Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface

TL;DR

This paper analyzes the supersymmetric ground states of 3d $ =4$ gauge theories on Riemann surfaces, revealing geometric structures and symmetries, and verifying results through index calculations and mirror symmetry.

Contribution

It provides a detailed computation of supersymmetric ground states for 3d $ =4$ theories on Riemann surfaces, connecting them to Higgs branch geometry and symmetry considerations.

Findings

Ground states split into A and B type twist sectors.

Results match twisted index and mirror symmetry predictions.

Ground state spaces are described geometrically via Higgs branch.

Abstract

This paper studies supersymmetric ground states of 3d $\mathcal{N}=4$ supersymmetric gauge theories on a Riemann surface of genus $g$. There are two distinct spaces of supersymmetric ground states arising from the $A$ and $B$ type twists on the Riemann surface, which lead to effective supersymmetric quantum mechanics with four supercharges and supermultiplets of type $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ respectively. We compute the space of supersymmetric ground states in each case, graded by flavour and R-symmetries and in different chambers for real mass and FI parameters, for a large class of supersymmetric gauge theories. The results are formulated geometrically in terms of the Higgs branch geometry. We perform extensive checks of compatibility with the twisted index and mirror symmetry.