Partition Function of $N=2$ Gauge Theories on a Squashed $S^4$ with $SU(2)\times U(1)$ Isometry

TL;DR

This paper computes the partition function of N=2 supersymmetric gauge theories on a family of squashed 4-spheres with SU(2)×U(1) symmetry, showing it remains invariant under various squashing deformations.

Contribution

It determines the conditions for supersymmetry on squashed 4-spheres and demonstrates the partition function's independence from squashing parameters using localization.

Findings

Partition function is independent of squashing functions.

Supersymmetry conditions are established for the background.

Localization technique is used to compute the partition function.

Abstract

We study $N=2$ supersymmetric gauge theories on a large family of squashed 4-spheres preserving $SU(2)\times U(1)\subset SO(4)$ isometry and determine the conditions under which this background is supersymmetric. We then compute the partition function of the theories by using localization technique. The results indicate that for $N=2$ SUSY, including both vector-multiplets and hypermultiplets, the partition function is independent of the arbitrary squashing functions as well as of the other supergravity background fields.