# Twisted Indices of 3d ${\mathcal N} = 4$ Gauge Theories and Enumerative Geometry of Quasi-Maps

## TL;DR

This paper connects the twisted index of 3d N=4 gauge theories on a Riemann surface to the enumerative geometry of quasi-maps, revealing geometric and mirror symmetry insights through localization and algebraic interpretation.

## Contribution

It establishes a geometric interpretation of the twisted index as a virtual Euler characteristic of moduli spaces of quasi-maps and explores mirror symmetry implications.

## Key findings

- Twisted index equals the virtual Euler characteristic of quasi-map moduli spaces.
- Localization reduces the path integral to moduli spaces of vortex equations.
- Mirror symmetry exchanges enumerative invariants of mirror Higgs branches.

## Abstract

We explore the geometric interpretation of the twisted index of 3d ${\mathcal N} =4$ gauge theories on $S^1\times \Sigma$ where $\Sigma$ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d ${\mathcal N} = 4$ mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters.