Localization on round sphere revisited

TL;DR

This work develops a framework for supersymmetric localization of gauge theories on round $S^3$ with a background $U(1)$ gauge field that breaks isometries, achieved by employing unusual charged Killing spinors to preserve $ abla$-SUSY. Localization reduces the path integral to Gaussian fluctuations around a fixed locus, yielding one-loop determinants that combine into the double sine function, mirroring results on squashed spheres via a Weyl transformation. The authors relate round-sphere constructions to known squashed-sphere geometries, show that $rac{1}{2}$-BPS Wilson loops become torus-knots for rational $s$, and discuss implications for large-$N$ reductions and potential M-theory embeddings. Overall, the paper demonstrates that round $S^3$ can support nontrivial localization data equivalent to squashed cases, enriching the landscape of exactly calculable observables in three dimensions.

Abstract

We consider supersymmetric gauge theories on round 3-sphere with a certain background gauge field. The Lagrangians break the usual symmetry because the background gauge field which we have turned on violates the isometry. In order to maintain the supersymmetry, we choose unfamiliar charged Killing spinors as N = 2 SUSY parameters. We perform localization calculous within this setup and find the double sine function as we expected. We comment on more direct relationship between theories on round sphere and squashed sphere via Weyl transformation.