Gauge Theories on ALE Space and Super Liouville Correlation Functions

TL;DR

The work extends the AGT paradigm to gauge theories on the ALE space $\mathcal{O}_{\mathbb{P}^1}(-2)$ by establishing a precise link to $\mathcal{N}=1$ super Liouville CFT correlators. It derives a blow-up formula expressing the full ALE partition function as a convolution of flat-space Nekrasov functions, and demonstrates that the $\mathcal{N}=2^*$ case factorizes into the product of the $\widehat{SU}(2)_2$ character and the super Virasoro torus block (up to a $U(1)$ factor). The sphere four- and five-point correlators and the torus one-point function are shown to correspond to specific quiver gauge theories, with perturbative Liouville three-point functions matching gauge theory perturbative contributions. The results hint at a broader orbifold-AGN correspondence, suggesting extensions to ADE quotients and connections to topological strings and M-theory.

Abstract

We present a relation between N=2 quiver gauge theories on the ALE space O_{P^1}(-2) and correlators of N=1 super Liouville conformal field theory, providing checks in the case of punctured spheres and tori. We derive a blow-up formula for the full Nekrasov partition function and show that, up to a U(1) factor, the N=2^* instanton partition function is given by the product of the character of \hat{SU}(2)_2 times the super Virasoro conformal block on the torus with one puncture. Moreover, we match the perturbative gauge theory contribution with super Liouville three-point functions.