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 $B^{\, p, p^{\prime}, H}_n$ 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 $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ $\otimes$ $F$, where $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ 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 $Y^{i}_{2, {\tt R}} - Y^{i}_{1, {\tt R} + s_{i} - 1} \geq 1 - r_{i}$, and $Y^{i}_{1, {\tt R}} - Y^{i}_{2, {\tt R} + p^{\prime} - s_{i} - 1} \geq 1 - p + r_{i}$, where $Y^{i}_{j, {\tt R}}$ is row-${\tt R}$ of $Y^{i}_j, j \in \{1, 2\}$, we obtain a well-defined expression that we identify with $B^{\, p, p^{\prime}, H}_n$. We check the correctness of this expression for ${\bf 1.}$ Any 1-point $B^{\, p, p^{\prime}, H}_1$ on the torus, when the operator insertion is the identity, and ${\bf 2.}$ The 6-point $B^{\, 3, 4, H}_3$ on the sphere that involves six Ising magnetic operators.