Notes on SUSY Gauge Theories on Three-Sphere

TL;DR

The paper develops a generalized localization framework for three-dimensional ${ m N}=2$ gauge theories on the three-sphere $S^3$, incorporating chiral matter with arbitrary R-charges and providing an explicit one-loop determinant in terms of the double sine function $s_{b=1}$. It constructs the necessary SUSY transformations and Lagrangians for vectormultiplets and matter, and derives a concrete integral formula for the partition function that includes nonabelian gauge groups, Chern-Simons and FI terms, and arbitrary representations. As an application, it analyzes mass-deformed ${ m N}=4$ SQED (the $T[SU(2)]$ theory) to show the self-mirror property is preserved under the deformation and to connect the $S^3$ partition function with mirror symmetry and, more broadly, to 4D $\mathcal{N}=2^*$ S-duality structures and Virasoro torus blocks.

Abstract

We extend the formula for partition functions of N=2 superconformal gauge theories on S^3 obtained recently by Kapustin, Willett and Yaakov, to incorporate matter fields with arbitrary R-charge assignments. We use the result to check that the self-mirror property of N=4 SQED with two electron hypermultiplets is preserved under a certain mass deformation which breaks the supersymmetry to N=2.