AGT, N-Burge partitions and W_N minimal models

Abstract

Let ${\mathcal{B}}_{N,n}^{p,p^{\prime },\mathcal{H}}$ be a conformal block, with $n$ consecutive channels ${\chi }_{ı}$, $ı=1,\cdots ,n$, in the conformal field theory ${\mathcal{M}}_N^{p,p^{\prime }}\times {\mathcal{M}}^{\mathcal{H}}$, where ${\mathcal{M}}_N^{p,p^{\prime }}$ is a ${\mathcal{W}}_N$ minimal model, generated by chiral fields of spin $1,\cdots ,N$, and labeled by two co-prime integers $p$ and $p^{\prime }$, $1<p<p^{\prime }$, while ${\mathcal{M}}^{\mathcal{H}}$ is a free boson conformal field theory. ${\mathcal{B}}_{N,n}^{p,p^{\prime },\mathcal{H}}$ is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra $A_{N-1}$, spanned by weight vectors ${\omega }_1,\cdots ,{\omega }_{N-1}$. We restrict our attention to conformal blocks with vertex operators whose charge vectors point along ${\omega }_1$. The charge vectors that label the initial and final states can point in any direction. Following the ${\mathcal{W}}_N$ AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute ${\mathcal{B}}_{N,n}^{p,p^{\prime },\mathcal{H}}$, leads to ill-defined expressions. We show that restricting the states that flow in the channels ${\chi }_{ı}$, $ı=1,\cdots ,n$, to states labeled by $N$ partitions that satisfy conditions that we call $N$-Burge partitions, leads to well-defined expressions that we identify with ${\mathcal{B}}_{N,n}^{p,p^{\prime },\mathcal{H}}$. We check our identification by showing that a specific non-trivial conformal block that we compute, using the $N$-Burge conditions satisfies the expected differential equation.

34 pages. More references, same content

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