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

TL;DR

This work applies rigid supersymmetry localization to a 3D N=2 Chern-Simons-Matter theory on a squashed S^3 with SU(2)×U(1) isometry and a class of complex backgrounds. By solving generalized Killing spinor equations and employing Q-exact deformations, the partition function is reduced to a matrix-model form Z = (1/|W|) ∫ d^rσ Z_class Z_mat^{1-loop} Z_g^{1-loop}, where Z_class encodes FI and CS contributions and the 1-loop factors arise from matter and gauge sectors. The 1-loop determinants are expressed via the double-sine function s_b with parameters Q and b determined by the squashing and a complex background angle Θ; these results reproduce Jap-2 and IY in appropriate limits, and illustrate that κ shifts affect only the classical piece while Θ controls the 1-loop structure. Overall, the study provides a unified framework relating various squashed-S^3 localization results and highlights how complex backgrounds influence the partition function, with potential implications for holography and gravity duals.

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.