Exact SUSY Wilson loops on $S^3$ from $q$-Virasoro constraints

TL;DR

This paper derives an explicit recursive formula for calculating supersymmetric Wilson loop averages in 3d $ abla$2 Yang-Mills-Chern-Simons theory on a squashed sphere, connecting to $q$-Virasoro constraints and enabling computations on the round sphere.

Contribution

It introduces a novel recursive approach for Wilson loop calculations in 3d supersymmetric gauge theories using $q$-Virasoro constraints, applicable to both squashed and round spheres.

Findings

Explicit recursive formula for Wilson loops derived

Factorization of Wilson loops at specific Chern-Simons levels

Results applicable to both squashed and round spheres

Abstract

Using the ideas from the BPS/CFT correspondence, we give an explicit recursive formula for computing supersymmetric Wilson loop averages in 3d $\mathcal{N}=2$ Yang-Mills-Chern-Simons $U(N)$ theory on the squashed sphere $S^3_b$ with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level $\kappa_2$ and Fayet-Illiopoulos parameter $\kappa_1$. For these values of $\kappa_1$ and $\kappa_2$ the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent $D^2 \times S^1$ theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.