Диссертация (1137342), страница 26
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Relation to Nekrasov functionsThis expression can be represented aslsgn (Z) =: lsgn (p, q) + lsgn (p0 , q0 ) + m.To get the desired formula, one has to absorb m by adding extra shift eiδ2 η+ = −1.Combining this shift with the previous formulas (4.115), we deduce the full shift(4.107) of the Fourier transformation parameters. The final formula for the relativesign islsgn Z/Ẑ = lsgn (p, q) + lsgn (p0 , q0 ) ,which completes our calculation.1195Exact conformal blocks for the W-algebras,twist fields and isomonodromicdeformationsAbstractWe consider the conformal blocks in the theories with extended conformal W-symmetryfor the integer Virasoro central charges.
We show that these blocks for the generalizedtwist fields on sphere can be computed exactly in terms of the free field theory on thecovering Riemann surface, even for a non-abelian monodromy group. The generalizedtwist fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how theseexact conformal blocks can be effectively computed using the technique arisen fromthe gauge theory/CFT correspondence. We discuss also their direct relation withthe isomonodromic tau-function for the quasipermutation monodromy data, whichcan be an encouraging step on the way of definition of generic conformal blocks forW-algebra using the isomonodromy/CFT correspondence.IntroductionAn interest to conformal field theories (CFT) with extended nonlinear W-symmetrygenerated by the higher spin holomorphic currents has long history, starting from theoriginal work [ZamW].
These theories resemble many features of ordinary CFT (withonly Virasoro symmetry), like free field representation and degenerate fields [FZ, FL],but it already turns to be impossible to construct in generic situation their conformalblocks [BW] (or the blocks for the algebra of higher spin W-currents) which are themain ingredients in the bootstrap definition of the physical correlation functions.This interest has been seriously supported in the context of rather nontrivial correspondence between two-dimensional CFT and four-dimensional supersymmetric gaugetheory [LMN, NO, AGT], where the conformal blocks have to be compared with theNekrasov instanton partition functions [Nek, NP] producing in the quasiclassical limitthe Seiberg-Witten prepotentials [SW].
This correspondence meets serious difficultiesbeyond the level of the SU (2) gauge quivers on gauge theory side, i.e. for the higherrank gauge groups, which should correspond to the not yet defined generic blocks of1215. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsthe W-conformal theories. It is already clear, however, that the technique developedin two-dimensional CFT can be applied to four-dimensional gauge theories, and viceversa. Following [KriW, Mtau, GMqui] we are going to demonstrate how it can saveefforts for the computation of the exact conformal blocks for the twist fields in theorieswith W-symmetry.Even in the Virasoro case generic conformal block is a very nontrivial special function [BPZ], but there exists two important particular cases where the answer is knownalmost in explicit form – the correlation functions containing degenerate fields (whichare related to the integrals of hypergeometric type) and the exact Zamolodchikovblocks for a nontrivial (though c = 1) theory [ZamAT86, ZamAT87, ApiZam] 1 .
Thefirst class can be generalized to the case of W-algebras, where similar hypergeometricformulas arise in the case of so-called completely degenerate fields [FLitv07]. Thealgebraic definition still exists when degeneracy is not complete, and in this casethe most effective way of computation comes from use of the gauge theory Nekrasovfunctions.Below we are going to study the W-analogs of the Zamolodchikov conformal blocks,which do not belong to the class of algebraic ones. They can be nevertheless computedexactly, partially using the methods of gauge theories and corresponding integrablesystems.
We are going to demonstrate also their direct relations with exactly knownisomonodromic τ -functions [SMJ], which confirms therefore their role as an important example of a generic W-block which can be possible defined (for integer centralcharges) in terms of corresponding isomonodromic problem [Gav].The exact conformal blocks of the W-algebras are closely related to the correlationfunctions of the twist fields, studied long ago in the context of perturbative stringtheory (see e.g.
[Knizhnik, BR, DFMS]). However, unlike [ZamAT87], the correlatorsof the twist fields in these papers were not really expressed through the conformalblocks, and therefore their relation to the W-algebras remained out of interest, so weare going to fill partially this gap.The chapter is organized as follows. In sect. 6.4 we define the correlators of currents on sphere in presence of the twist fields, and show how they can be computedin terms of free conformal field theory on the cover.
In sect. 5.3 we identify the twistfields with the primary fields of the W-algebra and propose a way to extract the valuesof their quantum numbers from the previously computed correlation functions of thecurrents. We also show there that these W-charges have obvious meaning in terms ofthe eigenvalues of the quasipermutation monodromy matrices. In sect. 5.4 we definethe result for the exact conformal block in terms of integrable systems.
In particular,we show that the main classical contribution to the result satisfies the well-knownSeiberg-Witten (SW) period equation [SW, KriW], moreover, in this case they can beimmediately solved, which gives the most effective way to express the answer throughthe period matrix and the prime form on the covering surface. Next, in sect. 5.5 wediscuss the connection of the W-algebra conformal blocks with the τ -function of theisomonodromic problem, and show that the W-blocks we have constructed correspondin this context to the τ -function for the case of quasipermutation monodromy data.In sect.
5.6 we construct some explicit examples, and some extra technical information1Strictly speaking the CFT-Painlevé correspondence [GIL12] gives rise to a collection of newexact conformal blocks, coming from the algebraic solutions of Painlevé VI.1225.2. Twist fields and branched covers(the recursion procedure we have used for construction of correlators of the higherW-currents, the discussion of their OPE with the stress-tensor, and the computationof the asymptotics of the period matrix on the cover and its relation with the structure constants in the expansion of the isomonodromic τ -function) is located in theAppendix.Twist fields and branched coversDefinitionWe start now with the construction of the conformal blocks of W (slN ) = WN algebra at integer Virasoro central charges c = N − 1 following the lines of [ZamAT87,Knizhnik, BR].
It is well-known [FL] that WN algebra has free-field representation interms of N − 1 bosonic fields with the currents J a (z) = i∂φa (z) satisfying operatorproduct expansion (OPE)J a (z)J b (z 0 ) =K ab+ reg.(z − z 0 )2(5.1)where K ab is the scalar product in the Cartan subalgebra h ⊂ g = slN . For theNP−1current J(z) =ha J a (z) = i∂φ(z), where ha is the basis in h, it is useful toa=1introduce explicit componentsJi (z) = (ei , J(z)),i = 1, . . .
, N(5.2)with {ei } being the weights of the first fundamental or vector representation, so thatJi (z)Jj (z 0 ) =δij − N1(ei , ej )+reg.=+ reg.(z − z 0 )2(z − z 0 )2(5.3)All high-spin currentsPof the WN -algebra at c = N − 1 are elementary symmetricpolynomials of Ji (z) ( Ji (z) = 0), e.g.
the first three arei11X: (J(z), J(z)) :=: Ji (z)2 :22 iX1XW (z) = W3 (z) =: Ji (z)Jj (z)Jk (z) :=: Ji (z)3 :(5.4)3ii<j<k!2XX1X 41W4 (z) =: Ji (z)Jj (z)Jk (z)Jl (z) := :Ji2 (z) : −: Ji (z) :84 iii<j<k<lT (z) = −W2 (z) =and the primary fields for the current algebra are exponentials of φ(z) ∈ hVθ (z) = ei(θ,φ(z))(5.5)with the corresponding eigenvalues wk (θ) of the zero modes of the Wk (z)-generatorsgiven by symmetric functions of (ei , θ).1235. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsNow we are going to introduce new fields Os (z), which are still primary for allhigh-spin currents {Wk (z)}, but not for the currents Ji (z).
They can be realized asmonodromy fieldsγq : J(z)Os (q) 7→ s(J(z))Os (q)(5.6)for some contours γq encircling the point q on the base curve, where s ∈ WslN = SN isan element of the corresponding Weyl group. The particular cases of this constructionwere known for the Abelian monodromy group of the cover [BR, ZamAT87], but eventhere in the cases with N > 2 they were not identified with WN primary fields.Now we are going to construct the particular conformal block (on P1 with globalcoordinate z), where all monodromy fields can be grouped as Os (q2i+1 )Os−1 (q2i+2 ) atq2i+1 → q2i+2 , so that one can take an OPEXOs (z)Os−1 (z 0 ) =Cs,θ (z − z 0 )∆(θ)−2∆(s) (Vθ (z 0 ) + descendants)(5.7)θand fix the quantum numbers in the intermediate¸ channels, where there are only1N −1the fields with definite h = u(1)charges 2πi z dζJ(ζ)Vθ (z) = θ · Vθ (z).
In order to do this consider G0 (q) = G0 (q1 , ..., q2L ), together with 1-form G1i (z|q)dz =G1i (z|q1 , ..., q2L )dz and bidifferential G2ij (z, z 0 |q)dzdz 0 = G2ij (z, z 0 |q1 , ..., q2L )dzdz 0 , whereG0 (q1 , ..., q2L ) = hOs1 (q1 )Os−1(q2 )...OsL (q2L−1 )Os−1 (q2L )i1LG1i (z|q1 , ..., q2L )G2ij (z, z 0 |q1 , ..., q2L )= hJi (z)Os1 (q1 )Os−1(q2 )...OsL (q2L−1 )Os−1 (q2L )i1L(5.8)= hJi (z)Jj (z 0 )Os1 (q1 )Os−1(q2 )...OsL (q2L−1 )Os−1 (q2L )i1Lwhich become single-valued on the cover π : C → P1 with the branch points qα andcorresponding monodromies sα .
The indices i, j are just labels of the sheets of thiscover, so the multi-valued differentials (5.8) on P1 are now expressed in terms of thesingle-valued G1 (ξ|q1 , ..., q2L )dξ and G2 (ξ, ξ 0 |q1 , ..., q2L )dξdξ 0 on the covering surface C:G1i (z|q1 , ..., q2L )dz = G1 (z i |q1 , ..., q2L )dz iG2ij (z, z 0 |q1 , ..., q2L )dzdz 0 = G2 (z i , z 0j |q1 , ..., q2L )dz i dz 0j(5.9)where z i = π −1 (z)i is the coordinate at i’th preimage of the point z, not the power(note that number i is not defined globally due to the presence of monodromies).
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