SUSY Gauge Theories on Squashed Three-Spheres

TL;DR

The paper builds a complete framework for 3D ${\cal N}=2$ gauge theories on squashed $S^3$ by introducing a background $U(1)$ gauge field to support charged Killing spinors, enabling SUSY on both familiar ($SU(2)\times U(1)$) and less symmetric ($U(1)\times U(1)$) squashed geometries. Using localization, it computes the partition functions by reducing to a Coulomb-branch integral and a one-loop determinant; for the familiar squashing the result mirrors the round $S^3$, while for the less symmetric case the determinant yields the double sine function $s_b(x)$ with a general squashing parameter $b$, linking the 3D partition function to Liouville/Toda CFT structure constants. The work thus provides a concrete $b$-deformation of 3D-AGT-type correspondences and suggests a path toward analogous deformations on $S^4$. These findings illuminate the role of geometry and background fields in shaping SUSY partition functions and their CFT interpretations, with potential extensions to more general dualities and higher-dimensional analogs.

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.