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On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

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Abstract

In their recent paper, Alday et al. (Lett Math Phys 91:167–197, 2010) 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 by the exact form of the 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 \({\mathcal{A}={\sf Vir} \otimes\mathcal{H}}\) 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 \({Z_{{\sf bif}}}\) 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 Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin–Ono integrable hierarchy.

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia

    Vasyl A. Alba, Vladimir A. Fateev, Alexey V. Litvinov & Grigory M. Tarnopolskiy

  2. Bogolyubov Institute for Theoretical Physics NASU, 03680, Kiev, Ukraine

    Vasyl A. Alba

  3. Department of General and Applied Physics, Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Russia

    Vasyl A. Alba & Grigory M. Tarnopolskiy

  4. Institute for Theoretical and Experimental Physics, 117218, Moscow, Russia

    Vasyl A. Alba

  5. Laboratoire de Physique Théorique et Astroparticules, Université Montpellier II, UMR5207 CNRS-UM2, 34095, Montpellier, France

    Vladimir A. Fateev

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  1. Vasyl A. Alba
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  2. Vladimir A. Fateev
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  3. Alexey V. Litvinov
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  4. Grigory M. Tarnopolskiy
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Correspondence to Alexey V. Litvinov.

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Alba, V.A., Fateev, V.A., Litvinov, A.V. et al. On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture. Lett Math Phys 98, 33–64 (2011). https://doi.org/10.1007/s11005-011-0503-z

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