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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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We continue our study of the AGT correspondence between instanton counting on \( {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_p}}} \right.} \) and Conformal field theories with the symmetry algebra \( \mathcal{A}\left( {r,p} \right) \). In the cases r = 1, p = 2 and r = 2, p = 2 this algebra specialized to: \( \mathcal{A}\left( {1,2} \right)=\mathcal{H}\oplus \widehat{\mathfrak{sl}}{(2)_1} \) and \( \mathcal{A}\left( {2,2} \right)=\mathcal{H}\oplus \widehat{\mathfrak{sl}}{(2)_2}\oplus \mathrm{NSR} \).

As the main tool we use a new construction of the algebra A(r, 2) as the limit of the toroidal \( \mathfrak{g}\mathfrak{l}(1) \) algebra for q, t tend to −1. We claim that the basis of the representation of the algebra \( \mathcal{A}\left( {r,2} \right) \) (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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Correspondence to M. A. Bershtein.

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ArXiv ePrint: 1211.2788

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Belavin, A.A., Bershtein, M.A. & Tarnopolsky, G.M. Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity. J. High Energ. Phys. 2013, 19 (2013).

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