Generalized Rogers Ramanujan Identities from AGT Correspondence

TL;DR

This paper derives generalized Rogers-Ramanujan identities for certain conformal field theory cosets related to the AGT correspondence, specifically providing explicit identities for the case of r=2, p, and n=1.

Contribution

It introduces new generalized Rogers-Ramanujan identities for a broad class of cosets associated with the AGT correspondence, extending known identities to all positive integers n, r, and p.

Findings

Derived GRR identities for the case r=2, n=1, p for all characters.

Proposed existence of GRR identities for all positive integers n, r, p.

Connected identities to the structure of conformal blocks in AGT correspondence.

Abstract

AGT correspondence and its generalizations attracted a great deal of attention recently. In particular it was suggested that $U(r)$ instantons on $R^4/Z_p$ describe the conformal blocks of the coset ${\cal A}(r,p)=U(1)\times sl(p)_r\times {sl(r)_p\times sl(r)_n\over sl(r)_{n+p}}$, where $n$ is a parameter. Our purpose here is to describe Generalized Rogers Ramanujan (GRR) identities for these cosets, which expresses the characters as certain $q$ series. We propose that such identities exist for the coset ${\cal A}(r,p)$ for all positive integers $n$ and all $r$ and $p$. We treat here the case of $n=1$ and $r=2$, finding GRR identities for all the characters.