Диссертация (1137342), страница 6
Текст из файла (страница 6)
These algebras werefirst introduced by A. Zamolodchikov in [ZamW], and their study was substantiallydeveloped in [FZ] (for the first nontrivial W3 -case) and [FL] (for generic WN ). Otherdevelopments in the theory of W -algebras are discussed in the review [BS].The most condensed form of the commutation relations of W3 is given by theoperator product expansions (OPEs) of the energy-momentum tensor T (z) and theW -current W (z):2T z+wc2++ reg. ,T (z)T (w) =2(z − w)4(z − w)23W (w)∂W (w)T (z)W (w) =++ reg.
,(z − w)2z−w2T z+wc2W (z)W (w) =++3(z − w)6(z − w)4132z+w1 2z+w+Λ+ ∂ T+ reg.(z − w)2 22 + 5c2202where Λ(z) = (T T )(z) −(2.1)3 2∂ T (z).10The representation theory of this algebra is very similar to that of the Virasoroalgebra. In the generic case one has the Verma module with the highest vector |∆, wisuch that L0 |∆, wi = ∆|∆, wi, W0 |∆, wi = w|∆, wi. Hence the representation spaceis spanned by the vectorsL−m1 L−m2 . .
. L−mk W−n1 W−n2 . . . |∆, wi,m1 ≥ m2 ≥ . . . ≥ mk , n1 ≥ n2 ≥ . . . ≥ nk ,(2.2)while the set of the highest weight vectors themselves corresponds to primary fields(vertex operators) of the 2d CFT. As in the Virasoro case, these fields can be determined by their OPEs with higher-spin currents T (z) and W (z):162.2. Isomonodromic deformations and moduli spaces of flat connections∂φ(w)∆φ(w)+ reg.+2(z − w)z−wwφ(w)(W−1 φ)(w) (W−2 φ)(w)W (z)φ(w) =+ reg.++3(z − w)(z − w)2z−wT (z)φ(w) =(2.3)However, the W-descendants such as (W−1 φ) and (W−2 φ) are not defined in general(this is to be contrasted with the Virasoro case where one has e.g.
(L−1 φ)(w) =∂φ(w)), which means that the 3-point functions involving such fields are not reallydefined. As a consequence, one cannot express the matrix elementsh∆∞ , w∞ |φ(1)L−m1 L−m2 . . . L−mk W−n1 W−n2 . . . |∆0 , w0 iin terms of h∆∞ , w∞ |φ(1)|∆0 , w0 i only. It was shown in [BW] in an elegant way thatall such 3-point functions can be expressed in terms of an infinite number of unknownconstantsk(2.4)Ck = h∆∞ , w∞ |φ(1)W−1|∆0 , w0 i,k = 1, 2, .
. .The problem is that having this infinite number of constants (which for the 4-pointconformal block actually becomes doubly infinite) one can adjust them as to obtainany function as a result. In this chapter we show that the isomonodromic approachcan fix this ambiguity in such a way that all these parameters become functions onthe moduli space of the flat connections on the sphere with 3 punctures. In the sl3case this space is 2-dimensional (we denote the corresponding coordinates by µ andν), so all Ck = Ck (µ, ν).Note that for the WN algebra one would have the set of constants Ck1 ,...,kl withl = 21 (N − 1)(N − 2) non-negative indices (e.g., this easily follows from analysis of[BW]), which is half of the dimension of the moduli space of flat slN connections onthe 3-punctured sphere.The chapter is organized as follows.
In Section 2 we briefly discuss the origins ofthe Schlesinger system and the space of flat connections on the punctured Riemannsphere. Then we introduce a collection of convenient local coordinates on this space,which are related to pants decomposition of the sphere. In Section 3 an iterativealgorithm of the solution of the Schlesinger system is proposed. We then present a setof non-trivial properties of this solution, discovered experimentally, and put forwarda conjecture about isomonodromy-CFT correspondence in higher rank, which relatesWN conformal blocks to the isomonodromic tau function. In particular, for a collectionof known W3 conformal blocks we present the 3-point functions that can be used toconstruct the τ -function in the form of explicit expansion.
In Section 4 we describethe problems of definition of the general W3 conformal block and discuss how theycan be addressed using the global analytic structure induced by crossing symmetry.We conclude with a brief discussion of open questions.Isomonodromic deformations and moduli spaces offlat connectionsThe main object of our study will be the Fuchsian linear system172. Isomonodromic τ -functions and WN conformal blocksnX AνdΦ(z) =Φ(z) = A(z)Φ(z) ,dzz − zνν=1XAν = 0 .(2.5)νHere Aν are traceless matrices with distinct eigenvalues, Φ(z) is the matrix of Nindependent solutions of the system normalized as Φ(z0 ) = 1.
It is obvious that uponanalytic continuation of the solutions along a contour γν encircling zν they transforminto some linear combination of themselves:γν : Φ(z) 7→ Φ(z)Mν ,(2.6)where Mν ∈ GLN (C). The relation γn . . . γ1 = 1 from π1 (CP 1 \{z1 , . . . , zn }, z0 ) imposes the conditionM1 . . .
Mn = 1 .(2.7)The well-known Riemann-Hilbert problem is to find the correspondence{M1 , . . . , Mn } → {A1 , . . . , An } .(2.8)It is easy to see that the conjugacy classes of Mν areMν ∼ exp (2πiAν ) .(2.9)The eigenvalues of Aν determine the asymptotics of the fundamental matrix solutionnear the singularities, so one can fix even this asymptotics and study the correspondingrefined Riemann-Hilbert problem.
We will work only with traceless matrices Aν sincethe scalar part trivially decouples.Schlesinger systemSince it is difficult to solve the generic Riemann-Hilbert problem exactly, one can firstask a simpler question: how to deform simultaneously the positions of the singularitieszν and matrices Aν but preserve the monodromies Mν . The answer follows from theinfinitesimal gauge transformationAνΦ(z) ,Φ(z) 7→ 1 + z − zν(2.10)AνAνA(z) 7→ A(z) + −, A(z) ,(z − zν )2z − zνthat iszν 7→ zν + ,[Aν , Aµ ]Aµ6=ν 7→ Aµ + ,zν − zµX [Aν , Aµ ]Aν 7→ Aν − ,zν − zµµ6=ν18(2.11)2.2. Isomonodromic deformations and moduli spaces of flat connectionsleading to the Schlesinger system of non-linear equations∂Aµ[Aµ , Aν ]=,∂zνzµ − zνX [Aµ , Aν ]∂Aν=−.∂zνzµ − zνµ6=νNote that one can fix zn = ∞, then the corresponding matrix A∞ = −(2.12)n−1PAν will beν=1constant.
A non-trivial statement is that the relationsX tr Aµ Aν∂log τ =∂zµzµ − zνν6=µ(2.13)are compatible and define the τ -function τ (z1 , . . . , zn ) of the Schlesinger system. It iseasy to see that the 3-point τ -function is given by a simple expression:τ (z1 , z2 , z3 ) = const · (z1 − z2 )∆3 −∆1 −∆2 (z2 − z3 )∆1 −∆2 −∆3 (z1 − z3 )∆2 −∆1 −∆3 ,where ∆ν = 21 tr A2ν . Let us now attempt to solve the Schlesinger system for the 4-pointcase and compute the corresponding τ -function in the form of certain expansion.Moduli spaces of flat connectionsThe main object of our interest is the τ -function. It depends on monodromy datawhich provide the full set of integrals of motion for the Schlesinger system. It will beuseful to start by introducing a convenient parametrizaton of this space.One starts with n matrices Mν ∈ SLN , with fixed nondegenerate eigenvalues, i.e.there are n(N 2 −N ) parameters. These matrices are constrained by one equation (2.7)and are considered up to an overall SLN conjugation, which decreases the number ofparameters by 2(N 2 − 1).
So the resulting number of parameters isdim Msln N (θ 1 , . . . , θ n ) = (n − 2)N 2 − nN + 2 .(2.14)Here θ ν ∈ h (h is the Cartan subalgebra) define the conjugacy classes: Mν ∼ e2πiθν .It is obvious that θ ν is equivalent to θ ν + hν , such that for all weights of the firstfundamental representation ei one has (ei , hν ) ∈ Z. It means that hν ∈ ⊕ri=1 Zαi∨ ,where αi∨ ∈ h are simple coroots (for the simply-laced case they coincide with theroots).For the general Lie algebra this formula can be written asdim Mgn (θ 1 , .
. . , θ n ) = (n − 2) dim g − n · rank g .(2.15)In particular, for n = 3 punctures on the spheredim Mg3 (θ 1 , θ 2 , θ 3 ) = dim g − 3 · rank g .This formula gives the number of non-simple roots of g. In the slN case it specializesto(2.16)dim Msl3 N (θ 1 , θ 2 , θ 3 ) = (N − 1)(N − 2) .192. Isomonodromic τ -functions and WN conformal blocksThis expression vanishes for sl2 , which drastically simplifies the study of the corresponding isomonodromic problem.
However already for sl3 this dimension is equal to2, i.e. it is nonvanishing. One way to simplify the problem is to set θ 2 = ae1 : in thiscase the orbit of the adjoint actione2πiae1 7→ g −1 e2πiae1 ghas the dimension dim Oae1 = dim g − dim stab(e1 ) = N 2 − 1 − (N − 1)2 = 2N − 2.The total dimension is 2(N 2 − N ) + (2N − 2) − 2(N 2 − 1) = 0. In this calculationthe first two terms correspond to the dimensions of orbits: two generic and one witha large stabilizer.
The last term corresponds to one equation and one factorization.Hence(2.17)dim Msl3 N (θ 1 , ae1 , θ 3 ) = 0 .This case is the best known on the side of W -algebras [FLitv07], [FLitv09], [FLitv12].In the mathematical framework, this situation corresponds to rigid local systems.Pants decomposition of Mg4We begin our consideration with an arbitrary Lie group G containing a Cartan torusH ⊂ G.
The corresponding Lie algebras are g and h, respectively. At some point wewill switch to G = SLN (C) case.The moduli space Mg4 is described by 4 matrices satisfying M1 M2 M3 M4 = 1,defined up to conjugation:Mg4 = {(M1 , M2 , M3 , M4 )}/G .(2.18)Let us introduce S = M1 M2 and consider two triples{(M1 , M2 , S −1 ), (S, M3 , M4 )} .(2.19)Note that the products inside each of these triples are equal to the identity. Let usnow choose the submanifold with fixed eigenvalues of M1 , . . . , M4 , S. One may alsouse the freedom of the adjoint action to diagonalize SS = e2πiσ ,where σ ∈ h. We thereby obtain a submanifoldMg4 (θ 1 , θ 2 ; σ; θ 3 , θ 4 ) = {(M1 , M2 , e−2πiσ ), (e2πiσ , M3 , M4 )}/H ⊂ Mg4 (θ 1 , θ 2 , θ 3 , θ 4 ) ,(2.20)where the remaining factorization is performed over the Cartan torus H ⊂ G.
It isvery similar to what happens for Mg3 :Mg3 = {(M1 , M2 , M3 )}/G = {(M1 , M2 , e2πiθ3 )}/H ,(2.21)except that the conjugation is simultaneous for both triples. To relax this condition,let us define an extra Cartan torus acting on Mg4 :h : {(M1 , M2 , e−2πiσ ), (e2πiσ , M3 , M4 )} 7→ {(M1 , M2 , e−2πiσ ), h−1 (e2πiσ , M3 , M4 )h} ,(2.22)202.2. Isomonodromic deformations and moduli spaces of flat connectionswhich looks like a relative twist of one part of the sphere with respect to another (inthe sl2 case it will be exactly the geodesic flow).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
