AGT, Burge pairs and minimal models

Abstract

We consider the AGT correspondence in the context of the conformal field theory $M^{p,p^{\prime }}$ $\otimes$ $M^H$, where $M^{p,p^{\prime }}$ is the minimal model based on the Virasoro algebra $V^{p,p^{\prime }}$ labeled by two co-prime integers $\{p,p^{\prime }\}$, $1<p<p^{\prime }$, and $M^H$ is the free boson theory based on the Heisenberg algebra $H$. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in $M^{p,p^{\prime }}$ $\otimes$ $M^H$ leads to ill-defined or incorrect expressions. Let $B_n^{p,p^{\prime },H}$ be a conformal block in $M^{p,p^{\prime }}$ $\otimes$ $M^H$, with $n$ consecutive channels ${\chi }_i$, $i=1,\cdots ,n$, and let ${\chi }_i$ carry states from $H_{r_i,s_i}^{p,p^{\prime }}$ $\otimes$ $F$, where $H_{r_i,s_i}^{p,p^{\prime }}$ is an irreducible highest-weight $V^{p,p^{\prime }}$-representation, labeled by two integers $\{r_i,s_i\}$, $0<r_i<p$, $0<s_i<p^{\prime }$, and $F$ is the Fock space of $H$. We show that restricting the states that flow in ${\chi }_i$ to states labeled by a partition pair $\{Y_1^i,Y_2^i\}$ such that $Y_{2,\mathtt{R}}^i-Y_{1,\mathtt{R}+s_i-1}^i\ge 1-r_i$, and $Y_{1,\mathtt{R}}^i-Y_{2,\mathtt{R}+p^{\prime }-s_i-1}^i\ge 1-p+r_i$, where $Y_{j,\mathtt{R}}^i$ is row-$\mathtt{R}$ of $Y_j^i,j\in \{1,2\}$, we obtain a well-defined expression that we identify with $B_n^{p,p^{\prime },H}$. We check the correctness of this expression for $1.$ Any 1-point $B_1^{p,p^{\prime },H}$ on the torus, when the operator insertion is the identity, and $2.$ The 6-point $B_3^{3,4,H}$ on the sphere that involves six Ising magnetic operators.

22 pages. Simplified the presentation

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