Multi instanton calculus on ALE spaces

TL;DR

This paper computes the prepotential and gravitational corrections for supersymmetric gauge theories on ALE spaces, deriving homological data and the N=4 partition function, confirming duality predictions.

Contribution

It introduces a localization-based method to analyze gauge theories on ALE spaces, providing new explicit calculations of prepotentials and moduli space homologies.

Findings

Prepotential and gravitational corrections computed for SU(N) theories on ALE spaces.

Derived the homology Poincaré polynomial of moduli spaces of self-dual connections.

Confirmed the N=4 partition function as a modular form consistent with SL(2,Z) duality.

Abstract

We study SYM gauge theories living on ALE spaces. Using localization formulae we compute the prepotential (and its gravitational corrections) for SU(N) supersymmetric ${\cal N}=2, 2^*$ gauge theories on ALE spaces of the $A_n$ type. Furthermore we derive the Poincar\'{e} polynomial describing the homologies of the corresponding moduli spaces of self-dual gauge connections. From these results we extract the ${\cal N}=4$ partition function which is a modular form in agreement with the expectations of $SL(2,\Z)$ duality.