Comments on twisted indices in 3d supersymmetric gauge theories

TL;DR

This paper explores the exact computation of twisted indices in 3d ${ m extbf{N}=2}$ supersymmetric gauge theories on Riemann surfaces times a circle, revealing insights into Wilson loop algebras, dualities, and mirror symmetry.

Contribution

It provides a unified framework for calculating twisted indices and Wilson loop algebras, and applies these to study dualities and mirror symmetry in 3d supersymmetric theories.

Findings

Exact formulas for twisted indices on ${ m extbf{ extit{ ext{Riemann surfaces}}}}$ times $S^1$.

Derivation of quantum Wilson loop algebra via Bethe equations.

Application of indices to analyze 3d Seiberg dualities and mirror symmetry.

Abstract

We study three-dimensional ${\mathcal N}=2$ supersymmetric gauge theories on ${\Sigma_g \times S^1}$ with a topological twist along $\Sigma_g$, a genus-$g$ Riemann surface. The twisted supersymmetric index at genus $g$ and the correlation functions of half-BPS loop operators on $S^1$ can be computed exactly by supersymmetric localization. For $g=1$, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe equations for the theory on ${\mathbb R}^2 \times S^1$. This also provides a powerful and simple tool to study 3d ${\mathcal N}=2$ Seiberg dualities. Finally, we study A- and B-twisted indices for ${\mathcal N}=4$ supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the $S^2 \times S^1$ twisted indices and the Hilbert series of ${\mathcal N}=4$ moduli spaces.