1001.0029.txt

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+e2γ(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=−2κ2i(¯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
(or G-P) equation in which κispositive.
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