Localization of Supersymmetric Chern-Simons-Matter Theory on a Squashed $S^3$ with $SU(2)\times U(1)$ Isometry

TL;DR

This paper advances the localization of supersymmetric Chern-Simons-Matter theories on a squashed three-sphere with specific isometry, revealing the importance of background gauge field shifts and unifying previous results as special cases.

Contribution

It introduces a new localization framework on a squashed $S^3$ with complex backgrounds, highlighting the role of gauge field shifts and integrating prior findings.

Findings

Localization on squashed $S^3$ with $SU(2)\times U(1)$ isometry achieved.

Background gauge field shifts are crucial for nontrivial results.

Previous results are recovered as special limits of the new framework.

Abstract

Localization of supersymmetric $\mathcal{N}=2$ Chern-Simons-Matter theory on a squashed $S^3$ with $SU(2)\times U(1)$ isometry has been studied by different groups of authors. In this paper, we localize the theory on a squashed $S^3$ with $SU(2)\times U(1)$ isometry and a class of complex background. We see that certain kinds of shifts of the background gauge fields are crucial in obtaining nontrivial results, and the previously found results on this manifold can be incorporated in our results as special limits.