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      Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

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          Abstract

          We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(1) algebra for q,t tend to -1. We claim that the basis of the representation of the algebra A(r,2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q,t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the \(\mathcal{N}=1\) Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1.

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          Liouville Correlation Functions from Four-dimensional Gauge Theories

          , , (2010)
          We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1.
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            Basic representations of affine Lie algebras and dual resonance models

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              On combinatorial expansion of the conformal blocks arising from AGT conjecture

              In their recent paper \cite{Alday:2009aq} Alday, Gaiotto and Tachikawa proposed a relation between \(\mathcal{N}=2\) four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired from the exact form of instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra \(\mathcal{A}=\text{\sf Vir}\otimes\mathcal{H}\) which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of \(\mathcal{A}\). The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called \(Z_{\text{\sf{bif}}}\) appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in \cite{Alday:2009aq}. We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin-Ono integrable hierarchy.
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                Author and article information

                Journal
                2012-11-12
                2013-02-21
                Article
                10.1007/JHEP03(2013)019
                1211.2788
                6b700f6c-fead-4251-9320-3437bedf62f8

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                JHEP 1303:019, 2013
                34 pages, v2: references added, misprints corrected; v3: exposition improved, new section inserted; v4: misprints corrected, Propositions 3.1,3.2,4.1 are called Conjectures, new subsection 5.3 were included, reference added. Version for JHEP
                hep-th math-ph math.MP math.QA math.RT

                Mathematical physics,High energy & Particle physics,Mathematical & Computational physics,Algebra

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