Towards the localization of SUSY gauge theory on a curved space

TL;DR

This paper extends the localization techniques used in supersymmetric gauge theories from the four-sphere to more general curved spaces, enabling new exact calculations in these settings.

Contribution

It generalizes Pestun's localization method to a broader class of curved spaces for N=4 supersymmetric gauge theories.

Findings

Derived the Q-exact term for localization on various curved spaces

Identified conditions for the positivity of the Q-exact term

Evaluated the super Yang-Mills action in localized configurations

Abstract

We consider an N=4 supersymmetric gauge theory on a curved space. We try to generalize Pestun's localization calculation on the four-sphere to a more general class of curved spaces. We calculated the Q-exact term to localize the path-integral, and when it becomes positive definite, we obtain a configuration where the path-integral localizes. We also evaluate the super Yang-Mills action in this configuration.