Диссертация (Isomonodromic deformations and quantum field theory), страница 5

. , 2L, dΩ =dΩrα and other ingredients in the r.h.s. are givenαby (5.16), (5.20) and the period matrix of C.This theorem gives the solution for a Seiberg-Witten system.P• Formula 5.78: Q(r) =i 6=q jqαβi P1/lαα)rαi rβj log Θ∗ (A(qαi )−A(qβj ))− (rαi )2 lαi log d(z(q)−qh2∗ (q)iqαThis formula gives the “r-charge contribution” to the exact conformal block.PP11• Formula 5.87: G0 (q|a) = τB (q) exp 2 aI TIJ (q)aJ + aI UI (q, r) + 2 Q(r)IJIThis is the general formula for the conformal block of twist fields (generalizationof Zamolodchikov’s formula).• Theorem 6.2: The characters of the twisted representations are given by theformulas (6.85), (6.88), (6.95), (6.97).• Theorem 6.3:If g1 ∼ g2 in G for different g1 , g2 ∈ NG (h), then χg1 (q) =χg2 (q).This theorem generalizes the Gauss identity from Zamolodchikov’s construction.• Theorem 6.4: The conformal blocks (6.163) for generic W (o(2N )) twist fieldsare given byG0 (a, r, q) = τB (Σ|q)τB−1 (Σ̃|q)τSW (a, r, q)where∂qi log τB (Σ|q) =XRes tz (ξ)dξ,∂qi log τB (Σ̃|q) =π2N (ξ)=qiXRes t̃z (ζ)dζπN (ζ)=qii = 1, .

. . , 2Mand∂qi log τSW (a, r, q) =∂log τSW =∂aI14˛dS,Xπ2N (ξ)=qiRes(dS)2,dzAI ◦ BJ = δIJ ,BI12i = 1, . . . , 2MI, J = 1, . . . , g−iq=qα1.2. OutlineOrganization of the thesisAll the parts of this thesis are self-contained papers with their own introductions, sothey can be read independently. But nevertheless, there are some logical dependenciesbetween different parts. I show them on the following diagram:Chapter5Chapter6Chapter2Chapter3Chapter4Chapter 2 is devoted to the numerical solution of the Schlesinger system ofthe rank 3 and to the computation of corresponding isomonodromic tau-function.Its main task was to formulate and check the main conjectures for the higher-rankisomonodromy-CFT correspondence, which are then proved in the next chapters.Chapter 3 deals with the free-fermionic construction of monodromy fields.

Axiomatically such fields are defined by:1) it is a fermionic group-like element2) its two-particle matrix elements are expressed through the solution of 3-pointFuchsian system.Then we prove that such fields are W-primaries, and at the same time their correlationfunction can be given as some Fredholm determinant. In this way we give the freefermionic proof of the conjectures from Chapter 2. Also we get some integrablehierarchies which are related to such fields.Chapter 4 is written in a pure mathematical language and absolutely rigorously,so it does not require any field-theory background. In this chapter we develop theframework in which the Fredholm determinant formula can be proved rigorously.

Todo this first we cut the sphere with n punctures into n − 2 three-punctured sphere,and then introduce the spaces of functions on the obtained boundaries. Then weconstruct two projectors onto the space of functions that can be continued betweenthe different boundaries, PΣ and P⊕ . After that we restrict these projectors on someother space H+ : PΣ,+ , P⊕,+ . Thanks to this procedure they become non-degenerate.−1Then we define an infinite-dimensional determinant τ = det PΣ,+P⊕,+ . Next we provethat:1) derivatives of this determinant coincide with the derivatives of isomonodromictau-function,2) it is a Fredholm determinant, whose kernel in the 4-point case reproduces theone obtained in Chapter 3,131. Introduction3) the minor expansion of the determinant reproduces a combinatorial formulawhich in the known cases can be obtained from the isomonodromy-CFT + AGTcorrespondences. In this sense it gives one more independent proof of AGT forc = N.Chapter 5 is devoted to the study of W-twist fields.

The main techniques hereare the free-field conformal theory and the algebraic geometry of complex curves.From the field theory we get equations that are satisfied by the correlation functionsof twist fields, and then solve them in terms of period matrices, Abel maps and thetheta-function of some branched cover of the punctured sphere. This constructiongeneralizes the conformal block of Zamolodchikov.Chapter 6 is also devoted to W-twist fields, but from the algebraic point of view.Here we consider the W-algebras for the orthogonal series, too. We start from the freefermionic definition of W-algebras, their vertex operators in the spirit of Chapter 3,and then show that for quasi-permutation monodromy a lot of other bosonic constructions of these algebra can be obtained with the help of various exotic bosonizations.We also find a sequence of character identities that come from equivalences betweendifferent representations.

In addition, we give a simple generalization of the exactconformal block from Chapter 5 to the orthogonal series.ReferencesThe content of Chapters 2-6 is based on the following papers in order. Almost nochanges were done to avoid producing mistakes. Therefore some mathematical objectsare introduced several times, but any time the only properties that are needed for agiven chapter are introduced, so it would not confuse the reader.• M. Bershtein, P.

Gavrylenko, A. Marshakov, Twist-field representations of Walgebras, exact conformal blocks and character identities , [hep-th/1705.00957],Under review in Communications in Mathematical Physics• P. Gavrylenko, O. Lisovyy, Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions, [math-ph/1608.00958], Submitted toCommunications in Mathematical Physics• P. Gavrylenko, A. Marshakov, Free fermions, W-algebras and isomonodromicdeformations, Theor. Math. Phys. 2016, 187:2, 649–677, [hep-th/1605.04554]• P. Gavrylenko, A.

Marshakov, Exact conformal blocks for the W-algebras, twistfields and isomonodromic deformations, JHEP02(2016)181,[hep-th/1507.08794]• P. Gavrylenko, Isomonodromic τ -functions and WN conformal blocks, JHEP09(2015)167,[hep-th/1505.00259]142Isomonodromic τ -functions and WNconformal blocksAbstractWe study the solution of the Schlesinger system for the 4-point slN isomonodromyproblem and conjecture an expression for the isomonodromic τ -function in terms of2d conformal field theory beyond the known N = 2 Painlevé VI case.

We showthat this relation can be used as an alternative definition of conformal blocks for theWN algebra and argue that the infinite number of arbitrary constants arising in thealgebraic construction of WN conformal block can be expressed in terms of only afinite set of parameters of the monodromy data of rank N Fuchsian system with threeregular singular points.

We check this definition explicitly for the known conformalblocks of the W3 algebra and demonstrate its consistency with the conjectured formof the structure constants.IntroductionThere are two topics in the mathematical physics that remained independent for a longtime: the theory of isomonodromic deformations, initiated by R.

Fuchs, P. Painlevéand L. Schlesinger in the beginning of 20th century (see [IN] and references therein),and the 2d conformal field theory (CFT) founded by A. Belavin, A. Polyakov andA. Zamolodchikov in 1984 [BPZ]. Both theories have wide range of applications.Conformal field theory describes perturbative string theory and second order phasetransitions in the 2d systems. The theory of isomonodromic deformations gives rise tonon-linear special functions such as Painlevé transcendents, which appear in differentproblems of mathematical physics: for example, in the random matrix theory andgeneral relativity.First relations between the theory of isomonodromic deformations and 2d quantumfield theory have been established in 1978-80 by M.

Sato, M. Jimbo and T. Miwa[SMJ]. More recently, O. Gamayun, N. Iorgov and O. Lisovyy have discovered thatthe τ -function of the Painlevé VI equation (related to the rank two Fuchsian systemwith four regular singular points on the Riemann sphere) can be expressed as a sum ofc = 1 conformal blocks, multiplied by certain ratios of the Barnes functions – a typicalexpansion of the correlation function in CFT [GIL12]. Their formula gives the general152. Isomonodromic τ -functions and WN conformal blockssolution of Painlevé VI equation. This conjecture has already been proved in two ways:one proof is purely representation-theoretic and adapted initially for the 4-point τ function [BShch] but can provide us with a collection of nontrivial bilinear relationsfor the n-point conformal blocks, whereas another one is based on the computationof monodromies of conformal blocks with degenerate fields and allows to consideran arbitrary number of regular singular points on the Riemann sphere [ILTe].

Thecorrespondence also extends to the irregular case: for instance, it gives exact solutionsof the Painlevé V and III equations [GIL13], [ILT14], which are known to describecorrelation functions in certain massive field theories.The present chapter is concerned with the extension of the isomonodromy-CFTcorrespondence to higher rank. Already in [GIL12] there was a suggestion that themonodromy preserving deformations of Fuchsian systems of rank N should be relatedto 2d CFT with central charge c = N − 1. One obvious and natural candidate forsuch a theory is the Toda CFT with WN algebra of extended conformal symmetry.We show that indeed the N × N isomonodromic problem corresponds to the WNalgebra, whose Virasoro part has central charge c = N − 1.