AGT, Burge pairs and minimal models

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Article

Abstract

We consider the AGT correspondence in the context of the conformal field theory \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime}\otimes {\mathrm{\mathcal{M}}}^{\mathcal{H}} \), where \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime } \) is the minimal model based on the Virasoro algebra \( {{\mathcal{V}}^{p, p}}^{\prime } \) labeled by two co-prime integers {p, p′}, 1 < p < p′, and \( {\mathrm{\mathcal{M}}}^{\mathcal{H}} \) is the free boson theory based on the Heisenberg algebra \( \mathcal{H} \). Using Nekrasov’s instanton partition functions without modification to compute conformal blocks in \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime}\otimes {\mathrm{\mathcal{M}}}^{\mathcal{H}} \) leads to ill-defined or incorrect expressions.

Let \( {\mathrm{\mathcal{B}}}_n^{p, p\prime, \mathcal{H}} \) be a conformal block in \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime}\otimes {\mathrm{\mathcal{M}}}^{\mathcal{H}} \), with n consecutive channels χι, ι\( = 1, \cdot p \cdot p \cdot p, \)n, and let χι carry states from \( {\mathcal{H}}_{r_{\iota},{s}_{\iota}}^{p, p\prime}\otimes \mathrm{\mathcal{F}} \), where \( {\mathcal{H}}_{r_{\iota},{s}_{\iota}}^{p, p\prime } \) is an irreducible highest- weight \( {\mathcal{V}}^{p, p\prime } \) -representation, labeled by two integers {rι, sι}, 0 < rι< p, 0 < sι< p′, and \( \mathrm{\mathcal{F}} \) is the Fock space of \( \mathcal{H} \).

We show that restricting the states that flow in χι, ι = 1, · · · , n, to states labeled by partition pairs \( \left\{{Y}_1^{\iota},{Y}_2^{\iota}\right\} \) that satisfy \( {Y}_{2,\sigma}^{{}_{\iota, \mathrm{T}}}-{Y}_{1,\sigma +{r}_{\iota}-1}^{\iota, \mathrm{T}}\ge 1-{s}_{\iota} \), and \( {Y}_{1,\sigma}^{\iota, \mathrm{T}}-{Y}_{2,\sigma + p-{r}_{\iota}-1}^{\iota, \mathrm{T}}\ge 1- p^{\prime }+{s}_{\iota} \),where \( {Y}_{i,\sigma}^{\iota, \mathrm{T}} \) is the σ-column of \( {Y}_i^{\iota},\kern0.5em i\in \left\{1,2\right\} \), we obtain a well-defined expression that we identify with \( {\mathrm{\mathcal{B}}}_n^{p, p\prime, \mathcal{H}} \). We check the correctness of this expression for 1. Any 1-point \( {\mathrm{\mathcal{B}}}_1^{p, p\prime, \mathcal{H}} \) on the torus, when the operator insertion is the identity, and 2. The 6-point \( {\mathrm{\mathcal{B}}}_3^{3,4,\mathcal{H}} \) on the sphere that involves six Ising magnetic operators.

Keywords

Supersymmetric gauge theory Conformal and W Symmetry 

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Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical PhysicsIndependent University of MoscowMoscowRussia
  4. 4.Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia

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