AGT, Burge pairs and minimal models

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 χ ι , ι $$ = 1, \cdot p \cdot p \cdot p, $$ n, and let χ ι 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 {r ι , s ι }, 0 < r ι < p, 0 < s ι < p′, and $$ \mathrm{\mathcal{F}} $$ is the Fock space of $$ \mathcal{H} $$ . We show that restricting the states that flow in χ ι , 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.