SUSY Gauge Theories on Squashed Three-Spheres

TL;DR

This paper investigates 3D N=2 supersymmetric gauge theories on squashed three-spheres, demonstrating how to preserve supersymmetry with background fields and computing exact partition functions linked to conformal field theories.

Contribution

It provides explicit Lagrangians, supersymmetry rules, and localization-based partition functions for gauge theories on squashed spheres with different isometries, connecting to Liouville and Toda theories.

Findings

Supported charged Killing spinors on squashed spheres with background gauge fields

Derived explicit partition functions via localization as Coulomb branch integrals

Linked the measure and integrand to Liouville and Toda conformal field theories

Abstract

We study Euclidean 3D N=2 supersymmetric gauge theories on squashed three-spheres preserving isometries SU(2) x U(1) or U(1) x U(1). We show that, when a suitable background U(1) gauge field is turned on, these squashed spheres support charged Killing spinors and therefore N=2 supersymmetric gauge theories. We present the Lagrangian and supersymmetry rules for general gauge theories. The partition functions are computed using localization principle, and are expressed as integrals over Coulomb branch. For the squashed sphere with U(1) x U(1) isometry, its measure and integrand are identified with the building blocks of structure constants in Liouville or Toda conformal field theories with b \neq 1.