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Thisformula is obtained as a solution of system of integrable equations, so called SeibergWitten equations:II∂log τSW =dSaI =dS,∂aIBIAISimilar equations describe low-energy behaviour of N = 2 supersymmetric gauge theories,that are also closely related to CFT due to AGT correspondence.Another result of this chapter is identification between the Fourier transformation ofconformal block and explicit formula for the tau-function for quasi-permutational monodromy known due to Korotkin: Xa(n,b)1(U )τIM (q|a, b) =G0 (q|a + n)e= τB (q) exp 2 Q(r) Θbgn∈ZThis fact gives one more evidence for the correspondence between W-algebras and isomonodromic deformations.Chapter 6This chapter is also devoted to W-twist fields, but from more 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 generators in the B- and D-series may be written5in terms of complexified real fermions as follows:Uk (z) =1 k−1D2 zNX(ψα∗ (z) · ψα (z) + ψα (z) · ψα∗ (z)) + 21 Dzk−1 Ψ(z) · Ψ(z)α=1NYV (z) =: ψα∗ (z)ψα (z) : Ψ(z)α=1where Dz is Hirota derivative. We reformulate construction of the twist fields in termsof fermions. It turns out that now correct object, that parametrizes twist fields, is thenormalizer of Cartan algebra NG (h). Different conjugacy classes in NG (h) give differenttwist fields.Main subject of this chapter is the computation of characters of modules, constructedabove the twist fields. Typical example of such character is the formula for the twist fieldwhich corresponds to element g consisting of K cycles of lengths li with extra diagonalmultipliers ri :KP1(r l +ni )2P2li i ii=12KqP lj −124lj n1 +...+nK =0j=1χg (q) = qK Q∞Q(1 − q k/lj )j=1 k=1In the numerator we see the lattice AK−1 theta-function.One of the parts of this chapter is devoted to the situation when g1 ∼ g2 are conjugated in G for inequivalent g1 , g2 ∈ NG (h).

We show that in this case two differentcharacters coincide χg1 (q) = χg2 (q). This gives a series of non-trivial character identities,which we also prove explicitly. Some of them coincide with Macdonald identity, and someof them seem to be new. One of the tools of character computations are exotic bosonization formulas that relate bosons and fermions with different boundary conditions: likebosonization of periodic and anti-periodic fermions into single anti-periodic boson.In this chapter we also compute conformal blocks of the twist fields in D-series.

Mainfeature of this case is the structure of the 2N -fold cover, which can be shown on thefollowing commutative diagram:π2NσΣπ2Σ̃πNCP1This cover has an involution σ, and its factor over this involution is smaller N -fold cover.Most of the objects that are used in the construction are σ-antisymmetric: for example,the only sufficient part of the period matrix is Prym period matrix. Desired formula forconformal block in this case has the structure similar to A-case:G0 (a, r, q) = τB (Σ|q)τB−1 (Σ̃|q)τSW (a, r, q)In some cases it can also reduce to the formula for A-series.6ConclusionThis thesis contains some number of constructions that give explicit formulas for isomonodromic tau-functions, for conformal blocks of W-algebras, and relate some of them toeach other.

The main technical tools are free-field constructions of the vertex operators,use of Seiberg-Witten integrable system, and manipulations with projection-like operatorsin functional spaces.ReferencesThe content of Chapters 2-5 is based on the following papers in order:2.

P. Gavrylenko, Isomonodromic τ -functions and WN conformal blocks, JHEP09(2015)167,[hep-th/1505.00259]3. P. Gavrylenko, A. Marshakov, Free fermions, W-algebras and isomonodromic deformations, Theor. Math. Phys. 2016, 187:2, 649–677, [hep-th/1605.04554]4. P. Gavrylenko, O. Lisovyy, Fredholm determinant and Nekrasov sum representationsof isomonodromic tau functions, [math-ph/1608.00958], Under review in Communications in Mathematical Physics5. P.

Gavrylenko, A. Marshakov, Exact conformal blocks for the W-algebras, twistfields and isomonodromic deformations, JHEP02(2016)181,[hep-th/1507.08794]6. 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 Physics7.

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