Localization of four-dimensional super Yang-Mills theories compactified on Riemann surface

TL;DR

This paper applies localization techniques to compute the partition function of 4D super Yang-Mills theories on a Riemann surface, revealing a trivial elliptic genus and deriving an effective theory on the surface.

Contribution

It demonstrates the use of localization and elliptic genus evaluation to analyze super Yang-Mills theories on curved surfaces, leading to new insights into their reduced form.

Findings

Elliptic genus evaluated as trivial in this context

Partition function reduces to a theory on the Riemann surface

Method provides a pathway for analyzing similar gauge theories

Abstract

We consider the partition function of super Yang-Mills theories defined on $T^2 \times \Sigma_g$. This path integral can be computed by the localization. The one-loop determinant is evaluated by the elliptic genus. This elliptic genus gives trivial result in our calculation. As a result, we obtain a theory defined on the Riemann surface.