Dualities and loops on squashed $S^3$

TL;DR

This paper studies supersymmetric gauge theories on a squashed three-sphere, analyzing Wilson and vortex loops, their supersymmetry preservation, and the matching of partition functions with mirror duals, including numerical evaluations for ABJM theory.

Contribution

It extends known results from the round sphere to the squashed sphere, demonstrating supersymmetry preservation, partition function matching, and conjecturing equality of ABJM and super-Yang-Mills partition functions.

Findings

Wilson and vortex loops preserve up to four supercharges

Partition functions match between dual theories on squashed sphere

Numerical evidence supports conjectured equality of ABJM and super-Yang-Mills partition functions

Abstract

We consider $\mathcal{N}=4$ supersymmetric gauge theories on the squashed three-sphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values of these $\frac{2}{3}$-BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual super-Yang-Mills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the Lee-Yang zeros and of the non-perturbative corrections to the all order large $N$ expression for the ABJM partition function.