# Twisted Hilbert Spaces of 3d Supersymmetric Gauge Theories

## TL;DR

This paper explores the structure of supersymmetric ground states in 3d N=2 gauge theories on a line times a Riemann surface, revealing new insights into mirror symmetry through cohomological constructions.

## Contribution

It introduces a cohomological framework for the space of ground states in twisted 3d supersymmetric theories, enabling new tests of mirror symmetry.

## Key findings

- Constructed the space of ground states via cohomology of vortex moduli spaces.
- Demonstrated dependence of ground states on deformation parameters like superpotentials and mass parameters.
- Provided new checks of 3d abelian mirror symmetry.

## Abstract

We study aspects of 3d $\mathcal{N}=2$ supersymmetric gauge theories on the product of a line and a Riemann surface. Performing a topological twist along the Riemann surface leads to an effective supersymmetric quantum mechanics on the line. We propose a construction of the space of supersymmetric ground states as a graded vector space in terms of a certain cohomology of generalized vortex moduli spaces on the Riemann surface. This exhibits a rich dependence on deformation parameters compatible with the topological twist, including superpotentials, real mass parameters, and background vector bundles associated to flavour symmetries. By matching spaces of supersymmetric ground states, we perform new checks of 3d abelian mirror symmetry.