Cohomological Localization of $\mathcal N = 2$ Gauge Theories with Matter

TL;DR

This paper develops a cohomological framework for a broad class of $ abla=2$ supersymmetric gauge theories on four-manifolds, enabling exact partition function calculations through localization.

Contribution

It extends localization techniques from super Yang-Mills to general $ abla=2$ theories with matter, including hypermultiplets, on diverse four-manifolds.

Findings

Constructed a general cohomological formulation for $ abla=2$ theories.

Derived explicit partition functions for various topologies.

Unified Donaldson-Witten and Pestun's theories within a common framework.

Abstract

We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $\mathcal{N}=2$ gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun's theory on $S^4$ as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.