Instantons and 2d superconformal field theory

Abstract

A recently proposed correspondence between 4-dimensional $\mathcal{N} = 2$ SUSY SU(k) gauge theories on ${{{{\mathbb{R}^4}}} \left/ {{{\mathbb{Z}_m}}} \right.}$ and SU(k) Toda-like theories with Z _m parafermionic symmetry is used to construct four-point $\mathcal{N} = 1$ super Liouville conformal block, which corresponds to the particular case k = m = 2.

The construction is based on the conjectural relation between moduli spaces of SU(2) instantons on ${{{{\mathbb{R}^4}}} \left/ {{{\mathbb{Z}_2}}} \right.}$ and algebras like $\widehat{\text{gl}} {(2)_2} \times \mathcal{N}\mathcal{S}\mathcal{R}$. This conjecture is confirmed by checking the coincidence of number of fixed points on such instanton moduli space with given instanton number N and dimension of subspace degree N in the representation of such algebra.