$\widehat{\mathfrak{sl}}(n)_N$ WZW conformal blocks from $SU(N)$ instanton partition functions on ${\mathbb {C}}^2/{\mathbb {Z}}_n$

TL;DR

This paper establishes a connection between 4D $SU(N)$ instanton partition functions on orbifolded spaces and 2D $ ext{WZW}$ conformal blocks, extending the AGT correspondence to include $ ext{WZW}$ models.

Contribution

It demonstrates how to extract $ ext{WZW}$ conformal blocks from 4D $SU(N)$ instanton partition functions on ${f C}^2/{f Z}_n$, generalizing previous AGT-related results.

Findings

Derived explicit relations between instanton partition functions and $ ext{WZW}$ conformal blocks.

Extended the AGT correspondence to include $ ext{WZW}$ models on orbifolded spaces.

Provided a method to trivialize parafermionic factors in the partition functions.

Abstract

Generalizations of the AGT correspondence between 4D $\mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${\mathbb {C}}^2$ with $\Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $\mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, \ldots$, on ${\mathbb {C}}^2/{\mathbb {Z}}_n$, $n = 2, 3, \ldots$, with $\Omega$-deformation and 2D conformal field theories with $\mathcal{W}^{\, para}_{N, n}$ ($n$-th parafermion $\mathcal{W}_N$) symmetry and $\widehat{\mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $\mathcal{W}^{\, para}_{N, n}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${\mathbb {C}}^2/{\mathbb {Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $\widehat{\mathfrak{sl}}(n)_N$ WZW conformal blocks, $n = 2, 3, \ldots$, $N = 1, 2, \ldots\, .$