On Supersymmetric Gauge Theories on S^4 x S^1

TL;DR

This work constructs supersymmetric gauge theories on $S^4 \times S^1$ by formulating Killing spinors and SUSY transformations that reproduce Pestun's $N=2$ structure upon dimensional reduction on $S^1$. It shows that a conventional Yang–Mills action does not extend to the full space, except in the 4D limit, but provides a SUSY-exact regulator suitable for localization. Localization reveals that the saddle points coincide with the Pestun setup on $S^4$ with KK-mode contributions, and the partition function on $S^4 \times S^1$ factors into a KK-summed Pestun-like determinant times Nekrasov’s instanton piece, with $a_0$ periodic and $R\to0$ recovering the Pestun result. These results illuminate the 5D/4D connection relevant for M5-brane constructions and offer a concrete framework for exact computations in curved backgrounds beyond $S^4$.

Abstract

We construct supersymmetric gauge theory on S^4 x S^1. We find a consistent supersymmetry transformations which reduced to the 4D N=2 supersymmetry transformation studied by Pestun by the dimensional reduction on S^1. We find there is no analogue of the usual Yang-Mills action except in the 4D limit. We also apply the localization technique to the partition function of the theories.