Wilson loops and free energies in $3d$ $\mathcal{N}=4$ SYM: exact results, exponential asymptotics and duality

TL;DR

This paper provides exact analytical results for Wilson loops and free energies in 3d $ ext{N}=4$ supersymmetric gauge theories, exploring their asymptotics and dualities using generalized Selberg integrals.

Contribution

It introduces explicit formulas for partition functions and Wilson loops in various representations for 3d $ ext{N}=4$ theories, including asymptotic and duality analyses.

Findings

Exact expressions for Wilson loops in multiple representations.

Explicit free energy formulas for orthogonal and symplectic groups.

Analysis of exponential asymptotics and duality checks.

Abstract

We show that $U(N)$ $3d$ $\mathcal{N}=4$ supersymmetric gauge theories on $S^{3}$ with $N_{f}$ massive fundamental hypermultiplets and with a Fayet-Iliopoulos (FI) term are solvable in terms of generalized Selberg integrals. Finite $N$ expressions for the partition function and Wilson loop in arbitrary representations are given. We obtain explicit analytical expressions for Wilson loops with symmetric, antisymmetric, rectangular and hook representations, in terms of Gamma functions of complex argument. The free energy for orthogonal and symplectic gauge group is also given. The asymptotic expansion of the free energy is also presented, including a discussion of the appearance of exponentially small contributions. Duality checks of the analytical expressions for the partition functions are also carried out explicitly.