Parafermionic Liouville field theory and instantons on ALE spaces

TL;DR

This paper explores the correspondence between certain coset conformal field theories and $ ext{SU}(n)$ gauge theories on orbifold spaces, providing evidence for a generalized AGT relation involving parafermionic algebras and instanton partition functions.

Contribution

It demonstrates the equivalence of Nekrasov partition functions and conformal blocks for $ ext{SU}(2)$ on $ ext{R}^4/ ext{Z}_p$ with parafermionic algebra structures, extending AGT correspondence insights.

Findings

Nekrasov partition function matches conformal blocks up to U(1)-factor.

Evidence supports the correspondence for arbitrary p values.

Structure of instanton partition functions suggests a generalized AGT relation.

Abstract

In this paper we study the correspondence between the $\hat{\textrm{su}}(n)_{k}\oplus \hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p}$ coset conformal field theories and $\mathcal{N}=2$ SU(n) gauge theories on $\mathbb{R}^{4}/\mathbb{Z}_{p}$. Namely we check the correspondence between the SU(2) Nekrasov partition function on $\mathbb{R}^{4}/\mathbb{Z}_{4}$ and the conformal blocks of the $S_{3}$ parafermion algebra (in $S$ and $D$ modules). We find that they are equal up to the U(1)-factor as it was in all cases of AGT-like relations. Studying the structure of the instanton partition function on $\mathbb{R}^4/\mathbb{Z}_p$ we also find some evidence that this correspondence with arbitrary $p$ takes place up to the U(1)-factor.