Localizing gauge theories on $S^d$

TL;DR

This paper conjectures one-loop determinants for localized supersymmetric gauge theories on spheres of various dimensions, analyzes their large N behavior, and proposes a regularization method for divergences in specific supersymmetric cases.

Contribution

It provides a conjectured form of one-loop determinants for localized gauge theories on $S^d$ and explores their strong coupling behavior across dimensions.

Findings

Derived N dependence of free energy in 3<d<4 and 3<d<6.

Proposed regularization approach for divergences in 4d and 6d supersymmetric theories.

Analyzed strong coupling limits for large N gauge theories.

Abstract

We conjecture the form of the one-loop determinants for localized gauge theories with eight supersymmetries on $d$-dimensional spheres. Combining this with results for the localized action, we investigate the strong coupling behavior in the large $N$ limit for a continuous range of $d$. In particular, we find the $N$ dependence of the free energy for supersymmetric Yang-Mills with only a vector multiplet in $3<d<4$ and for maximally supersymmetric Yang-Mills in $3< d<6$. We also argue that this gives an effective way to regularize divergences after localization in $d=4$ for ${\mathcal N}=2$ gauge theories and $d=6$ for the maximally supersymmetric case.