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

Understand

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.