dblϕ/ba∇dbl2
+p
+11We shall postpone detailed discussion of this topic till Sec tion 6.
+15also discussed in [50]. Because of similarity of these two quotients the question
+arises: Can they be equal to each other? The affirmative answerto this question
+is obtained in Ref.[58]. It can be formulated as
+Theorem [58]. Suppose that γis a conformal class on Mwhichdoes not
+contain metric of positive scalar curvature. Then
+Yγ= sup
+g∈γλgV(g)2
+d≡¯λ(M), (3.25a)
+where¯λ(M) is Perelman’s ¯λinvariant. Equivalently,
+λgV(g)2
+d≤Yγ, (3.25b)
+whereV(g) =/integraltext
+ddx√ˆgis the volume.
+The equality happens when gis the Yamabe minimizer. It is metric of
+unit volume for manifold Mof constant scalar curvature (which, according to
+theorem above, should be negative so that Mis hyperbolic 3-manifold). Only
+for hyperbolic 3-manifolds whose Yamabe invariant Y−(M) = supγYγthe
+gravitational Cauchy problem for source-free gravitational field is well posed
+[45,46]. For gwhich is Yamabe minimizer we have SH−E[ˆg]≤Yγ.This result
+can be further extended by noticing that NSH−E[ˆg] =CS(A),whereNis some
+constant whose value depends upon the explicit form of the gauge fi eldA,and
+CS(A) is the Chern-Simons invariant to be described in the next section.
+To demonstrate that NSH−E[ˆg] =CS(A) it is sufficient to use some re-
+sults from works by Chern and Simons [59] and by Chern [60]. In [59] it w as
+proven that: a) the Chern-Simons (C-S) functional CS(A) (to be defined in
+next section) is a conformal invariant of M(Theorem 6.3. of [59]) and, b) that
+the critical points of CS(A) correspond to 3-manifolds which are (at least lo-
+cally) conformally flat (Corollary 6.14 of [59]). Subsequently, these r esults were
+reobtained by Chern, Ref.[60], in much simpler and more physically sugg es-
+tive way. In view of these facts, at least for Yamabe minimizers we ob tain,
+CS(A) =NSY[ϕ],whereNis some constant (different for different gauge
+groups). That this is the case should come as not too big of a surpris e since
+for Lorentzian 2+1 gravity Witten, Ref.[61], demonstrated the equ ivalence of
+the Hilbert-Einstein and C-S functionals without reference to resu lts of Chern
+and Simons just cited. At the same time, the Euclidean/Hyperbolic 3d gravity
+was discussed only much more recently, for instance, in the paper b y Gukov,
+Ref.[62]. To avoid duplications we refer our readers to these papers for further
+details.
+4 Floer-style nonperturbative treatment of Y-
+M fields
+4.1 Physical content of the Floer’s theory
+Strikingresemblancebetweenresultsof nonperturbativetreatm entof4-dimensional
+Y-M fields and two dimensional nonlinear sigma model at the classical le vel is
+16well documented in Ref.[63 ]. Zero curvature equations, e.g. Eq.(2.7), can be
+obtained either by using the two- dimensional nonlinear sigma model o r three-
+dimensional C-S functional. As discussed in previous section, the se lf-duality
+condition for Y-M fields also leads (upon reduction) to zero curvatu re condition.
+Since the Ernst equation describing static gravitational (and elect rovacuum)
+fields is obtainable both from conditions of self-duality for the Y-M fie ld and
+from minimization of 3-dimensional nonlinear sigma model, it follows that 3-d
+gravitational nonlinear sigma model, Eq.(3.8), contains nonperturb ative infor-
+mation about Y-M fields. Furthermore, in view of results of Appendix A, it
+also should contain information about the static electromagnetic fie lds, for the
+combined gravitational and electromagnetic waves and, with minor m odifica-
+tions, for the combined gravitational, electromagnetic and neutrin o fields. The
+nonperturbative treatment of Y-M fields is usually associated eithe r with the in-
+stanton ormonopole calculations. This observation leads to the con clusion that,
+at least in some cases, zero curvature equation should carry all no nperurbative
+information about Y-M fields. This point of view is advocatedand deve loped by
+Floer [11,21 ]. Below,we shall discuss Floer’s point of view now in the language
+used in physics literature. For the sake of illustration, it is convenien t to present
+our arguments for Abelian Y-M (that is electromagnetic) fields first .
+The action functional Sin this case is given by12
+S=1
+2t/integraldisplay
+0dt/integraldisplay
+Mdv[E2−B2], (4.1)
+whereB=∇×AandE=−∇ϕ−∂
+∂tA,ϕ≡A0.It is known that, at least
+for electromagnetic waves, it is sufficient to put A0= 0 (temporal gauge). In
+such a case the above action can be rewritten as
+S[A] =1
+2t/integraldisplay
+0dt/integraldisplay
+Mdv[˙A2−(∇×A)2], (4.2)
+where˙A=∂
+∂tA.From the condition δS/δA= 0 we obtain∂E
+∂t=∇×B. The
+definition of Bguarantees the validity of the condition ∇·B= 0 while from the
+definition of Ewe get another Maxwell equation∂B
+∂t=−∇×E. The question
+arises: will these results imply the remaining Maxwell’s equation ∇·E= 0
+essential for correct formulation of the Cauchy problem? If such a constraint
+satisfied at t= 0,naturally, it will be satisfied for t>0. Unfortunately, for t= 0
+the existence of such a constraint does not follow from the above e quations and
+should be introduced as independent. This causes decomposition of the field
+AasA=A/bardbl+A⊥.Taking into account that E=−∂
+∂tA,we obtain as well
+∇·(E/bardbl+E⊥).Then, by design ∇·E⊥= 0,while∇·E/bardblremains to be defined
+by the initial and boundary data. Because of this, it is alwayspossible to choose
+A/bardbl= 0 and to use only A⊥for description of the field propagation [64]. Hence,
+12Up to an unimportant scale factor.
+17the action functional Scan be finally rewritten as
+S[A⊥] =1
+2t/integraldisplay
+0dt/integraldisplay
+Mdv[˙A2
+⊥−(∇×A⊥)2]. (4.3)
+In such a form it can be used as action in the path integrals, e.g. see R ef.[64],
+page 152, describing free electromagnetic field. Such path integra l can be eval-
+uated both in Minkowski and Eucldean spaces by the saddle point met hod.
+There is, however, a closely related method more suitable for our pu rposes. It
+is described in the monograph by Donaldson, Ref.[11]. Following this ref erence,
+we replace time variable tby−iτin the functional S[A⊥] .Consider now this
+replacement in some detail. We have13
+1
+2T/integraldisplay
+0dτ/integraldisplay
+Mdv[˙A2
+⊥+(∇×A⊥)2]
+=1
+2T/integraldisplay
+0dτ(/integraldisplay
+Mdv[[˙A⊥+(∇×A⊥)]2−∂
+∂τ(A⊥·∇×A⊥)]).(4.4)
+Since variation of A⊥is fixed at the ends of τintegral, the last term can be
+dropped so that we are left with the condition
+∂
+∂τA⊥=−B⊥ (4.5)
+extremizing the Euclidean action SE[A⊥].The above results are transferable
+to the non Abelian Y-M field by continuity and complementarity. Since in
+the Abelian case fields EandBare dual to each other, by applying the curl
+operator to both sides of Eq.(4.5) (and removing the subscript ⊥) we obtain
+the equivalent form of self-duality equations in accord with those on page 33 of
+Ref.[6]. This calculation provides an independent check of Donaldson’s method
+of computation. Since the (anti)self-duality condition in the Abelian c ase can
+be written as B=∓E[9].and since E=−∂
+∂τA, we conclude that Eq.(4.5)
+is the self-duality equation. This conclusion is immediately transferab le to the
+non Abelian Y-M case where the analog of Eq. (4.5) is
+∂
+∂τA=∗F(A(τ)), (4.6)
+in accord with Floer. The symbol * denotes the Hodge star operatio n in 3 di-
+mensions. Following Donaldson [11] this result should be understood a s follows.
+Introduce a connection matrix A=A0dτ+3/summationtext
+i=1Aidxisuch that both A0andAi
+depend upon all four variables τ,x1,x2andx3.In the temporal gauge A0should
+13We shall assume (without loss of generality) that ˙A⊥is collinear with A⊥.
+18be discarded so that τbecomes a parameter in the remaining A′
+is.Evidently,
+it can be associated with the spectral parameter (e.g. see previou s section).
+The Hodge star operator in Eq.(4.6) is needed to make this equation a s an
+equation for one-forms The obtained results fit nicely into Cauchy f ormulation
+of dynamics of both Y-M and gravity. Indeed, under conditions ana logous to
+that discussed in [45,46]the space-time (4-manifold) is decomposab le into direct
+productM×R(a trivial fiber bundle) in such a way that all differential op-
+erations acting on 4-manifold are been projected down to 3-manifo ldM. This
+is essential part of Floer’s theory. Furthermore, since δCS(A)/δA=F(A) the
+above Eq.(4.6) can be equivalently rewritten as
+∂
+∂τA=∗[δCS(A)/δA] (4.7)
+so that the Chern-Simons functional is playing a role of a ”Hamiltonian ” in
+Eq.s(4.7). From the theory of dynamical systems it follows then tha t the dy-
+namics is taking place between the points of equilibria defined by zero c urvature
+condition F(A) = 0.At the same time, using our work, Ref.[50], it is easily rec-
+ognize Eq.(4.7) as an equation for the gradient flow, e.g. see Eq.s(3.1 4). For the
+sake of space we shall not discuss this topic any further. Interes ted readers are
+encouraged to consult Ref.[65]. For supersymmetric Y-M fields part icipating
+in Seiberg-Witten theory the gradient flow equations are discussed in detail in
+Ref.[66]
+The mechanical system described by Eq.(4.7) should be eventually qu an-
+tized. Since the quantization procedure is outlined in Ref.[67], to avoid du-
+plications, we shall concentrate attention of our readers on aspe cts of Floer’s
+theory not covered in [67] but still relevant to this paper. To do so, we follow
+Donaldson [11]. This is accomplished in several steps.
+First, in the previous section we noticed that the axially symmetric self-du al
+solution for Y-M fields does not depend on x0(orτ) variable. Therefore, if such
+solution is substituted back into Y-M functional, it produces diverge nt result.
+Although the cure for this issue is discussed in subsection 4.4, in this s ubsection
+we provide needed background. For this purpose, following Ref.[68] we consider
+the Y-M action S[F] for the pure Y-M field14
+S[F] =−1
+8/integraldisplay
+R4d4xtr(FµνFµν). (4.8)
+The duality condition15∗Fµν=1
+2εµναβFαβallows us then to rewrite this action
+as follows
+S[F] =−1
+16/integraldisplay
+R4d4x[tr((Fµν∓∗Fµν)(Fµν∓∗Fµν))±2tr(Fµν∗Fµν)] (4.9)
+14Strictly following notations of Ref.[68] we do not indicate that in general the integration
+should be made over some 4-manifold M. In physics literature, and inEq.s(2.11), itisassumed
+that we are dealing with R4(orS4upon compactification). In Floer’s theory it is essential
+that the 4-manifold is decomposable as M×R. This decomposition should be treated with
+care as described in the Donaldson’s book [11]
+15With the convention that ε1234=−1.
+19sincetr(FµνFµν) =tr(∗Fµν∗Fµν).The winding number Nfor SU(2) gauge
+field is defined as16
+N=−1
+8π2/integraldisplay
+R4d4xtr(Fµν∗Fµν)≡ −1
+8π2/integraldisplay
+R4tr(Fµν∧Fµν) (4.10)
+so that use of this definition in Eq.s(4.8),(4.9) produces
+S[F]≥π2|N| (4.11)
+with the equality taking place when the (anti) self-duality condition (e .g. see
+Eq.(2.10) ) holds. In such a case the saddle point action is becoming ju st a
+winding number (up to a constant).
+Second, if our space-time 4-manifold Mcan be decomposed as M×[0,1],
+the following identity can be used [11]
+/integraldisplay
+M×[0,1]tr(Fµν∧Fµν) =/integraldisplay
+Mtr(A∧dA+2
+3A∧A∧A)/equalsdotsCS(A).(4.12)
+Here the symbol /equalsdotsmeans ”up to a constant”. The decomposition M×[0,1]
+reflects the fact that the C-S functional is defined up to mod Z. This ambiguity
+can be removed if we agree to consider C-S functional as a quotient R/Z. Ac-
+cordingly, this allows us to replace M×RbyM×[0,1].Details can be found
+in Ref.[11]. Thus, one way or another the winding number Nin Eq.(4.10) can
+be replaced by the Chern-Simons functional.
+Third, since the equation of motion for the C-S functional is zero curvat ure
+condition F= 0, i.e.
+dA+A∧A= 0, (4.13)
+implying that the connection Ais flat, we can use this result in Eq.(4.12) in
+order to rewrite it as (e.g. for SU(2))
+1
+8π2/integraldisplay
+Mtr(A∧dA+2
+3A∧A∧A) =−1
+24π2/integraldisplay
+Mtr(A∧A∧A).(4.14)
+For other groups the prefactor and the domain of integration will b e different
+in general.
+Fourth, zero curvature Eq.(4.13) involves connections which are function s of
+three arguments and a spectral/time parameter. In such setting minimization
+of Y-M functional is not divergent in view of Eq.(4.11).
+Fifth, the obtained result, Eq.(4.14), coincides with that known for the
+winding number for SU(2) instantons in physics literature[8,19] wher e it was
+obtained with help of entirely different arguments. It should be note d though
+that in spite of apparent simplicity of these results, actual calculat ions of C-S
+functionals (invariants) for different 3-manifolds are, in fact, ver y sophisticated
+[69,70]. In accord with Floer and Ref.[67], we conclude that nonpertur batively
+16We follows notations of Ref. [68] in which R4is actually standing for S4∈SU(2)
+20the 4-dimensional pure Y-M quantum field theory is a topological field theory
+of C-S type.
+Sixth, the isomorphism noticed by Louis Witten acquires now natural expla -
+nation. It becomes possible in view of results just presented, on on e hand, and
+the fact that NSH−E[ˆg] =CS(A) (previous section), on another. For fields
+with axial symmetry, equations of motion, Eq.(4.13), for gravity an d Y-M fields
+coincide.
+Seventh, the instantons in Floer’s theory are notthe same as considered
+in physics literature [8,19]. To understand this, we must take into acc ount
+that in Floer’s theory manifolds under consideration are 4-manifolds Mwith
+tubular ends. Such manifolds are complete Riemannian manifolds with fi nite
+number of tubular ends made of half tubes (0 ,∞) so that locally each such
+manifold lookslike Ui=Li×(0,∞) withLibeing a compact 3-manifold(called
+a”crossectionofatube”)and inumberingthetubes. Theclosureof M\/uniontextn
+i=1Ui
+is a compact manifold with boundary. If the crossection is S3,thenUis
+conformallyequivalentto apunctured ball B4\{0}.Thisimplies that amanifold
+Mwith tubular ends is conformally equivalent to a punctured manifold ˜M \
+{p1,...,pn}where˜Mis compact. The instanton moduli problem for Mis
+equivalent to that for the punctured manifold [11]. Recall that the m oduli space
+of instantons is defined as set of solutions of anti self-dual equat ions modulo
+gauge equivalence.
+Being armed with these definitions and taking into account that the ( anti)
+self-duality Eq.(4.7) we can interpret the instanton as a path conne cting one
+flat connection F= 0 at ”time” τ=−∞with another flat connection at ”time”
+τ=∞[11]. It is permissible for the path to begin at one flat connection, to
+wind around a tube (modulo gauge equivalence) and to end up at the s ame flat
+connection, Ref.[11], page 22. Evidently, this caseinvolves4-manifo lds with just
+one tubular end. Physically, each flat connection F= 0 represents the vacuum
+state so that the instantons discussed in the Introduction should be connecting
+different vacua. In this sense there is a difference between the inte rpretation of
+instantons in mathematics and physics literature. As in the case of s tandard
+quantum mechanics, only imposition of some additional physical cons traints
+permitsustoselectbetweenallpossiblesolutionsonlythosewhichar ephysically
+relevant. In the present context it is known that all exactly integr able systems
+are described by the zero curvature equation F= 0 [5,6 ]. It is also known
+that differences between these equations are caused in part by diff erences in a
+way the spectral parameter enters into these equations. Since f or the Floer’s
+instantons F/\e}atio\slash= 0,it means that the curvature Fshould be parametrized in such
+a way that the ”time” parameter should become a spectral parame ter when
+F= 0.In this work we do not investigate this problem17. Instead, we shall
+focus our attention on different vacua, that is on different (knot- like) solutions
+of zero curvature equation F= 0.18
+17See Ref.[7] for introduction into this topic.
+18A complement of each knot in S3is 3-manifold. Floer’s instantons are in fact connecting
+various three-manifolds. These 3- manifolds (with tubular ends) should be glued together to
+formM.The gluing procedure is extremely delicate mathematical op eration [11]. It is above
+214.2 The Faddeev-Skyrme model and vacuum states of the
+Y-M functional
+In the light of results just presented, we would like to argue that th e F-S model
+is indeed capable of representing the vacuum states of pure Y-M fie lds. For
+this purpose it is sufficient to recall the key results of the paper by A uckly and
+Kapitansky [71]. These authors were able to rewrite the Faddeev fu nctional
+E[n] =/integraldisplay
+S3dv{|dn|2+|dn∧dn|2} (4.15)
+in the equivalent form given by
+Eϕ[a] =/integraldisplay
+S3dv{|Daϕ|2+|Daϕ∧Daϕ|2}. (4.16)
+In this expression, the covariant derivative Daϕ=dϕ+[a,ϕ]. Evidently, Eϕ[a]
+acquires its minimum when ϕ=aand the connection becomes flat (that is
+covariant derivative becomes zero). Since this result is compatible w ith those
+discussed in previous subsection, it implies that, indeed, Faddeev’s m odel can
+be used for description of vacuum states for pure Y-M fields. The o nly question
+remains: Is this model the onlymodel describingQCDvacuum? In view ofEq.s
+(3.18),(3.19) it should be clear that this is not the case. The full expla nation
+is given below, in Sections 5,6. In addition, the disadvantage of the F- S
+model as such (that is without modifications) lies in the absence of ga p upon
+its quantization as was recognized already by Faddeev and Niemi in Re f.[25].
+In Sections 5,6 we shall eliminate this deficiency in a way different from t hat
+described in the Introduction (e.g. in Ref.s[25,26]). In the meantime, we would
+like to find the place for monopoles in our calculations.
+4.3 Monopoles and the Ernst equation
+4.3.1 Monopoles versus instantons
+To introduce notations and for the sake of uninterrupted reading , we need to
+describebrieflythealternativepointofviewattheresultsofprevio ussubsection.
+For this purpose, following Manton [49], we need to make a comparison between
+the Lagrangiansfor SU(2) Y-M and the Y-M-Higgs fields described r espectively
+by
+LY−M=−1
+4tr(FµνFµν) (4.17)
+and
+LY−M−H=−1
+4tr(FµνFµν)−1
+2tr(DµΦ·DµΦ)−λ
+2(1−Φ·Φ)2(4.18)
+the level of rigor of this paper. To imagine the connected sum of knots [20] is much easier
+than the connected sum of 3-manifolds. This sum has physical meaning discussed in Section
+6.
+22with covariant derivative for the Higgs field defined as DµΦ=∂µΦ+[Aµ,Φ]
+and with connection Aµusedtodefine the Y-M curvaturetensor Fµν=∂µAν−
+∂νAµ+[Aµ,Aν],provided that Φ=Φata,Aµ=Aa
+µta,and [ta,tb] =−2εabctc.
+Now the self-duality condition F=∗Fcan be equivalently rewritten as Fij=
+−εijkFk0with indices i,j,krunning over 1,2,3. Incidentally, in the temporal
+gaugethisresultisequivalenttoFloer’sEq.(4.6)Considernowthelimit λ→0in
+Eq.(4.18). In the Minkowski spacetime the field equations originating from the
+Y-M-Higgs Lagrangian can be solved by using the Bogomolny ansatz e quations
+Fij=−εijkDkΦin which A0= 0 (temporal gauge). Instead of imposing the
+temporal gauge condition, we can identify the Higgs field ΦwithA0so that
+the Bogomolny equations read now as follows:
+Fij=−εijkDkA0. (4.19)
+Bogomolny demonstrated that the Prasad-Sommerfield monopole s olution can
+be obtained using Eq.(4.19). Thus, any static (that is time-independ ent) so-
+lution of self-duality equations is leading to Bogomolny-Prasad-Somm erfield
+(BPS) monopole solution of the Y-M fields, provided that we interpre t the
+component A0as the Higgs field. Suppose now that there is an axial symme-
+try. Forgacs, Horvath and Palla [72] (FHP) demonstrated equivale nce of the
+set of axially symmetric Bogomolny Eq.s (4.19) to the Ernst equation. The
+static monopole solution is time-independent self-dual gauge field. B ecause of
+this time independence, its four-dimensional action is infinite(because of the
+time translational invariance) while that for instantons is finite. Furthermore,
+the boundary conditions for monopoles and instantons are differen t. The infin-
+ity problem for monopoles can be cured somehow by considering the m onopole
+dynamics [68] but this topic at this moment ”is more art than science” , e.g.
+read [68], page 309. For the same reason we avoid in this section talkin g about
+dyons (pseudo particles having both electric and magnetic charge) . Hence, we
+would like to conclude our discussion with description of more mathema tically
+rigorous treatments. By doing so we shall establish connections wit h results
+presented in previous sections
+4.4 Calorons .
+Calorons are instantons on R3×S1.From this definition it follows that, phys-
+ically, these are just instantons at finite temperature19. Calorons are related
+to both instantons on R4(orS4) and monopoles on R3(orS3).Heuristically,
+the large period calorons are instantons while the small period caloro ns are
+monopoles [73,74]. These results do not account yet for the fact th at both the
+Y-M action and the self-duality equations are conformally invariant. Atiyah
+[75]. noticed that the Euclidean metric can be represented either as
+ds2
+E=/parenleftbig
+dx1/parenrightbig2+/parenleftbig
+dx3/parenrightbig2+/parenleftbig
+dx3/parenrightbig2+/parenleftbig
+dx4/parenrightbig2(4.20a)
+19This explains the word ”caloron”.
+23or as
+ds2=r2
+R2[R2(/parenleftbig
+dx1/parenrightbig2+/parenleftbig
+dx2/parenrightbig2+(dr)2
+r2)+R2dϕ2] (4.20b)
+withRbeing some constant. The above representation involves polar r,ϕ
+coordinates in the ( x3,x4) plane thus implying some kind of axial symmetry.
+Since self-duality equations are conformally invariant, the prefact orr2
+R2can be
+dropped so that the Euclidean space R4becomes conformally equivalent to
+the product H3×S1.For such manifold the constant scalar curvature of the
+hyperbolic 3-space H3is−1/R2.Furthermore, the remaining term represents
+the metric on a circle of radius R. Following Ref.[73], let ( x1,x2,x3) be coor-
+dinates for the hyperbolic ball model of H3so that the radial coordinate be
+R=/radicalBig
+(x1)2+(x2)2+(x3)2. Let 0≤ R ≤R.Letτbe a coordinate on S1
+with period β,then the metric on H3×S1can be represented as
+ds2
+H=dτ2+Λ2(dR2+R2dΩ2), (4.21a)
+where Λ = (1 − R/R)−1anddΩ2is the metric on 2-dimensional sphere. If
+we introduce an auxiliary coordinate µ= (R/2)arctanh( R/R),and complex
+coordinate z=µ+iτ,the above hyperbolic metric can be rewritten as
+ds2
+H=dτ2+dµ2+Ξ2dΩ2(4.21b)
+with Ξ = ( R/2)sinh(2µ/R).By analogy with transition from Eq.(4.20a) to
+(4.20b)wecanproceedasfollows. Let r=/radicalBig
+(y1)2+(y2)2+(y3)2with(y1,y2,y3,y0)
+being coordinates on R4.By lettingt=y0the Euclidean metric can be written
+as usual, i.e.
+ds2
+E=dt2+dr2+r2dΩ2(4.22a)
+so that
+ds2
+H=ξ2ds2
+E, (4.22b)
+withξ= (R/2)[cosh(2µ/R)+cos(2τ/R)].This correspondencebetween R4/integerdivideR2
+andH3×S1is made with help of the mapping w=tanh(z/R) (withw=r+it
+andβ=πR).LetM=H3×S1(orH3×R)then, inviewofconformalinvariance,
+we can rewrite Eq.(4.8) as
+S[F] =−1
+8/integraldisplay
+Mtr(FµνFµν)Ξ2dτdµdΩ. (4.23)
+We have to rewrite the winding number, Eq.(4.10), accordingly. Since it is
+a topological invariant, this means that the self-duality equations m ust be ad-
+justed accordingly. For instance, for the hyperbolic calorons onH3×S1the
+self-duality equation reads
+F0i=1
+2ΛεijkFjk. (4.24)
+The action S[F] nowis finite with tr(FµνFµν)→0whenµ→ ∞.For hyperbolic
+instantons we have finite action with tr(FµνFµν)→0 whenµ2+τ2→ ∞.
+24The results just described match nicely with the results by Witten [76 ] on
+Euclidean SU(2) instantons invariant under the action of SO(3) Lie g roup. His
+results will be discussed in detail in the next section. Notice, that Eu clidean
+metric, Eq.(4.22a), becomes that for H2×S2if we rewrite it as
+ds2
+E=r2(dt2+dr2
+r2+dΩ2) (4.25a)
+and, as before, we drop the conformal factor r2so that it becomes
+ds2
+H=dt2+dr2
+r2+dΩ2. (4.25b)
+Interestingly enough, that results by Witten initially developed for H2×S2
+can be also used without change for H3×S1andH3×Rsince the action of
+SO(3)pullsbacktothesemanifolds[73]. Thisfactisofimportancesincesuchan
+extensionmakeshis resultscompatible with bothFloer’s method ofca lculations
+for Y-M fields and with results of Section 3. Omitting all details, the action,
+Eq.(4.23), is reduced to that known for two dimensional Abelian Ginzb urg-
+Landau (G-L) model ”living” on the hyperbolic 2 manifold Xcoordinatized by
+µandτwith the metric
+ds2
+H=dµ2+dτ2
+Ξ2. (4.26)
+Explicitly, such G-L action functional SG−Lis given by [73]
+SG−L=π
+2/integraldisplay
+Xdτdµ[Ξ2(∇×A)2+2|(∇+iA)φ|2+Ξ−2(1−|φ|2)] (4.27)
+withAandφbeing respectively the Abelian gauge and the Higgs fields, φ=
+φ1+iφ2so that|φ|2=φ2
+1+φ2
+2.This functional is obtained upon substitution of
+solution of the self-duality equations into the Y-M action functional, Eq.(4.23).
+We refer our readers to the original paper, Ref.[73] for details. In the limit
+β→ ∞the above functional coincides with that obtained by Witten [76]. The
+self-dualityequationsobtainedbyWittendescribeinstantonswhich liealongthe
+fixed axis while Fairlie, Corrigan, ’t Hooft, and Wilczek [77]developed an ansatz
+(CFtHWansatz)fortheself-dualityequationsproducinginstanto nsatarbitrary
+locations. Manton[78]demonstratedthatWitten’sandCFtHWmulti- instanton
+solutions are gauge equivalent while Harland [73,74 ]demonstrated how these
+instantons and monopoles can be obtained from calorons in various lim its. The
+obtained results provide needed background information for solut ion of the gap
+problem. This solution is discussed in the next section.
+5 Solution of the gap problem
+5.1 Idea of the proof
+By cleverly using symmetry of the problem Witten [76] reduced the no n Abelian
+Y-M action functional to that for the Abelian G-L model ”living” in the hyper-
+bolic plane. This is one of examples of the Abelian reduction of QCD discu ssed
+25in our paper, Ref.[79]. Vortices existing in the G-L model could be visua lized
+as made of some two-dimensional surfaces (closed strings) living in t he ambi-
+ent space-time. These are known as Nambu-Gotto strings. Their t reatment
+by Polyakov [80] made them to exist in spaces of higher dimensionality. In or-
+der for them to be useful for QCD, Polyakov suggested to modify s tring action
+by adding an extra (rigidity) term into string action functional. By do ing so
+the problem was created of reproducing Polyakov rigid string model from QCD
+action functional. The latest proposal by Polyakov [81] involves con sideration
+of spin chain models while that by Kondo [82] involves the F-S model der ived
+directly from QCD action functional. As explained in [79], in the case of scatter-
+ing processes of high energy physics one is confronted essentially w ith the same
+combinatorialproblems as were encountered at the birth ofquant um mechanics.
+In Ref.[83] we explained in detail why Heisenberg’s (combinatorial) met hod of
+developing quantum mechanical formalism is superior to that by Schr ¨ odinger.
+In Ref.[74] using these general results we demonstrated how the c ombinatorial
+analysis of scattering data leads to spin chain models as microscopic m odels
+describing excitation spectrum of QCD. Thus, the mass gap problem can be
+considered as already solved in principle. Nevertheless, in [ 94] such a solution
+is obtained ”externally”, just based on the rules of combinatorics. As with
+quantum mechanics, where atomic model is used to test Heisenberg ’s ideas,
+there is a need to reproduce this combinatorial result ”internally” b y using mi-
+croscopic model of QCD. For this purpose, we shall use the G-L fun ctional,
+Eq.(4.27). By analogy with the flat case, we expect that it can be rew ritten in
+terms of interacting vortices. In the present case, in view of Eq.(4 .26), vortices
+”live” not in the Euclidean plane but in 3+1 Minkowski space-time. This is
+easy to understand if we recall the SO(3) ⇄SU(2) correspondence and take into
+account the analogous correspondence between SU(1,1) and SO( 2,1).
+Within such a picture it is sufficient to look at evolution dynamics of the
+individual vortex. Typically, it is well described by the dynamics of the continu-
+ousHeisenbergspin chainmodel [84,85]in Euclidean space. In the pre sent case,
+this formalism should be extended to the Minkowski space and, even tually, to
+hyperbolic space (that is to the case of Abelian model discovered by Witten).
+Details of such an extension are summarized in Appendix B. After tha t, the
+next task lies in connecting these results with the Ernst equation. I n the next
+subsection we initiate this study.
+5.2 Heisenberg spin chain model and the Ernst equation
+For the sake of space, this subsection is written under assumption that our read-
+ers are familiar with the book ”Hamiltonian methods in the theory of so litons”
+[86] (or its equivalent) where all needed details can be found. The co ntin-
+uous XXX Heisenberg spin chain is described with help of the spin vecto r20
+20In compliance with [86] we suppress the time-dependence.
+26/vectorS(x) = (S1(x),S2(x),S3(x)) restricted to live on the unit sphere S2:
+/vectorS2(x) =3/summationdisplay
+i=1S2
+i(x) = 1 (5.1)
+while obeying the equation of motion
+∂
+∂t/vectorS=/vectorS×∂2
+∂x2/vectorS (5.2)
+known as the Landau-Lifshitz (L-L) equation. By introducing matr icesU(λ)
+andV(λ) via
+U(λ) =λ
+2iS,V(λ) =iλ2
+2S+λ
+2S∂
+∂xS,S=/vectorS·/vector σ (5.3)
+so thatσiis one of Pauli’s spin matrices and λis the spectral parameter and
+requiring that S2=I,whereIis the unit matrix, the zero curvature condition
+∂
+∂tU−∂
+∂xV+[U,V] = 0 (5.4)
+is obtained. With account of the constraint S2=Iit can be converted into
+equation
+∂
+∂tS=1
+2i[S,∂2
+∂x2S] (5.5)
+equivalent to Eq.(5.2). The correspondence between Eq.s(5.2) and (5.5) can be
+made forS(x,t) matrices of arbitrary dimension.
+Having in mind Witten’s result [76], we want now to extend these Euclidea n
+results to the case of noncompact Heisenberg spin chain model ”livin g” either
+in Minkowski or hyperbolic space. In doing so we follow, in part, Ref.[ 56] and
+Appendix B. For this purpose we need to remind our readers some fa cts about
+the Lie group SU(1,1). Since this group is related to SO(2,1), very mu ch like
+SU(2) is related to SO(3), we can proceed by employing the noticed a nalogy.
+In particular, since S=/vectorS·/vector σ,we can preserve this relation by writing now
+S=/vectorS·/vector τ.Using this result we obtain,
+S=/parenleftbiggSziS−
+iS+−iSz/parenrightbigg
+∈su(1,1), S±=Sx±iSy, (5.6)
+where the form of matrices generating su(1,1) Lie algebra is similar to that
+for Pauli matrices. This time, however, det S=−1 even though S2=I.
+Explicitly, ( Sz)2−(Sx)2−(Sy)2= 1,that is the motion is taking place on the
+unit pseudosphere S1,1.Matricesτigeneratingsu(1,1) are fully characterized
+by the following two properties
+tr(τατβ) = 2gαβ, [τα,τβ] = 2ifαβγτγ;gαβ=diag(−1,−1,1);α,β,γ= 1,2,3
+(5.7)
+27withfαβγbeing structure constants for su(1,1) algebra. An analog of the
+equation of motion, Eq.(5.5), now reads
+∂
+∂tSα=/summationdisplay
+β,γfαβγSβ∂2
+∂x2Sγ. (5.8)
+If we defines the Poisson brackets as {Sα(x),Sβ(y)}=−fαβγSγ(x)δ(x−y),
+then the above equation of motion can be rewritten in the Hamiltonian form
+∂
+∂tSα={H,Sα}, (5.9)
+provided that the Hamiltonian His given by
+H=1
+2∞/integraldisplay
+−∞dx(∇xSα)gαβ/parenleftbig
+∇xSβ/parenrightbig
+≡1
+4tr∞/integraldisplay
+−∞dx(∇xS)2. (5.10)
+Since now the motion takes place on pseudosphere ˇS2, it is convenient to intro-
+duce the pseudospherical coordinates by analogy with spherical, e .g.
+Sx(x,t) = sinhθ(x,t)cosϕ(x,t),Sy(x,t) = sinhθ(x,t)sinϕ(x,t),Sz(x,t) = coshθ(x,t).
+(5.11)
+Also, by analogy with spherical case we can use the stereographic p rojection
+: from pseudosphere to hyperbolic plane. Recall [ 102], that in the case of a
+sphereS2the inverse stereographic projection: from complex plane CtoS2is
+given by
+S+=2z
+1+|z|2,S−=2z∗
+1+|z|2,Sz=1−|z|2
+1+|z|2. (5.12)
+The mapping from CtoH2is obtained with help of Eq.(5.12) in a straightfor-
+ward way as
+S+=2ξ
+1−|ξ|2,S−=2ξ∗
+1−|ξ|2,Sz=1+|ξ|2
+1−|ξ|2. (5.13)
+Using this correspondence the equations of motion, Eq.(5.10), rew ritten in
+terms ofξandξ∗variables (while keeping in mind that they are parametrized
+byxandt) are given by
+i∂
+∂tξ+∂2
+∂x2ξ+2ξ∗
+1−|ξ|2/parenleftbigg∂
+∂xξ/parenrightbigg2
+= 0. (5.14)
+In the static ( t−independent) case the above equation is reduced to
+(|ξ|2−1)∇2
+xξ= 2ξ∗(∇xξ)2(5.15)
+easily recognizable as the Ernst equation. In his paper, Ref. [42], Er nst used
+variational principle applied to the functional Eq.(3.18). From Appen dix B we
+28know that both the L-L equation and its hyperbolic version describe the motion
+of (could be knotted) vortex filament. Because ofthis, the funct ional, Eq.(3.18),
+should undergo the same reduction as was made in going from Eq.(B1.a ) to
+(B1.b). Explicitly, this means that the functional, Eq.(3.18), should b e reduced
+in such a way that the Hamiltonian, Eq.(5.10), should be replaced by
+H=−2∞/integraldisplay
+−∞dx|∇xξ|2
+(1−|ξ|2)2, (5.16)
+where the sign in front is chosen in accord with Ref.[87] and our Eq.(3.2 2).
+The Hamiltonian equation of motion, Eq.(5.9), reproducing Eq.(5.14) c an be
+obtained if the Poisson bracket is defined as by {ξ(x),ξ∗(y)}= (1−|ξ|2)2δ(x−
+y).The obtained results set up the stage for quantization. It will be dis cussed in
+subsection 5.4. In the meantime, we need to connect results of Witt en’s work,
+Ref.[76], with those we just obtained.
+5.3 From Abelian Higgs to Heisenberg spin chain model
+5.3.1 The Abelian Higgs model
+The work by Witten [76] had been further analyzed in the paper by Fo rgacs
+and Manton [88]. The major outcome of their work lies in demonstratio n
+of uniqueness of the self-duality ansatz proposed by Witten. The s elf-duality
+equations obtained in Witten’s work are reduced to the system of th ree coupled
+equations describing interaction between the Abelian Y-M and Higgs fi elds
+∂0ϕ1+A0ϕ2=∂1ϕ2−A1ϕ1, (5.17a)
+∂1ϕ1+A1ϕ2=−(∂0ϕ2−A0ϕ1), (5.17b)
+r2(∂0A1−∂1A0) = 1−ϕ2
+1−ϕ2
+2. (5.17c)
+To analyze these equations, we recall that the original self-duality equations for
+theY-M fieldsareconformallyinvariant. We cantakeadvantageoft hisfact now
+by temporarily dropping the conformal factor r2in Eq.(5.17c). Then, the above
+equations become the Bogomolny equations for the flat space Abelia n Higgs
+model, e.g. for the model described by the action functional, Eq.(4.2 4), with the
+conformal factor Ξ = 1 [89]. Such obtained equations contain all info rmation
+about the Abelian Higgs model and, hence, they are equivalent to th is model.
+It is of importance for us to demonstrate this explicitly for both Euc lidean
+and hyperbolic spaces. For this purpose we introduce a covariant d erivative
+Dµ=∂µ−iAµ,µ= 0,1,and the complex field φ=φ1+iφ2. Consider the
+Bogomolny equation following [68]:
+D0φ+iD1φ= 0. (5.18)
+Usingtheabovedefinitionsstraightforwardcomputationreprodu cesEq.s(5.17a,b).
+These equations can be used to obtain
+r2(D0−iD1)(D0+iD1)φ= 0 (5.19)
+29implying
+r2(D0D0+D1D1)φ=−ir2[D0,D1]φ=−r2(∂0A1−∂1A0)φ=−(1−ϕ2
+1−ϕ2
+2)φ,
+(5.20)
+where the last equality was obtained with help of Eq.(5.17c). Evidently , the
+equation
+(D0D0+D1D1)φ+1
+r2(1−ϕ2
+1−ϕ2
+2)φ= 0 (5.21)
+isoneoftheequationsof”motion”fortheG-Lmodelon H2, e.g. seeRef.[89](Eq.(11.3)
+page 98). The second is the Ampere’s equation
+εµν∂µ(r2B) =i(φ¯Dν¯φ−¯φDνφ) (5.22)
+with the ”magnetic field” B=∂0A1−∂1A0.Details of derivation are given in
+Ref.[68], pages 198-199. Eq.(5.22) also coincides with that given in the book by
+Taubs and Jaffe, Ref.[ 105] (Eq.(11.3) page 98).
+Corollary 1 .Since both equations can be obtained by mimization of the
+functional, Eq. (4.27), they are equivalent to the Abelian Higgs model which, in
+turn, is the reduced form of the Y-M functional for pure gauge fie lds.
+We continue with the discussion of Witten’s treatment of Eq.s(5.17) s ince
+we shall need his results later on. First, he selects physically conven ient gauge
+condition via ∂µAµ= 0.This leads to the choice: Aµ=εµν∂νψ(for some scalar
+ψ). With such a choice for Aµthe first two of Eq.s(5.17) can be rewritten as
+(∂0−∂0ψ)ϕ1= (∂1−∂1ψ)ϕ2, (5.23a)
+(∂1−∂1ψ)ϕ1=−(∂0−∂0ψ)ϕ2. (5.23b)
+Let nowϕ1=eψχ1andϕ2=eψχ2.Then the above equations are reduced to
+the Cauchy-Riemann-type equations: ∂0χ1=∂1χ2and∂1χ1=∂0χ2.Introduce
+the function f=χ1−iχ2. Then, the last of Eq.s(5.17) acquires the form
+−r2∇2ψ= 1−ff∗e2ψ. (5.24)
+Notice that −r2∇2=−r2(∂2
+∂t2+∂2
+∂r2) is the hyperbolic Laplacian [90]. Eq.(5.24)
+is still gauge invariant in the sense that by changing f→fhandψ→ψ−
+1
+2ln(hh∗) in this equation we observe that it preserves its original form. This
+is so because ∇2ln(hh∗) = 0 for any analytic function which does not have
+zeros. Ifhdoes have zeros for r >0, then substitution of ψ→ψ−1
+2ln(hh∗)
+intoEq.(5.24)producesisolatedsingularitiesatthesezeros. After theseremarks,
+Eq.(5.24)canbe simplified further. Forthispurpose, let ψ= lnr−1
+2ln(ff∗)+ρ,
+provided that ∇2ln(ff∗) = 0 for any analytic function fwhich does not have
+zeros21. Under such conditions we end up with the Liouville equation
+∇2ρ=e2ρ. (5.25)
+It is of major importance for what follows.
+21In the case if it does, the treatment is also possible as expla ined by Witten. Following
+his work, we shall temporarily ignore this option.
+305.3.2 The Heisenberg spin chain model
+The results of Appendix B imply that the L-L Eq.(5.2) (or their hyperb olic
+equivalent, Eq.(5.8)) could be interpreted in terms of equations for the Serret-
+Frenet moving triad. Treatment along these lines suitable for immedia te appli-
+cations is given in papers by Lee and Pashaev [91] and Pashaev [92]. Be low we
+superimpose their results with those of our work, Ref.[84], to achiev e our goals.
+We begin with definitions. A collection of smooth vector fields nµ(x,t),
+µ= 0−2, forming an orthogonal basis is called the ”moving frame”. If x∈ S
+whereSis some two dimensional surface, then let n1(x,t) and n2(x,t) form
+basis for the tangent plane to S ∀x∈ S. Then, the Gauss map (that is the
+map from Sto two dimensional sphere S2or pseudosphere S1,1) is given by
+n2(x,t)≡s. By design, it should obey Eq.(5.1). This observation provides
+needed link between the spin and the moving frame vectors. Details a re given
+in [91,92] and Appendix B .It should be clear that since one can draw curves
+on surfaces both formalisms should involve the same elements. The r estriction
+for the curve to lie at the surface causes additional complications in general but
+nonessential in the present case.
+Next, we introduce the combinations n±=n0±in1possessing the following
+properties
+(n+,n+) = (n−,n−) = 0 , (n+,n−) = 2/κ2, (5.26)
+whereκ2= 1 forS2andκ2=−1 forS1,1andH2.Furthermore, ( ..,..) defines
+the scalar product (in Euclidean or pseudo-Euclidean spaces). Also ,
+n+×s=in+,n−×s=−in−,n−×n+=2iκ2s. (5.27)
+In addition, we shall use the vectors
+qµ=κ2
+2(∂µs,n+) and ¯qµ=κ2
+2(∂µs,n−) (5.28)
+in terms of which the equations of motion for the moving frame vecto rs look as
+follows:
+Dµn+=−2κ2qµs, (5.29)
+∂µs=qµn−+ ¯qµn+, (5.30)
+with covariant derivative Dµ=∂µ−i
+2VµandVµ=−2κ2(n1,∂µn0).Consider
+now Eq.(5.30) for µ= 1.Apply to it the operator ∂1and use the equations of
+motion and the definitions just introduced in order to obtain
+∂2
+1s=(D1q1)n−+/parenleftbig¯D1¯q1/parenrightbig
+n+−4
+κ2|q1|2s. (5.31)
+It can be shown that q0=iD1q1.In view of this, Eq.(5.30) for µ= 0 acquires
+the following form:
+∂0s=iD1q1n−−i¯D1¯q1n+. (5.32)
+This equation happens to be of major importance because of the fo llowing.
+Multiply (from the left) Eq.(5.31) by s×and use Eq.s(5.27). Then (depending
+31on signature of κ2) the obtained result is equivalent to the L-L Eq.(5.2) or its
+pseudoeuclidean version, Eq.(5.8). Furthermore, for this to happ en the fields Vµ
+andqµmust be subject to the following constraint equations obtainable dir ectly
+from Eq.s (5.29)
+Dµqν=Dνqµ, (5.33a)
+[Dµ,Dν] =−2κ2(¯qµqν−¯qνqµ). (5.33b)
+Wearegoingtodemonstratenowthattheseequationsareequivale nttoEq.s(5.17)
+obtained by Witten.
+We begin with the following observation. Let indices µandνbe respectively
+1 and 0. Then, taking into account that q0=iD1q1we can rewrite Eq.(5.33b)
+as
+F10=B1=−2��2i(¯q1D1q1−q1¯D1¯q1). (5.34)
+Surely, by symmetry we could use as well: q1=−iD0q0. This would give us
+an equation similar to Eq.(5.34). Take now the case κ2=−1 (that is consider
+S1,1) in these equations and compare them with the Ampere’s law, Eq.(5.22 ).
+We notice that these equations are not the same. However, since t he G-L model
+was originally designed for phenomenological (thermodynamical) des cription of
+superconductivity(asexplainedindetailinourwork,Ref.[84]),wekn owthatthe
+underlying equations (obtainable from the G-L functional) contain t he London
+equation which reads22
+∇×B=CB (5.35)
+withCbeing someconstant(determined byphysicalconsiderations). Ev idently,
+in view ofthe London(5.35), Eq.s(5.22)and (5.34) become equivalent . Consider
+now Eq.(5.33a). To understand better this equation, it is useful to rewrite
+Eq.(5.18) as follows
+D0φ=−iD1φorD0φ1=D1φ0, (5.36)
+whereφ1=φandφ0=−iφ.Take into account now that φ=a+iband
+identifyφ1withq1andφ0withq0.Then, Eq.(5.33b) acquires the following
+form (κ2=−1) :
+(∂0V1−∂1V0) =−i4(¯φ0φ1−φ0¯φ1) =−4(a2+b2). (5.37)
+Looking at Eq.(5.17c) we can make the following identifications: V1=A1,V0=
+A0,±2a=ϕ1,±2b=ϕ2.Then, comparison between Eq.s(5.17c) and (5.37)
+indicates that we are still missing a factor of r2in the l.h.s. and 1 in the r.h.s.
+Looking at Witten’s derivation of the Liouville Eq.(5.25), we realize that these
+two factors are interdependent. By clever choice of the function ψthey can
+be made to disappear. This makes physical sense since locally the und erlying
+surface is almost flat. This observation makes Eq.s(5.37) and (5.24) (or 5.17c)
+equivalent.
+22This is not the form of the London equation one can find in textb ooks. But in our work,
+Ref.[84], we demonstrated that Eq.(5.35) is equivalent to t he London equation.
+32Corollary 2 .The L-L and 2 dimensional G-L models are essentially equiv-
+alent in the sense just described both in Euclidean and in Minkowski sp aces.
+Corollary 3 .The ”hyperbolc” L-L Eq. (5.14)or its Euclidean analog should
+be identified with Floer’s Eq. (4.6).
+These results play an important role in the rest of this work and, in pa rtic-
+ular, in the next subsection.
+5.4 The proof (implementation)
+5.4.1 General remarks
+In Ref.[79], we demonstrated how treatment of combinatorial data associated
+with real scattering experiments leads to restoration of the unde rlying quantum
+mechanical model reproducing the meson spectrum. It was estab lished that
+the underlying microscopic model is the Richardson-Gaudin (R-G) XX X spin
+chain model originally designed for description of spectrum of excita tions in the
+Bardeen-Cooper-Schriefer (BCS) model of superconductivity. Subsequently,
+the same model was used for description of spectra of the atomic n uclei. Since
+the energy spectrum of the BCS model has the famous gap betwee n the ground
+and the first excited state, the problem emerges :
+Can spectral properties of nonperturbative quantum Y-M fie ld
+theory be described by the R-G model ?
+To answer this question affirmatively the ”equivalence principle” disco vered
+by L.Witten is very helpful. Using it, we can proceed with quantization o f pure
+Y-M fields by using results by Korotkin and Nicolai, Ref.[31], for gravity . By
+comparing the main results of our paper, Ref.[79], done for QCD, with those of
+Ref.[31], done for gravity, we found a complete agreement. In part icular, the
+Knizhnik-Zamolodchikov Eq.s(4.14),(4.15) and the R-G Eq.(4.29) of Re f.[79]
+coincide respectively with Eq.s(4.27),(4.26) and (4.50) of Ref.[31] eve n though
+methods of deriving of these equations are entirely different! Both Ref.s [79]
+and [31] do not reveal the underlying physics sufficiently deeply thou gh. In the
+remainder of this section we shall explain why this is indeed so and demo nstrate
+ways this deficiency can be corrected. Experimentally the challenge lies in de-
+signing scattering experiments providing clean information about th e spectrum
+of glueballs. Thus far this task was accomplished only in lattice calculat ions
+done for unphysically large number of colors, e.g. Nc→ ∞.[23].When it comes
+to interpreting realexperiments (always having only three colors to consider23),
+the situation is even less clear, e.g. see Ref.[93]. Hence, the gap prob lem is full
+ofchallengesforboth theoryand experiment. Fortunately, at lea st theoretically,
+the problem does admit physically meaningful solution as we explained a lready.
+We continue with ramifications in the next subsection.
+23E.g. read Section 6 .
+335.4.2 From Landau-Lifshitz to Gross-Pitaevskiiequation v ia Hashimoto
+map
+Since the F-S model is believed to be capable of describing QCD vacua a nd
+is also capable of describing knotted/linked structures [17], two que stions arise:
+a) Is this the only model capable of describing QCD vacua? b) To what extent
+it matters that the F-S model is also capable of describing knots and links?
+The negative answer to the first question follows from Corollary 3 imp lying
+that, in principle, both Euclidean and hyperbolic versions of the L-L e quation
+are capable of describing QCD vacua: different vacua correspond t o different
+steady-state solutions of the L-L equations. The negative answe r to the second
+question can be found in a review, Ref.[85], by Annalisa Calini. From this
+reference it follows that, besides the F-S model, knotted/linked st ructures can
+be also obtained by using standard (that is Euclidean) L-L equation, e.g. see
+Eq.(B.4) of Appendix B. This fact still does not explain why knots/links are of
+importance to QCD. We address the above issues in more detail in Sec tion 6. In
+view of what is said above, wether or not the hyperbolic version of L- L equation
+is capable of describing knotted structures is not immediately import ant for
+us. Far more important is the connection between the hyperbolic L- L and the
+Ernst equation. Only with this connection it is possible to reproduce r esults by
+Korotkin and Nicolai [31].
+Eq.(3.19)is just the F-S functionalwithout winding numberterm. Wh en the
+commutation relations for su(1,1) introduced in subsection 5.2 are r eplaced by
+those for su(2) this leads to the standard L-L equation (instead o f Eq.(5.14)).
+This replacement causes us to abandon the connection with Ernst e quation
+and, ultimately, with the results of Ref.[31]. In such a case the gap pr oblem
+should be investigated from scratch. In Ref.[25] Faddeev and Niemi indicated
+that, unless some amendments to the F-S model are made, it is gaple ss. At the
+same time from Appendix B it is known that the L-L equation associate d with
+the F-S model can be transformed into the NLSE with help of the Has himoto
+map. Recently, Ding [94] and Ding and Inoguchi [95] were able to find a nalogs
+of the Hashimoto map for the vortex filaments in hyperbolic, de Sitte r and
+anti de Sitter spaces. It is helpful to describe their findings using t erminology
+familiar from physics literature [96].This leads us to the discussion of pr operties
+of the Gross-Pitaevskii equation known in mathematics as the NLSE . In the
+system of units in which ℏ= 1 andm= 1/2 this equation can be written as
+[86]
+iψt=−ψxx+2κ/parenleftBig
+|ψ|2−c2/parenrightBig
+ψ= 0. (5.38)
+Zakharov and Shabat [97,98] performed detailed investigation of th is equation
+for both positive and negative values of the coupling constant κ.Forκ <0
+the above equation is used for description of knots/links [85]. The st andard
+Hashimoto map brings the L-L equation associated with the truncat ed F-S
+model to the NLSE with κ <0 [94, 95]. From the same references it can be
+found that the Hashimoto-like map brings the (hyperbolic) L-L-like e quation to
+the NLSE for which κ >0.Zakharov and Shabat studied in detail differences
+34in treatments of the NLSE for both negative and positive coupling co nstants.
+This difference is caused by difference in underlying physics which in bot h cases
+can be explained in terms of the properties of non ideal Bose gas [99,1 00]. The
+attentive reader might have noticed at this point that Eq.(5.38) app arently
+contains no information about the number of particles in such a gas. This
+parameter, in fact, is hidden in the constant c(the chemical potential) or it can
+be obtained selfconsistently with help of Eq.(5.38)(from which cis removed in a
+way described in Appendix B) as explained in Ref.[100]. With this informat ion
+at our disposal we are ready to make the next step.
+5.4.3 From non ideal Bose gas to Richardson-Gaudin equation s
+Even though statistical mechanics of 1-d interacting Bose gas was considered in
+detailbyLiebandLinger[101],solidstatephysicsliteratureisfullofr efinements
+of their results up to moment of this writing. These refinements hav e been
+inspired by experimental and theoretical advancements in the the ory of Bose
+condensation [96]. Among this literature we selected Ref.s[102,103] a s the most
+relevant to our needs.
+Following [102], the Hamiltonian for Ninteracting bosons moving on the
+circle of length Lis given by
+H=−N/summationdisplay
+i=1∂2
+∂x2
+i+2ˇc/summationdisplay
+1≤i<j≤Nδ(xi−xj) (5.39)
+with constant 2ˇ ccoinciding with 2 κin the system of units ℏ= 1 andm= 1/2.
+The case ˇc >0 (repulsive Bose gas) corresponding to the L-L equation in the
+hyperbolicplane/spacehappens tobe ofimmediate relevance. Onlyforthis case
+it is possible to establish the connection with workby Korotkinand Nico lai[31]!
+We begin by noticing that in the standard BCS theory of supercondu ctivity
+electrons are paired into singlets (Cooper pairs) with zero centre o f mass mo-
+mentum. The pairing interaction term in this theory accounts only fo r pairs
+of attractive electrons with opposite spin and momenta so that the degener-
+acy for each energy state is a doublet, with level degeneracy Ω = 224. In the
+interacting repulsive Bose gas model byRichardson [104] to mimic this pairing
+he coupled two bosons with opposite momenta ±kjinto one (pseudo) Cooper
+pair. An assembly of such formed pairs forms repulsive Bose gas which in the
+simplest case is described by the Hamiltonian, Eq.(5.39). Hence, the fermionic
+BCS-type model with strong attractive pairing interaction can be m apped into
+bosonic repulsive model proposed by Richardson. Although the idea of such
+mapping looks very convincing, its actual implementation in Ref.[102] h as some
+flaws. Because of this, we shall use results of this reference selec tively. For this
+purpose, fist of all we need to make an explicit connection between t he repulsive
+Bose gas model described by Eq.(5.39) and the model proposed by R ichardson.
+In the weak coupling limit ˇ cL≪1 the Bethe ansatz equations for the repulsive
+24We use here the same notations as in our work, Ref.[ 94].
+35Bose gas model described by the Hamiltonian, Eq.(5.39), acquire the following
+form:
+ki=2πdi
+L+2ˇc
+LN/summationdisplay
+j=1
+(j/negationslash=i)1
+kj−ki,i= 1,...,N. (5.40)
+Heredi= 0,±1,±2,...denote the excited states for fixed N. To link this result
+with Richardson’s (repulsive boson) model, consider the case of eve n number of
+bosons and make N= 2M. Next, consider the ground state of this model first.
+To the first order in ˇ c, it is clear that we can write ki=±√Ei. Specifically,
+letk1,2=±√E1,k3,4=±√E3,...,k2M−1,2M=±√EM.Using these results in
+Eq.(5.40), with the accuracy just stated, the Bethe ansatz equa tions after some
+algebra are converted into the following form:
+L
+2ˇc+˜M/summationdisplay
+j=1
+(j/negationslash=i)2
+Ej−Ei=1
+2Ei,i= 1,...,M;˜M≤M. (5.41)
+To analyze these equations, we expect that our readers are familia r with works
+of both Richardson-Sherman, Ref.[105], and Richardson, Ref.[104]. In [105]
+diagonalization of the pairing force Hamiltonian describing the BCS-ty pe su-
+perconductivity was made. Such a Hamiltonian is given by
+H=/summationdisplay
+f2εfˆNf−g/summationdisplay′
+f/summationdisplay′
+f′b†
+fbf′, (5.42)
+whereˆNf=1
+2(a†
+f+af−+a†
+f−af−),bf=af−af+, witha†
+fσandafσbeingfermion
+creation and annihilation operators obeying usual anticommutation relations
+[afσ,a†
+f′σ′]+=δσσ′δff′, whereσ=±denotes states conjugate under time
+reversal. The above Hamiltonian is diagonalized along with the seniority oper-
+ators (taking care of the number of unpaired fermions at each leve lf) defined
+by
+ˆνf=a†
+f+af−−a†
+f−af−. (5.43)
+By construction, [ H,ˆNf] = [H,ˆνf] = 0.The classification of the energy levels
+is done in such a way that the eigenvalues νfof the operator ˆ νf(0 andσ) are
+appropriatefor the case when g= 0.This observation allowsus to subdivide the
+Hamiltonian into two parts: H1,i.e.that which does not contain Cooper pairs
+(for which νf=σ) andH2,i.e.that which may contain such pairs (for which
+νf= 0).The matrix elements of H2are calculated with help of the bosonic-type
+commutation relations
+[bf′,ˆNf′] =δff′bfand [bf,b†
+f′] =δff′(1−2ˆNf′). (5.44)
+These commutators are bosonic but nontraditional. In the traditio nal case we
+have [bf,b†
+f′] =δff′.We refer our readers to Ref.[105] for details of how this
+commutator difficulty is resolved. In the light of this resolution, in Ref .[104]
+36Richardson proposed to deal with the interacting bosons model fr om the be-
+ginning. Supposedly, such bosonic model can be designed to reproduce res ults
+of the fermionic pairing model of Ref.[105]. An attempt to do just this was
+made in Ref.[102]. In the repulsive boson model by Richardson the ”pa iring”
+Hamiltonian is given by25
+H=/summationdisplay
+l2εlˆnl+g
+2/summationdisplay′
+f/summationdisplay′
+f′A†
+fAf′. (5.45)
+in which ˆnlandAf′are bosonic analogs of ˆNfandbf.It is essential that
+the sign of the coupling constant gis nonnegative (repulsive bosons). Upon
+diagonalization, the total energy Eis given by
+E=n/summationdisplay
+l=1εlνl+m/summationdisplay
+j=1Ej (5.46)
+so that summation in the first sum takes place over the unpaired bos ons while
+in the second- over the paired bosons whose energies Ejare determined from
+the Richardson’s equation (Eq.(2.29) of Ref. [104])26
+1
+2g+n/summationdisplay
+l=1dl
+2εl−Ek+m/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek= 0,k= 1,...,m (5.47)
+in whichnis the total number of single particle (unpaired) levels, mis the total
+number of pairs, dl=1
+2(2νl+ Ωl).From [104] it follows that for the bosonic
+model to mimic the BCS-type pairing model the degeneracy factor Ω l= 1 and
+νl= 0.It should be noted though that such an identification is not of much
+help in comparing the repulsive bosonic model with the attractive BCS -type
+fermionic model (contrary to claims made in Ref.[102]). This can be eas ily
+seen by comparison between Eq.(5.47) (that is Eq.(2.29) of Ref.[104]) with such
+chosen Ω landνlwith Eq.(3.24) of Ref.[105]. By replacing gin Eq.(5.47) by
+−gwe still will not obtain the analog of the key Eq.(3.24) of Ref.[105]! This
+fact has group-theoretic origin to be explained in the next subsect ion. In the
+meantime, Eq.(5.47) still can be used to connect it with Eq.(5.41) origin ating
+from different bosonic model described by the Hamiltonian Eq.(5.39). To do so
+we follow the path different from that suggested in Ref.[102]. Instea d, following
+the original Richardson’s paper [104], we let n= 1 in Eq.(5.47) then, without
+loosing generality, we can put ε1= 0 so that Eq.(5.47) acquires the following
+form
+1
+Ek=1
+2g+M/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek, k= 1,...,M. (5.48)
+25To avoid ambiguities, our coupling constantg
+2is chosen exactly the same as in [104].
+26Since Gaudin’s equation is obtained in the limit |g| → ∞.from Eq.(5.47) the spin -like
+model described by this equation is known as the Richardson- Gaudin (R-G) model.
+37The rationale for replacing mbyMis given on page 1334 of [104]. Evidently,
+Eq.s (5.41) and (5.48) are identical. This observation allows us to use t he
+Richardson model instead of that described by Eq.(5.39). At first s ight such
+an identification looks a bit artificial. To convince our readers that it d oes
+make sense, we would like to use the work by Dhar and Shastry [106,10 7] on
+excitation spectrum of the ferromagnetic Heisenberg spin chain. B y analogy
+with Eq.(5.41) these authors derived a similar equation obtained by re ducing
+the Bethe ansatz equations for Heisenberg ferromagnetic chain. It reads27
+1
+El=πd−d
+n/summationdisplay
+i=1
+i/negationslash=l2
+Ei−El. (5.49)
+Even though Eq.s(5.48) and (5.49) look almost the same, they are no t the same!
+The crucial difference lies in the signs in front of the second term in th e r.h.s. of
+theseequations. BecauseofthisdifferenceHeisenberg’sferroma gneticspinchain
+model is mapped onto Bose gas model with attractive interaction in complete
+accord with what was said immediately after Eq.(5.38). Regrettably, this result
+is still not the same as for the BCS-type model investigated in Richar dson-
+Sherman’s paper, Ref.[105]. This fact was recognized and discussed in some
+detail already by Richardson [104]. For completeness, we mention th at the
+problem of BCS-Bose-Einstein condensation (BEC) crossover whic h follows
+exactly the qualitative picture just described was made quantitativ e only
+very recently in Ref.[108]. Fortunately, it is possible to by-pass this r esult as
+explained in the next subsection.
+5.4.4 From Richardson-Gaudin to Korotkin-Nicolai equatio ns
+In Ref.[109] bosonic and fermionic formalism for pairing models discuss ed in
+the previous subsection was developed. This formalism happens to b e the most
+helpful for investigation of the gap problem. Indeed, define three operators
+ˆnl=/summationtext
+ma†
+lmalm,A†
+l= (Al)†=/summationtext
+ma†
+lma†
+l¯m. Theycanbe used forconstruction
+of operators K0
+l=1
+2ˆnl±1
+4ΩlandK+
+l=1
+2A†
+l=/parenleftbig
+K−
+l/parenrightbig†such that they obey
+the following commutator algebra
+[K0
+l,K+
+l] =δllK+
+l,[K+
+l,K−
+l] =∓2δllK0
+l. (5.50)
+Inthisalgebraaswellasintheprecedingexpressions,theuppersig ncorresponds
+to bosons and the lower to fermions. In Ref.[79], we discussed such a n algebra
+for the fermionic case only, e.g. see Eq.s (4.31) of [79]. These results can
+be extended now for the bosonic case. In fact, such an extension is already
+developed in Ref.[109]. Unlike [79], where we used sl(2,C) Lie algebra, only
+its compact version, that is su(2), was used in [109] for representing fermions.
+For bosonic case the commutation relations, Eq.(5.50), are those f orsu(1,1) Lie
+algebra. Incidentally, in the paper by Korotkinand Nicolai, Ref.[31], ex actly the
+27The physical meaning of constants entering this equation is not important for us. It is
+given in Ref.[106]..
+38same Lie algebra was used. Furthermore, in the same paper it was ar gued that
+it is permissible to replace su(1,1) bysl(2,R) Lie algebrawhile constructing the
+K-Z-type equations, e.g. read p.428 of this reference. Since in [79] thesl(2,C)
+Lie algebra was used, that is a complexified version of sl(2,R),this allows us to
+use many results from our work. Thus, in this subsection we shall dis cuss only
+those results of [109] which are absent in our Ref.[79]. In particular, following
+this reference the set of Gaudin-like commuting Hamiltonians written in terms
+of operators K0
+l,K+
+landK−
+lis given by
+Hl=K0
+l+2g{/summationdisplay
+l′(/negationslash=l)Xll′
+2(K+
+lK−
+l′+K−
+lK+
+l′)∓Yll′K0
+lK0
+l′}.(5.51)
+HereXll′=Yll′= (εl−εl′)−1.Forg→ ∞the first term can be ignored and the
+remainder can be used in the K-Z-type equations. The semiclassical treatment
+of these equations discussed in detail in [79] is resulting in the following set of
+Bethe (or R-G) ansatz equations
+n/summationdisplay
+l=1dl
+2εl−Ek±m/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek= 0, k= 1,...,m (5.52)
+to be compared with Eq.(5.47). Unlike Eq.(5.47), in the present case dl=
+1
+2(2νl±Ωl).The bosonic version of Eq.(5.52) corresponding to su(1,1) Lie alge-
+bra coincides with Eq.(4.50) of Korotkin and Nicolai paper, Ref.[31], pr ovided
+that the following identifications are made: dl⇄sl, 2εl⇄γj. Unlike Ref.[31],
+where Eq.(5.52) was obtained by standard mathematical protocol, in this work
+it is obtained based on the underlying physics. Because of this, it is ap propriate
+to extend our physics-stype analysis by considering the case of fin iteg′s.Then,
+Eq.(5.52) should be replaced by
+1
+2g±n/summationdisplay
+l=1dl
+2εl−Ek±m/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek= 0,k= 1,...,m. (5.53)
+In Ref.[31] the gap problem was discussed in detail for the fermionic c ase when
+the coupling constant gis negative (BCS pairing Hamiltonian), e.g. see Eq.s
+(4.43)-(4.45) of Ref.[31]. In the present case we are dealing with the bosonic
+case for which the coupling constant is positive. Hence the gap prob lem should
+be re analyzed. For this purpose, it is convenient to consider both p ositive and
+negative coupling constants in parallel for reasons which will become apparent
+upon reading.
+5.4.5 Emergence of the gap and the gap dilemma
+Eq.s(5.53) cannot be solved without some physical input. Initially, su ch an
+input was coming from nuclear physics (e.g. read [110-112]for gene ral informa-
+tion on nuclear physics). Indeed, Richardson’s papers [104,105] we re written
+39having applications to nuclear physics in mind. Given this, the question arises
+about the place of the R-G model among other models describing nuc lear spec-
+tra and nuclear properties. We need an answer to this question to fi nish proof
+of the gap’s existence in QCD.
+Looking at the Gaudin-like Hamiltonian, Eq.(5.51), and comparing it with
+the Hamiltonian, Eq.(6), in Ref.[113]28it is easy to notice that they are almost
+the same! Because of this, it becomes possible to transfer the met hodology of
+Ref.[113]tothepresentcase. Thus, itmakessensetorecallbriefl ycircumstances
+at which the gap emerges in nuclear physics.
+As is well known, the nuclei are made of protons and neutrons. One can
+talk about the number Nof nucleons, the number Z of protons and the number
+N of neutrons in a given nucleus. Nuclear and atomic properties happ en to
+be interrelated. For instance, in analogy with atomic physics one can think of
+some effective nuclear potential in which nucleons can move ”indepen dently”.
+This assumption leads to the shell model of nuclei. Use of Pauli principle guides
+fillings of shells the same way as it guides these fillings in atomic physics. T his
+leads to emergence of magic numbers 2, 8, 20, 28, 50, 82 and 126 fo r either
+protons or neutrons for the totally filled shells. Accordingly, the mo st stable
+are the doubly magic (for both protons and neutrons) nuclei. It is o f interest to
+know what kinds of excitations are possible in such shell models? The s implest
+of these is when some nucleon is moving from the closed shell to the em pty shell
+thus forming a hole. When the numberof nucleonsincreases, the question about
+thevalidity ofthe shellmodel emerges,againin analogywith atomicph ysics. As
+in atomic physics, one can think about the Hartree-Fock(H-F) and other many-
+body computational schemes, including that developed by Richards on-Sherman
+and Gaudin. For our purposes, it is sufficient to use only the Tamm-Da nkoff
+(T-D) approximation to the H-F equations described, for example, in Ref.[112].
+The essence of this approximation lies in restricting the particle-hole interac-
+tions to nucleons lying in the same shell. The T-D approximation is obtainable
+from the isR-G Eq.s (5.53)when the last term (effectively taking care of Pauli
+principle) in these equations is dropped . The T-D approximation was success-
+fully applied for description of the giant nuclear dipole resonance [110 -112]. At
+the classical level the physics of this resonance was explained in the paper by
+Goldhaber and Teller [114]. The resonance is caused by two nuclear vib rational
+modes: one, when protons and neutrons move in the opposite direc tions and
+another- when they move in the same direction. Upon quantization o f such
+classical model and taking into account the isotopic spin of nucleons , the trun-
+cated Eq.s(5.53) are obtained in which both signs for the coupling con stant are
+allowed since the nucleon system is expected to be in two isospin state s :T= 1
+andT= 029. Details of these calculations are given in Ref.[112], page 221. So-
+lutions of the T-D equations can be obtained graphically in complete an alogy
+with that described in our work, Ref.[79]. These graphical solutions r eflect the
+particle-hole duality built into the T-D approximation. Because of this duality,
+28Published in 1961!
+29This can be easily understood based on the fact that isospin f or both particles and holes
+is equal to 1/2 [110-112].
+40the magnitude of the gap in both cases should be the same. To demon strate
+this, the seniority scheme described in [110-112] is helpful. The seniority opera-
+tor was defined by Eq.(5.43). It determines the number of unpaired particles in
+the nuclear system. Since it commutes with the Hamiltonian, the many -body
+states can be classified with help of its eigenvalues νf.Suppose at first that all
+single particle energies εfare the same (that is εf=ε) so that all seniority
+eigenvalues νfareν.Let then Nbe the total number of nucleons. Thus, the
+state for which ν= 0 contains only pairs, analogously, the state ν= 1 contains
+just one unpaired nucleon, ν= 2 has 2 unpaired nucleons and Nshould be
+even and so on. So, states ν= 0,ν= 2,ν= 4,...can exist only in even nuclei.
+For such nuclei the gap is nonzero. To see this, we follow Refs.[110-1 12] which
+we would like now to superimpose with the results of the Richardson-S herman
+paper, Ref.[105]. Specifically, on page 231 of this reference one can find the
+following result for the ground state ( ν= 0) energy
+Eν=0(N) = 2Nε−gN(Ω−N+1) (5.54)
+whereNis the number of pairs. To connect this result with that in Refs.[110-
+112], letN=N/2 and consider the difference
+Eν=0(N) =Eν=0(N/2)−Nε=−g
+4N(2Ω−N+2).(5.55)
+TheobtainedresultcoincideswithEq.(11.14)ofRef.[112]asrequired . Toobtain
+states of seniority ν= 2nwe use Eq.(3.2) of Ref.[105]. It reads
+Eν=2n(N) = 2Nε−g(N−n)(Ω−N−n+1), n= 0,...,N. (5.56)
+Repeating the same steps as in ν= 0 case we obtain,
+Eν(N) =−g
+4(N-ν)(2Ω−N-ν+2). (5.57)
+Finally, consider the difference
+Eν(N)−Eν=0(N)=g
+4ν(2Ω−ν+2). (5.58)
+This result is in accord with Eq.(11.22) of Ref.[112]. Since the obtained d if-
+ference is N-independent it can be used both ways: a) for calculations in the
+thermodynamic limit N → ∞ and b) for making accurate calculations in the
+opposite limit of very small number of nucleons. In the simplest case w e should
+consider only one shell and the first excited state of seniority 2 for this shell.
+Initially (the ground state) we have just one pair while finally (the firs t excited
+state) we have two independent particles occupying single particle le vels.
+Looking at Eq.s(5.53) and letting there m= 1(one pair) we recognize that
+the second sum in this set of equations disappears. Thus, by design , we are left
+with the T-D approximation. Using Eq.(5.58) for ν= 2 we obtain the following
+value of the gap ∆ :
+∆ =E2(N)−E0(N) =gΩ. (5.59a)
+41Notice, that since Ω is the degeneracy, there could be no more than N= Ω
+particles at the single particle level. Thus, in general we should have N ≤Ω.
+Because ofthe particle-holeduality, it is permissible to look alsoat the situation
+for which N ≥Ω.This is equivalent to changing the sign in front of the coupling
+constant. Repeating again all steps leads to the final result for th e gap
+∆ =E2(N)−E0(N)=|g|Ω. (5.59b)
+It is demonstrated in Ref.s [110-112]that in the limit N → ∞,when the contin-
+uum approximation (replacing summation by integration) can be used leading
+to a more familiar BCS-type equation for the gap, the result just ob tained sur-
+vives. Indeed, in Ref.[103] the BCS-type result is obtained in the con tinuum
+approximation for the attractive Bose gas. In view of the results just obtained,
+it should be clear that such a result should hold for both attractive a nd repul-
+sive Bose gases. This conclusion is in accord with accurate recent Be the ansatz
+calculations done in Ref.[115] for systems of finite size. Thus, we jus t arrived
+at the issue which we shall call the gap dilemma . While the results obtained
+above strongly favor use of the repulsive Bose gas model (not linked with the
+F-S model ),the results obtained in this subsection indicate that, after all, the
+F-S model (linked with the attractive Bose gasmodel )can also be used for de-
+scription of the ground and excited states of pure Y-M fields .The essence of
+the dilemma lies in deciding which of these results should ac tually
+be used .
+While the answer is provided in the next section, we are not yet done w ith
+the gap discussion. This is so because the seniority model is applicable only
+to the case when all single-particle levels have the same energy. This is too
+simplistic. We would like now to discuss more realistic case
+Before doing so, few comments are appropriate. In particular, wit h all suc-
+cesses of nuclear physics models, these models are much less convin cing than
+those in atomic physics. Indeed, all nuclei are made of hadrons whic h are made
+of quarks and gluons. Thus the excitations in nuclei are in fact the e xcitations
+of quark-gluon plasma. This observation qualitatively explains why th e R-G
+equations work well both in nuclear and particle physics. Some attem pts to
+look at the processes in nuclear physics from the standpoint of had ron physics
+can be found in Refs.[116,117].
+Now we can return to the discussion of the T-D equations. Fortuna tely, de-
+tailed analytical study of these equations was recently made in Ref.[1 18]. The
+same authors extended these results to the case of two pairs in [11 9]. Since
+the results obtained in [119] are in qualitative agreement with those o btained
+in Ref.[118], we shall focus attention of our readers only on results o f Ref.[118].
+Thus, we need to find some kind of analytic solution of the following T-D equa-
+tion
+L/summationdisplay
+i=1Ωi
+2εi−E=1
+g. (5.60)
+For different ε′
+isnormally it should have Leigenvalues Eµ(1≤µ≤L).Since
+we are interested in finding the gap, the above equation is written fo r just one
+42nucleon pair. Thus the seniority ν= 0.It is of interest to check first what
+happens when all ε′
+iscoalesce. In such a case we obtain,
+Ω
+2¯ε−E=1
+g, (5.61)
+where Ω =/summationtext
+iΩiandεi= ¯ε∀i= 1,...,L.Eq.(5.61) can be equivalently rewrit-
+ten as
+E0= 2¯ε−Ωg. (5.62)
+This result for the ground state is in agreement with Eq.(5.54) for N= 1. The
+first excited state is made of one broken pair so that the pairing disa ppears
+and the energy Eν=2= 2¯ε. From here, the value of the gap is obtained as
+Eν=2−E0=gΩ in agreement with Eq.(5.59). If now we make all energy
+levels different, then one can see that solutions to Eq.(5.60) are sub divided
+into those lying in between the single particle levels ( trapped solutions ) and
+those which lie outside these levels ( collectivized solutions ). For|g|sufficiently
+large the solution, Eq.(5.61), is the leading term (in the sense describ ed below)
+representing the collectivized solution. Since the trapped solutions represent
+corrections to energies of single particle states, they do not cont ribute directly
+to the value of the gap. They do contribute to this value indirectly. I ndeed,
+following Ref.[118] we rewrite Eq.(5.60) as
+L/summationdisplay
+i=1Ωi
+2εi−E=1
+2¯ε−E/summationdisplay
+iΩi
+1+2εi−¯ε
+2¯ε−E=1
+g(5.63)
+and expand the denominator of Eq.(5.63) in a power series. As result , the
+following expansion
+E−2¯ε
+gΩ=−1−α2+γα3+O(α4) (5.64)
+is obtained in which ¯ ε=1
+Ω/summationtext
+iΩiεi,α=2σ
+gΩ,σ=/radicalbigg
+1
+Ω/summationtext
+iΩi(εi−¯ε)2andγis
+related to the higher order moments ( details are in Ref.s[118,119]). U sing these
+results, the gap is obtained in the same way as before.
+The quality of computations in Ref.[118] was tested for 3-dimensiona l har-
+monic oscillator (by adjusting dimensionality of this oscillator it can be t hought
+of as ”closed string model” representing both the shell model for a tomic nu-
+cleus and the gluonic ring for the Y-M fields) for which εi= (i+ 3/2) (in
+the system of units in which /planckover2pi1ω= 1) and Ω i= (i+ 1)(i+ 2)/2.For this 3-
+dimensional oscillator correctionsto the collectivized energy, Eq.(5 .64), become
+negligible already for |g| ≥0.2,provided that L≥8.Obtained results allow us
+to close this section at this point. These results are of no help in solvin g the
+gap dilemma though. This task is accomplished in the next section.
+436 Resolution of the gap dilemma
+6.1 Motivation
+In the previous section we provided evidence linking the gap problem f or Y-M
+fields with the problem about the excitation spectrum of the repulsiv e Bose gas.
+The gap equation, Eq.(5.59), is also used in nuclear physics where it is k nown
+to produce the same value for the gap for both signs of the coupling constantg.
+Since both options are realizable in Nature in the case of nuclear phys ics, the
+question arises about such possibility in the present case. In the ca se of nuclear
+physicsexperimentalrealization(giantnucleardipoleresonance)o fboth options
+for the coupling constant is experimentally testable. Thus, in the pr esent case
+we have to find some alternative physical evidence. If, indeed, suc h evidence
+could be found, this would allow us to bring back into play the well studie d F-S
+model which microscopically is essentially equivalent to the XXX 1d Heise nberg
+ferromagnetas results of Appendix B and subsections 3.5 and 5.2 ind icate. The
+next subsection supplies us with the alternative physical evidence.
+6.2 Some facts about harmonic maps and their uses in
+general relativity
+Suppose we are interested in a map from m−dimensional Riemannian manifold
+Mwith coordinates xaand metric γab(x) ton-dimensional Riemannian man-
+ifoldNwith coordinates ϕAand metric GAB(ϕ). A map M → N is called
+harmonic ifϕA(xa) satisfies the Euler-Lagrange (E-L) equations originating
+from minimization of the following Lagrangian
+L=√γGAB(ϕ)γab(x)ϕA
+,aϕB
+,b (6.1)
+in whichγ= det(γab).Since such defined Lagrangian is part of the La-
+grangian given by Eq.(3.6), the E-L equations for Eq.(6.1), in fact, c oincide
+with Eq.s(3.10). In the most general form they can be written as [38 ]30
+ϕA;a
+,a+ΓA
+BCϕB
+,aϕC,a= 0. (6.2)
+In such a form we can look at transformations ϕA′=ϕA′(ϕB) keeping Lform-
+invariant. To find such transformations, following Neugebauer and Kramer [38],
+we introduce the auxiliary Riemannian space defined by the metric
+dS2=GAB(ϕ)dϕAdϕB. (6.3)
+Use of the above metric allows us to investigate the invariance of Lwith help of
+standardmethods of Riemannian geometry. In the present case, this means that
+one should study Killing’s equations in spaces with metric GAB.Specifically, let
+us consider the Lagrangian for source-free Einstein-Maxwell field s admitting at
+30We use the 1st edition of Ref. [38] for writing this equation. This means that we have to
+define ΓA
+BCas ΓA
+BC=1
+2Gad{∂
+∂ϕcGbd+∂
+∂ϕbGcd−∂
+∂ϕdGbc}.
+44least one non-null Killing vector ξ.To design such a Lagrangian we begin with
+the Ernst equation, Eq.(2.4), for pure gravity and replace the Ern st potential
+ǫ=−F+iω31by two complex potentials Eand Φ. Then, by symmetry, the
+equations for stationary Einstein-Maxwell fields can be written as f ollows [38]
+FE;a
+,a+γabE,a(E,b+2Φ,b¯Φ) = 0,FΦ;a
+,a+γabΦ,a(E,b+2Φ,b¯Φ) = 0.(6.4)
+These equations are obtained by minimization of the Lagrangian
+L=√γ[ˆRab+2F−1γabΦ,aΦ,b+1
+2F−2γab(E,a+2¯ΦΦ,a)(E,b+2¯ΦΦ,b)],(6.5)
+i.e. from equationsδL
+δγab= 0,δL
+δΦ= 0 andδL
+δE= 0.Taking these results into
+account, the auxiliary metric, Eq.(6.3), can now be written as
+dS2= 2F−1dΦd¯Φ+1
+2F−2/vextendsingle/vextendsingledE+2¯ΦdΦ/vextendsingle/vextendsingle2. (6.6)
+The analysis done by Neugebauer and Kramer [38] shows that there are eight
+independent Killing vectors leading to the following finite transformat ions :
+E′=α¯αE, Φ′=αΦ;
+E′=E+ib, Φ′= Φ;
+E′=E(1+icE)−1, Φ′= (1+icE)−1;
+E′=E −2¯βΦ−β¯β, Φ′= Φ+β;
+E′=E(1−2¯γΦ−γ¯γE)−1,Φ′= (Φ+γE)(1−2¯γΦ−γ¯γE)−1.(6.7)
+Complex parameters α,β,γas well as real parameters bandcare connected
+with these eight symmetries. Evidently, solutions E′,Φ′are also solutions of
+Eq.s(6.4), provided that γabstays the same. Therefore if, say, we choose some
+vacuumsolutionasa”seed”,wewouldobtain, say,theelectrovacu umsolutionin
+accord with Appendix A. Incidentally, the electrovacuum solutions o btained by
+Bonnor (Appendix A) cannot be obtained with help of transformatio ns given by
+Eq.s(6.7). They areconsideredseparatelybelow. These observat ionsallowus to
+reduce the Lagrangian Lto the absolute minimum without loss of information.
+In 1973 Kinnersley [38] found that the group of symmetry transfo rmations for
+theEinstein-Maxwellequationswithnonnull Killingvectoristhe group SU(2,1)
+which has eight independent generators. In view of the above ment ioned reduc-
+tion ofLit is sufficient to replace the metric in Eq.(6.6) by a collection of much
+simpler metric related to each other by transformations Eq.(6.7). A ll the possi-
+bilities are described in the Table 34.1 of Ref.[38]. For our needs we focu s only
+on three of these (much simpler/reduced) metric listed in this table. These are
+dS2=2dξd¯ξ
+(1−ξ¯ξ)2,E=1−ξ
+1+ξ, (6.8)
+dS2=2dΦd¯Φ
+(1−Φ¯Φ)2, (6.9)
+31Recall, that −F=Vaccording to notations introduced in connection with Eq.(2 .4).
+45and
+dS2=−2dΦd¯Φ
+(1+Φ¯Φ)2. (6.10)
+The first and the second of these metric correspond to the vacuu m state,
+respectively, with Φ = 0 and E=−1, of pure gravity associated with the
+subgroup SU(1,1) of SU(2,1). The third metric, Eq.(6.10), corresp onds to a
+subgroup SU(2). It is related to the electrostatic fields ( E= 1) such that the
+space-time becomes asymptotically flat for E→0.It is important that the metric,
+Eq.(6.10), is related to the vacuum metric, Eq.s(6.8),(6.9), via trans formations
+either listed in Eq.(6.7) or related to these transformations. In par ticular, the
+related transformations can be obtained as follows. Using Ref.[38], it is conve-
+nient to make the parameters bandcin Eq.s(6.7) complex and to consider all
+eight complex parameters as independent of their complex conjuga tes. Under
+suchconditionsthemetricgivenbyEq.(6.10) isrelatedtothatgivenb yEq.(6.8)
+by the simplest complex transformation: Φ′=iξand¯Φ′=i¯ξ. These transfor-
+mations indicate that, starting with real vacuum solution for pure g ravity as a
+seed, the above transformations are capable of reproducing som e electrovacuum
+solutions. Additional details are discussed below.
+These results can be interpreted as follows. While the Ernst functio nal,
+Eq.(3.18), is representing pure axially symmetric gravity, the F-S-t ype func-
+tional, Eq.(3.19), should describe some special case of electrovacu um (Maxwell-
+Einstein) gravity. In view of results of Appendix C, it is possible to use these
+transformations in reverse (see below), that is to obtain the resu lts for pure
+gravity from those for electrovacuum. This peculiar ”duality” prop erty of grav-
+itational fields provides physically motivated resolution of the gap dile mma and,
+in addition, it allows us to obtain many new results.
+6.3 Resolutionof thegap dilemma and SU(3) ×SU(2)×U(1)
+symmetry of the Standard Model
+The original F-S-type model thus far is limited only to SU(2) gauge th eory.
+SU(2) gauge theory is known to be used for description of electrow eak interac-
+tions where, in fact, one has to use the gauge group SU(2) ×SU(1) [19 ]. The
+hadron physics (that is QCD) requires us to use the gauge group SU (3). This
+is caused by the fact that quark model of hadrons uses flavors (e .g. u,d,s,c,b, t)
+labeling quarks of different masses. Each of these quarks can be in t hree differ-
+ent colors (r,g,b) standing for ”red”, ”green” and ”blue”. Presen ce of different
+colors leads to fractional charges for quarks. Far from the targ et the scattering
+products are always colorless. The gauge group SU(3) is used for d escription of
+these colors. Although theoretically the number of colors can be gr eater than
+three, this number is strictly three experimentally [19]. The results o f this work
+allow us to reproduce this number of colors. For this purpose we hav e to be
+able to provide the answer to the following fundamental question :
+46Can equivalence between gravity and Y-M fields (for SU(2)
+gauge group) discovered by Louis Witten be extended to the gr oup
+SU(3)?
+Very fortunately, this can be done! For the sake of space, we sha ll be brief
+whenever details can be found in literature, e.g. see Refs.[120-122].
+To proceed, first, we have to go back to Eq.s(2.14),(2.15) and to mo dify
+these equations in such a way that instead of the Ernst Eq.(2.4) for the vacuum
+(gravity) field we should be able to obtain Eq.s (6.4) for electrovacuu m. In
+the limit Φ = 0 the obtained set of equations should be reducible to Eq.(2 .4).
+As it was noticed by G¨ urses and Xanthopoulos [120], in general, this t ask can-
+not be accomplished. Indeed, these authors demonstrated that the self-duality
+Eq.s(2.14) for SU(2) and for SU(3) Lie groups look exactly the same for axi-
+ally symmetric fields. Nevertheless, in the last case, upon explicit com putation
+instead of the vacuum Ernst Eq.(2.4) one gets an electovacuum equ ations (e.g.
+see Eq.s(6.4)) which, following Ernst [43], can be explicitly written as
+/parenleftBig
+ReE+|Φ|2/parenrightBig
+∇2E= (∇E+2¯Φ∇Φ)·∇E, (6.11a)
+/parenleftBig
+ReE+|Φ|2/parenrightBig
+∇2Φ = (∇E+2¯Φ∇Φ)·∇Φ. (6.11b)
+These equations are obtained if, instead of the matrix Mgiven by Eq.(2.15),
+one uses
+M=f−1
+1√
+2Φ −i
+2(E −¯E −2Φ¯Φ)√
+2¯Φ −i
+2(E+¯E −2Φ¯Φ) −i√
+2¯ΦE
+i
+2(¯E −E −2Φ¯Φ)i√
+2¯EΦ E¯E
+(6.12)
+in which, instead of the one complex potential ǫ=−F+iωused for solution of
+the vacuum Ernst Eq.(2.4), two complex potentials Eand Φ are being used. In
+this expression the overbars denote the complex conjugation and f=−1
+2(ǫ+
+¯ǫ+ 2Φ¯Φ).Since the Einstein-Maxwell Eq.s(6.4) (or (6.11)) are invariant with
+respect to transformations given by Eq.s(6.7), there should be a m atrixAwith
+constant coefficients such that the M′=AMA†will have primed potentials E
+and Φ takenfrom those listed in the set Eq.(6.7). Authors of [120]fo und explicit
+form of such A-matrices. However, when instead of matrix Mwe substitute the
+matrixM′into self-duality Eq.s(2.14), the combination M′−1∂M′looses this
+information. As result, we are left with the following situation: while on the
+gravity side the matrix M′=AMA†does allow us to obtain new and physically
+meaningful solutions from the old ones, on the Y-M side all this inform ation
+is lost. Thus, the one-to-one correspondence discovered by L.Wit ten for SU(2)
+isapparently lost for SU(3). Very fortunately, this happens only apparently!
+This is so because the Neugebauer- Kramer (N-K) transformation s described
+by Eq.s(6.7) do not exhaust all possible transformations which can b e applied
+to the matrix M, Eq.(6.12). Among those which are not accounted by N-
+K transformations are those by Bonnor [38 ,123] whose work is mentioned in
+Appendix A. These are given by
+E=ǫ¯ǫ;Φ =1
+2(ǫ−¯ǫ) =iω, (6.13)
+47whereǫ=−F+iωis solution of the Ernst Eq.(2.4). In view of the results of
+Appendix A one can be sure that the potentials Eand Φ satisfy Eq.s(6.11).
+This means that one can use these (Bonnor’s) potentials in the matr ixMto
+reproduceEq.s(6.11). Thistime, thereisone-toonecorresponde ncebetweenthe
+self-duality Y-M and the Einstein-Maxwell equations. Even though t his is true,
+the question immediately arises about relevance of such solutions to the solution
+of the gap problem discussed in Section 5. In Section 5 the Ernst Eq.( 2.4) was
+used essentially for this purpose while Eq.s(6.11) are seemingly differe nt from
+Eq.(2.4). Again, fortunately, the difference is only apparent.
+From the definition of Bonnor transformations, Eq.(6.13), it follows that
+the potential Eis real. Also, from the same definition it follows that |Φ|2=
+ω2.Introduce now new potential Z=E+ω2. For it, we obtain
+∇Z=∇(E+ω2) =∇E+2ω∇ω=∇E+2¯Φ∇Φ. (6.14)
+Using this result, Eq.s(6.11) can be rewritten as follows
+/parenleftbig
+Z∇2−∇Z·∇/parenrightbig/parenleftbigg
+E
+ω/parenrightbigg
+= 0. (6.15)
+Furthermore, consider the related equation
+/parenleftbig
+Z∇2−∇Z·∇/parenrightbig
+ω2= 0. (6.16)
+Evidently, if it can be solved, then equation/parenleftbig
+Z∇2−∇Z·∇/parenrightbig
+ω= 0 can be
+solved as well. This being the case, the system of Eq.s(6.15) will be solv ed if
+the Ernst-type vacuum equation
+Z∇2=∇Z·∇Z (6.17)
+of the same type as Eq.(2.4) is solved. The obtained result is opposite to that
+derived by Bonnor, described in Appendix A (see also works Hauser a nd Ernst
+[124] and by Ivanov [125]). This means that, at least in some cases (h aving
+physical significance) the self-dual Y-M fields for both SU(2) and S U(3) gauge
+groups are obtainable as solutions of the Ernst Eq.(2.4). This means that all
+results of Section 5 obtained for SU(2) go through for the gauge g roup SU(3).
+With these results at our disposal we would like to discuss their applica tions
+to the Standard Model [19 ,126]. From Ref.[120] it is known that the matrix
+M∈SU(3) has subgroups which belong to SU(2). In particular, one of s uch
+subgroups is obtained if we let Φ = 0 in Eq.(6.12). Then, in view of Eq.(6.17 ),
+it is permissible to replace Ebyǫof Eq.(2.4). Thus, the obtained matrix Mis
+decomposable as M=M1+M2,where the matrix M1is given by
+M1=f−1
+1 0ω
+0 0 0
+ω0ǫ¯ǫ
+. (6.18)
+48in agreement with the matrix Mdefined by Eq.(2.15) since in this case f=
+−1
+2(ǫ+¯ǫ) =F.At the same time, the matrix M2is given by
+M2=
+0 0 0
+0 1 0
+0 0 0
+. (6.19)
+Using elementary operations with matrices we can represent matrix Min the
+form
+˜M=
+0 0 1
+a b0
+b c0
+ (6.20)
+wherea= 1/F,b=ω/Fandc= (F2+ω2)/F.Such a form of the matrix
+˜Mis typical for the semidirect product of groups (when group elemen ts are
+represented by matrices). In general case one should replace ˜Mby
+˜M=
+0 0 1
+a b α 1
+b c α 2
+
+Since the 2 ×2 submatrix belongs to SU(2) (because its determinant is 1) nor-
+mally describing a rotation in 3d space (in view of SU(2) ⇄SO(3) correspon-
+dence), the parameters α1andα2are responsible for translation. In this, more
+general case, the matrix Mdescribes the Galilean transformations, that is a
+combination of translations and rotations. If the translational mo tion is one
+dimensional it can be compactified to a circle in which case we obtain the cen-
+tralizerofSU(3) as SU(2) ×U(1). At the level ofLie algebrasu(3) this result was
+obtained in Ref.[127], pages 232 and 267. Its physical interpretatio n discussed
+in this reference is essentially the same as ours. The obtained centr alizer is
+the symmetry group of the Weinberg-Salam model (part of the sta ndard model
+describing electroweak interactions).
+All these arguments were meant only to demonstrate that the F-S -type
+model, Eq.(3.19), should be used for description of electroweak inte ractions.
+For description of strong interactions, in accord with Ref.[120], we c laim that
+the matrix Mgiven by Eq.(6.12) in which Eand Φ are taken from Bonnor’s
+Eq.s(6.13) is intrinsically of SU(3) type. That is, it cannot be obtained from
+the matrix M(in which Φ = 0) by applications of the N-K transformations,
+i.e. there are no transformations of the type M′(Φ) =AM(Φ = 0)A†.There-
+fore, this type of SU(3) matrix should be associated with QCD part o f the SM.
+Hence we have to use the Ernst functional, Eq.(3.18), instead of th e F-S-type,
+Eq.(3.19). These results provide resolution of the gap dilemma. Evidently, this
+resolution is equivalent to the statement that the symmetry group of the SM is
+SU(3)×SU(2)×U(1). This result should be taken into account in designing all
+possible grand unified theories (GUT). In the next subsection we sh all discuss
+the rigidity of this result.
+496.3.1 Remarkable rigidity of symmetries of the Standard Mod el and
+the extended Ricci flow
+In addition to Bonnor’s transformations there are many other tra nsformations
+from vacuum to electrovacuum. In particular, in Appendix A we ment ioned
+transformations discovered by Herlt. By looking at Eq.s(A.5)-(A.7) describing
+these transformations and comparing them with those by Bonnor, Eq.(6.13),
+it is an easy exercise to check that all arguments leading from Eq.s(6 .11) to
+(6.17) go through unchanged. By using superposition of N-K trans formations
+and those either by Bonnor or by Herlt it is possible to generate a cou ntable
+infinity of vacuum-to electrovacuum transformations such that t hey could be
+brought back to the vacuum Ernst solution, Eq.(6.17), using result s of previous
+subsection. This property of Einstein and Einstein -Maxwell equatio ns we shall
+call ”rigidity”. In view of results of previous subsection, this rigidity explains
+the remarkableempirical rigidity of symmetries of the SM. Indeed, s uppose that
+the color subgroup SU(3) can be replaced by SU(N), N >3. In such a case it
+is appropriate again to pose a question : Can self-dual Y-M fields-gr avity cor-
+respondence discovered by L.Witten for SU(2) be extended for SU (N), N>3?
+In Ref.[128] G¨ urses demonstrated that, indeed, this is possible bu t under non-
+physical conditions. Indeed, this correspondence requires for S U(n+1) self-dual
+Y-M fields to be in correspondence with the set of n-1 Einstein-Maxw ell fields.
+Sincen= 1 andn= 2 cases have been already described, we need only to
+worry about n >2. In such a case we shall have many-to-one correspondence
+between the replicas of electrovacuum and vacuum Einstein fields wh ich, while
+permissible mathematically, is not permissible physically since the Bonno r-type
+transformations require one-to-one correspondence between the vacuum and
+electrovacuum fields. Herrera-Aguillar and Kechkin, Ref.[129], foun d a way of
+transforming the compactified fields of heterotic string (e.g. see E q.(3.12)) into
+Einstein–multi-Maxwellfields of exactly the same type as discussed in the paper
+by G¨ urses [128]. While in the paper by G¨ urses these replicas of Maxw ell’s fields
+needed to be postulated, in [129] their stringy origin was found explic itly. From
+here, it follows that results obtained in this subsection make the minim al func-
+tional, Eq.(3.8), and the associated with it Perelman-like functional, E q.(3.13),
+universal. The universality of the associated with it Ricci flow, Eq.s (3 .14),
+has physical significance to be discussed below.
+7 Discussion
+7.1 Connections with loop quantum gravity
+A large portion of this paper was spent on justification, extension a nd exploita-
+tion of the remarkable correspondence between gravity and self- dual Y-M fields
+noticed by Louis Witten. Such correspondence is achievable only non per-
+turbatively. In a different form it was emphasized in the paper by Mas on and
+Newman [130] inspired by work by Ashtekar, Jacobson and Smolin [131 ]. It is
+not too difficult to notice that, in fact, papers [130,131] are compat ible with
+50Witten’s result since reobtaining of Nahm’s equations in the context o f grav-
+ity is the main result of Ref.[131]. In this context the Nahm equations a re
+just equations for moving triad on some 3-manifold. Since the conne ction of
+Nam’s equations with monopoles can be found in Ref.[68] and with instan tons
+in Ref.[132] the link with Witten’s results can be established, in principle. Since
+the authors of [131] are the main proponents of loop quantum grav ity (LQG)
+such refinements might be helpful for developments in the field of LQ G. We
+shall continue our discussion of LQG in the next subsection.
+7.2 Topology changing processes, the extended Ricci flow
+and the Higgs boson
+According to the existing opinion the SM does not account for effect s of grav-
+ity. At the same time, in the Introduction we mentioned that in recen t works
+by Smolin and collaborators [32-34] it was shown that ”topological fe atures of
+certain quantum gravity theories32can be interpreted as particles, matching
+known fermions and bosons of the first generation in the Standard Model”.
+Similar results were also independently obtained in works by Finkelstein , e.g.
+see Ref.[133] and references therein. In particular, Finkelstein re cognized that
+all quantum numbers describing basic building blocks(=particles) oft he SM can
+be neatly organized with help of numbers used for description of kno ts. More
+precisely, with projections of these knots onto some plane. It hap pens, that for
+description of all particles of the electroweak portion of the SM the numbers
+describing trefoil knot are sufficient. The task of topological/knot ty description
+of the entire SM was accomplished to some extent in Ref.[33]. This refe rence
+as well as Ref.s[32-34] in addition are capable of describing particle dy nam-
+ics/transformations. All these works share one common feature : calculations
+do notrequire Higgs boson. This fact is consistent with results discussed in
+subsection 4.3.1.
+The question arises: Is this feature a serious deficiency of these t opological
+methods or are these methods so superior to other, that the Higg s boson should
+be looked upon as an artifact of the previously existing perturbativ e methods
+used in SM calculations? To answer this self-imposed question require s several
+steps.
+First, we recall that according to the existing opinion the SM does no t
+account for effects of gravity. In such a case all the above result s should have
+nothing in common with the SM which is not true.
+Second, the results obtained in this paper indicate that knots/links /braids
+mentioned above have not only virtual (combinatorial/topological) b ut also
+differential-geometric description (Appendix B). Because of this, t opological
+description should be looked upon as complementary to that obtaina ble with
+help of the F-S-type models.
+Third, it is known that knot/link- describing Faddeev model can be co n-
+verted into Skyrme model [134]. It is also known that the Skyrme-ty pe models
+32That is LQG.
+51do not account for quarks explicitly , Ref.[68], page 349. This is not a serious
+drawback as we shall explain momentarily.
+Fourth, much more important for us is the fact that the Skyrme mo del can
+be used both in nuclear [135] and high energy [136 ]physics where it is used for
+description of both QCD ( nicely describing the entire known hadron spectra )
+and electroweak interactions.
+To account for quarks one has to go back to the Faddeev-type mo dels capa-
+ble of describing knots/links and to make a connection between thes ephysical
+knots/links and topological/combinatorial knots/ links discussed in Refs[32-
+34,133]. This is still insufficient! It is insufficient because Floer’s Eq.(4.7) co n-
+nects different vacua each is being described by the zero curvatur e condition
+Eq.(4.13). It is always possible to look at such a condition as describing some
+knot/link differential geometrically. With each knot, say in S3,some 3-manifold
+is associated. Furthermore such a manifold should be hyperbolic (su bsection
+3.6), that is either associated with hyperbolic-type knot/links [20,13 7] in S3
+or with knots/links ”living” in hyperboloid embedded in the Minkowski sp ace-
+time. Such a restriction is absent in Ref.s[32-34,133]. At the same time the Y-M
+functional, Eq.(4.12), is defined for a particular 3-manifold whose co nstruction
+is quite sophisticated. Eq.(4.7) describes processes of topology ch ange by con-
+necting different vacua. Such changes formally are not compatible w ith the fact
+that we are dealing with one and the same 3-manifold M ×[0,1]. From the math-
+ematical standpoint [11] no harm is made if one considers just this 3- manifold,
+e.g. read Ref.[11], page 22, bottom. Since particle dynamics is encode d in
+dynamics of transformations between knots/links, it causes us to consider tran-
+sitions between different 3-manifolds. These 3-manifolds should be c arefully
+glued together as described in Ref.[11]. In this picture particle dynam ics involv-
+ing particle scattering/transformation is synonymous with proces ses involving
+topology change. These are carried out naturally by instantons. S uch processes
+can be equivalently and more physically described in terms of the prop erties of
+the (extended) Ricci flow (subsection 3.4) following ideas of Perelma n’s proof of
+the Poincare′conjecture. Indeed, experimentally there is only finite number of
+stable particles. Without an exception, the end products of all sca ttering pro-
+cesses involve only stable particles. This observation matches perf ectly with the
+irreversibility of Ricci flow processes involving changes in topology: f rom more
+complex-to less complex 3-manifolds. Such Ricci flow model upon dev elopment
+could provide mathematical justification to otherwise rather vagu e statements
+by Finkelstein that ”more complicated knots ( particles) can theref ore dynami-
+cally decay to trefoils (stable particles)”, Ref.[133], page 10, botto m.
+7.3 Elementary particles as black holes
+In the paper [138] by Reina and Treves and also in [139] by Ernst it was found
+that for asymptotically flat Einstein-Maxwell fields generated from the vacuum
+fields by means of transformations of the type described above, in Section 6,
+the gyromagnetic factor g= 2. For the sake of space, we refer our readers to a
+recent review by Pfister and King [140] for definitions of gand many historical
+52facts and developments. In [140] it was noticed that such value of gis typical
+for most of stable particles of the SM. In view of the quantum gravit y-Y-M
+correspondence promoted in this paper, the interpretation of ele mentary par-
+ticles as black holes makes sense, especially in view of the following exce rpt
+from Ref.[38], page 526, ”There is one-to-one correspondence be tween station-
+ary vacuum fields with sources characterized by masses and angula r momenta
+and stationary Einstein-Maxwell fields with purely electromagnetic s ources, i.e.
+charges and currents.”
+Appendix A
+Peculiar interrelationship between gravitational, elect romagnetic
+and other fields
+Unification of gravity and electromagnetism was initiated by Nordstr ¨ om in
+1913- before general relativity was formulated by Einstein. Almost immediately
+after Einstein’s formulation, Kaluza, in 1921, and Klein, in 1926, prop osed uni-
+fication of electromagnetism and gravityby embedding Einstein’s 4-d imensional
+theory into 5 dimensional space in which 5th dimension is a circle. These results
+and their generalizations (up to 1987) can be found in the collection o f papers
+compiled by Applequist, Chodos and Freund [141]. Regrettably, this c ollection
+does not contain alternative theories of unification. Since such alte rnative theo-
+ries are much less known/popular to/with string and gravity theore ticians, here
+we provide a brief representative sketch of these alternative the ories.
+The 1st unified Einstein-Maxwell theory in 4-dimenssional space-tim e was
+proposed and solved by Rainich in 1925. It was discussed in great det ail by
+Misner and Wheeler [142]. After Rainich there appeared many other w orks on
+exact solutions of Einstein-Maxwell fields [38]. The most striking outc ome of
+these, more recent, works is the fact that multitude of exact solu tions of the
+combined Einstein-Maxwell equations can be obtained from solutions of the
+vacuumEinstein equations.
+In 1961 Bonnor [123]obtained the following remarkable result (e.g. re ad his
+Theorem 1). Suppose solutions of the vacuum Einstein equations ar e known.
+Using these solutions, it is possible to obtain a certain class of solution s of
+Einstein-Maxwell equations.
+In Section 6 we obtained the reverse result: Einstein’s solutions for pure
+gravity were obtained from solutions of the Einstein-Maxwell equat ions. With-
+out doing extra work, the electrovacuum solution obtained by Bonn or can be
+converted into that describing propagation of the combined cylindr ical gravita-
+tional and electromagnetic waves. With some additional efforts one can use the
+obtained results as an input for results describing the combined gra vitational,
+electromagnetic and neutrino wave propagation [143-144 ].
+The results by Bonnor comprise only a small portion of results conne cting
+static gravity fields with electromagnetic and neutrino fields. The ne xt example
+belongs to Herlt [38 ,145]. It provides a flavor of how this could be achieved.
+53We begin with Eq.(2.5). When written explicitly, this equation reads
+/parenleftbigg
+∂2
+ρ+1
+ρ∂ρ+∂2
+z/parenrightbigg
+u= 0. (A.1)
+Thistypeofsolutionistheresultofuseofthematrix M,Eq.(2.15),inEq.(2.14b).
+Nakamura [146] demonstrated that there is another matrix Qgiven by
+Q=/parenleftbiggf fω
+fω f2ω2−ρ2f−1/parenrightbigg
+(A.2)
+and the associated with it analog of Eq.(2.14b)
+∂ρ(ρ∂ρQ·Q−1)+∂z(ρ∂zQ·Q−1) = 0 (A.3)
+leading to the equation analogous to Eq.(A.1), that is
+/parenleftbigg
+∂2
+ρ−1
+ρ∂ρ+∂2
+z/parenrightbigg
+˜u= 0. (A.4)
+Nakamura demonstrated that the solution ˜ uis obtainable from solution of
+Eq.(A.1). and vice versa. Thus, instead of the Ernst Eq.(2.4) we can use
+Eq.(A.4). This fact plays crucial role in Hertl’s work. In it, he uses Eq.( A.4)
+to obtainuin Eq.(A.1) as follows
+exp(2u) =/parenleftbig
+˜u−1+G/parenrightbig2(A.5)
+withGgiven by
+G= ˜u,ρ[ρ(u2
+,ρ+u2
+,z)−˜u˜u,ρ]−1. (A.6)
+These results allow him to introduce a potential χvia
+χ= ˜u−1−G. (A.7)
+Using the original work of Ernst [43] as well as Ref.[38], we find that so lution
+of the static axially symmetric coupled Einstein-Maxwell equations is g iven in
+terms of complex potentials ǫand Φ.In particular, in purely electrostatic case
+one hasǫ=¯ǫ=e2u−χand Φ = ¯Φ =χwhile the magnetostatic case is obtained
+from the electrostatic by requiring -Φ = ¯Φ =ψandǫ=¯ǫ=e2u−ψ. In this
+caseψis just relabeled χ.Ref.[38] contains many other examples of the cou-
+pledEinstein-Maxwellequationsobtainedfromthevacuum solutions ofEinstein
+equations.
+The aboveresultsshouldbe lookedupon fromthe standpoint offun damental
+problemoftheenergy-momentumconservationingeneralrelativit yrequiringin-
+troduction(inthesimplestcase)oftheLandau-Lifshitz(L-L)ene rgy-momentum
+pseudotensor. The description of more complicated pseudotenso rs (incorporat-
+ing that by L-L) can be found in the monograph by Ortin [147]. To this o ne
+should add the problem about the positivity of mass in general relativ ity. The
+difficulties withthese conceptsstem fromtheverybasicobservatio n, lyingatthe
+54heart of general relativity, that at any given point of space-time g ravity field
+can be eliminated by moving in the appropriately chosen accelerating f rame
+(the equivalence principle). This fact leaves unexplained the origin of the tidal
+forces requiring observation of motion of at least two test particle s separated
+by some nonzero distance. The explanation of this phenomenon with in general
+relativity framework is nontrivial.It can be found in [148]. In turn, it lea ds to
+speculations about the limiting procedure leading to elimination of grav ity at a
+given point33. Apparently, this problem is still not solved rigorously[147]. An
+outstandig collection of rigorous results on general relativity can b e found in
+the recent monograph by Choquet-Bruhat [149] while [150] discuss es peculiar
+relationship between the Newtonian and Einsteinian gravities at the s cale of
+our Solar system.
+Conversely, one can think of other fields at the point/domain where gravity
+is absent as subtle manifestations of gravity. Interestingly enoug h, such an idea
+was originally put forward by Rainich already in 1925 ! Recent status o f these
+ideas is given in paper by Ivanov [125]. From such a standpoint, the fu nctional
+given by Eq.(3.13) (that is the Perelman-like entropy functional) is su fficient for
+description of all fields with integer spin. With minor modifications (e.g. in-
+volving either the Newman-Penrose formalism [143,144] or supersym meric for-
+malism used in calculation of Seiberg-Witten invariants [66]), it can be us ed for
+description of all known fields in nature.
+Appendix B
+Some facts about integrable dynamics of knotted vortex filam ents
+B.1Connection with the Landau-Lifshitz equation
+Following Ref.[85], we discuss motion of a vortex filament in the incompre ss-
+ible fluid. Some historical facts relating this problem to string theory are given
+in our recent work, Ref.[84]. Let ube a velocity field in the fluid such that
+divu= 0. Therefore, we can write u=∇×A. Next, we define the vorticity
+w=∇×uso that eventually,
+u=−1
+4π/integraldisplay
+d3x(x−x′)×w(x′)
+/ba∇dblx−x′/ba∇dbl3. (B.1a)
+This expression can be simplified by assuming that there is a linevortex which
+is modelled by a tube with a cross-sectionalarea dAand such that the vorticity
+wis everywhere tangent to the line vortex and has a constant magnit ude w.
+Let then Γ =/integraltext
+wdAso that
+u=−Γ
+4π/contintegraldisplay(x−x′)×dγ
+/ba∇dblx−x′/ba∇dbl3(B.1b)
+withdγbeing an infinitesimal line segment along the vortex. Such a model
+of a vortex resembles very much model used for description of dyn amics of
+ring polymers [84]. Because of this, it is convenient to make the followin g
+33The abundance of available energy-momentum pseudotensors is result of these specula-
+tions.
+55identification : u(γ(s,t)) =∂γ
+∂t(s,t), withsbeing a position along the vortex
+contour and t-time. This allows us to write
+∂γ
+∂t(s′,t) =−Γ
+4π/contintegraldisplay(γ(s′,t)−γ(s,t))
+/ba∇dblγ(s′,t)−γ(s,t)/ba∇dbl3×∂γ
+∂sds (B.1c)
+and to make a Taylor series expansion in order to rewrite Eq.(B1c) as
+∂γ
+∂t=Γ
+4π[∂γ
+∂s′×∂2γ
+∂s′2/integraldisplayds
+|s−s′|+...]. (B.1d)
+In this expression only the leading order result is written explicitly. By intro-
+ducing a cut off εsuch that |s−s′| ≥εand by rescaling time: t→Γ
+4πtln(ε−1)
+one finally arrives at the basic vortex filament equation
+∂γ
+∂t=∂γ
+∂s′×∂2γ
+∂s′2. (B.2)
+Introduce now the Serret-Frenet frame made of vectors B,TandNso that
+B=T×N,ˇκN=dT
+ds,T=∂γ
+∂s,where ˇκis a curvature of γ. Then, Eq.(B.2) can
+be equivalently rewritten as
+∂γ
+∂t= ˇκB (B.3)
+or, as
+∂T
+∂t=T×Txx. (B.4)
+In the last equation the replacement s⇌xwas made so that the obtained
+equationcoincides with the Landau-Lifshitz (L-L) equationdescrib ing dynamics
+of 1d Heisenberg ferromagnets [86].
+B.2Hashimoto map and the Gross-Pitaevskii equation
+Hashimoto [85] found ingenious way to transform the L-L equation in to the
+nonlinear Scr¨ odinger equation (NLSE) which is also widely known in con densed
+matter physics literature as the Gross-Pitaevskii (G-P) equation [96]. Because
+of its is uses in nonlinear optics and in condensed matter physics for d escrip-
+tion of the Bose-Einstein condensation (BEC) theory of this equat ion is well
+developed. Some facts from this theory are discussed in the main te xt. Here we
+provide a sketch of how Hashimoto arrived at his result.
+LetT,UandVbe another triad such that
+U= cos(x/integraldisplay
+τds)N−sin(x/integraldisplay
+τds)B,V= sin(x/integraldisplay
+τds)N+cos(x/integraldisplay
+τds)B(B.5)
+in whichτis the torsion of the curve. Introduce new curvatures κ1andκ2in
+such a way that
+κ1= ˇκcos(x/integraldisplay
+τds) andκ2= ˇκsin(x/integraldisplay
+τds),
+56then, it can be shown that
+∂γ
+∂t=−κ2U+κ1V (B.6a)
+and
+∂2γ
+∂x2=κ1U+κ2V. (B.6b)
+Using these equations and taking into account that UtV=−UVtafter some
+algebra one obtains the following equation
+iψt+ψxx+[1
+2|ψ|2−A(t)]ψ= 0 (B.7)
+in whichψ=κ1+iκ2andA(t) is some arbitrary x-independent function. By
+replacingψwithψexp(−it/integraltextdt′A(t′)) in this equation we arrive at the canonical
+form of the NLSE which is also known as focussing cubic NLSE.
+iψt+ψxx+1
+2|ψ|2ψ= 0 (B.8)
+Itcanbeshownthatitssolutionallowsustorestoretheshapeofth ecurve/filament
+γ(s,t).The G-P equation can be identified with Eq.(B.7) if we make A(t) time-
+independent. In its canonical form it is written as (in the system of u nits in
+whichℏ= 1,m= 1/2) [86]
+iψt=−ψxx+2κ/parenleftBig
+|ψ|2−c2/parenrightBig
+ψ= 0. (B.9)
+Ingeneral,the signofthecouplingconstant κcanbebothpositiveandnegative.
+In view ofEq.(B.8), when motion of the vortexfilament takes place in E uclidean
+space, the sign of κis negative. This is important if one is interested in dynamic
+ofknottedvortexfilaments[85]. Forpurposesofthisworkitis alsoofinterestt o
+study motion of vortex filaments in the Minkowski and related (hype rbolic, de
+Sitter ) spaces. This should be done with some caution since the tran sition from
+Eq.(B.1a) to (B.2) is specific for Euclidean space. Thus, study can be made at
+the level of Eq.s (B.3) and (B.4). Fortunately, such study was perf ormed quite
+recently [94,95]. The summary of results obtained in these papers ca n be made
+with help of the following definitions. Introduce a vector n={n1,n2,n3}so
+that the unit sphere S2is defined by
+S2:n2
+1+n2
+2+n2
+3= 1. (B.10)
+Respectively, the de Sitter space S1,1(or unit pseudo sphere in R2,1) is defined
+by
+S1,1:n2
+1+n2
+2−n2
+3= 1, (B.11)
+while the hyperbolic space H2(or hyperboloid embedded in R2,1) is defined by
+H2:n2
+1+n2
+2−n2
+3=−1,n3>0. (B.12)
+57Using these definitions, it was proven in [94,95] that: a) for both de S itter and
+H2spaces there are analogs of the L-L equation ( e.g. those discusse d in the
+main text, in subsection 5.2.); b) the Hasimoto map can be extended f or these
+spacesso that the respective L-L equationsare transformed int o the same NLSE
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