Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials

Морозов, А.; Smirnov, Andrey

doi:10.1007/s11005-014-0681-6

Cited by 68 publications

(65 citation statements)

References 77 publications

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“…All the quantities in matrix model are analytically continued from integer values of N and β, what is made unambiguously due to Selberg nature of the integrals [113]. This quadruple decomposition is recently presented in some detail in [62] based on the number of previous developments [59][60][61][113][114][115][116][117][118][119][120][121][122], see section 2.1.1 below. The group theory symmetry behind the whole picture [64] is encoded in the 2-site U q (gl 2 ) XXZ spin chain integrable system [20,74] (reduced to XXX in 4d, when q = 1 [71]).…”

Section: Jhep05(2016)121mentioning

confidence: 99%

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Миронов¹,

Морозов²,

Zenkevich³

2016

J. High Energ. Phys.

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We consider Dotsenko-Fateev matrix models associated with compactified Calabi-Yau threefolds. They can be constructed with the help of explicit expressions for refined topological vertex, i.e. are directly related to the corresponding topological string amplitudes. We describe a correspondence between these amplitudes, elliptic and affine type Selberg integrals and gauge theories in five and six dimensions with various matter content. We show that the theories of this type are connected by spectral dualities, which can be also seen at the level of elliptic Seiberg-Witten integrable systems. The most interesting are the spectral duality between the XYZ spin chain and the Ruijsenaars system, which is further lifted to self-duality of the double elliptic system.

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Section: Jhep05(2016)121mentioning

confidence: 99%

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Миронов¹,

Морозов²,

Zenkevich³

2016

J. High Energ. Phys.

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“…At the same time they are relatively new special functions, far from being thoroughly understood and clearly described. They are deformations of the generalized Jack polynomials introduced in [59,60]. Even the simplest questions about them are yet unanswered.…”

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confidence: 99%

On factorization of generalized Macdonald polynomials

Kononov

Морозов

2016

Eur. Phys. J. C

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A remarkable feature of Schur functions-the common eigenfunctions of cut-and-join operators from W ∞ -is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U q (SL N ) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorizationon a one-(rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding.Generalized Macdonald polynomials (GMPs) [1-3] play a constantly increasing role in modern studies of the 6d version of AGT relations [31][32][33][34][35][36][37][38][39][40] and spectral dualities [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. At the same time they are relatively new special functions, far from being thoroughly understood and clearly described. They are deformations of the generalized Jack polynomials introduced in [59,60]. Even the simplest questions about them are yet unanswered. In this letter we address one of them-what happens to the hook formulas for classical, quantum, and Macdonald dimensions at the level of GMPs? We find that they survive, but only partly-on a onedimensional line in the space of time variables. Lifting to the entire two-dimensional topological locus remains to be found.We begin by recalling that the Schur functions χ R { p}, depend on representation (Young diagram) R and on infinitely many time variables p k (actually, a particular χ R depends a e-mail: morozov@itep.ru only on p k with k ≤ |R| = # boxes in R). They get nicely factorized on a peculiar two-dimensional topological locus,Coarm a and coleg l are the ordinary coordinates of the box in the diagram. To keep the notation consistent throughout the text, in (1) we call the relevant parameter t, not q, from the very beginning.It is often convenient to ignore the simple overall coefficient and substitute the product formulas like (2) by a polynomial expression for a plethystic logarithm ⎛

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“…The change of basis is nontrivial in the sense that for the states on several parallel legs the matrix of this transformation does not factorize into a tensor product of matrices acting on each leg. Indeed, the transformation is given by the spectral duality, and the two basis sets are the standard Schur (or Macdonald) symmetric functions and the generalized Macdonald polynomials [41,[192][193][194][195]. This matrix was called generalized Kostka function in [31,32].…”

Section: Spectral Duality and Change Of Basismentioning

confidence: 99%

Explicit examples of DIM constraints for network matrix models

Awata

Kanno

Matsumoto

et al. 2016

J. High Energ. Phys.

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118

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Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.

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Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials

Cited by 68 publications

References 77 publications

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

On factorization of generalized Macdonald polynomials

Explicit examples of DIM constraints for network matrix models

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