Twisted indices of 3d $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps

Abstract

Abstract We explore the geometric interpretation of the twisted index of 3d $$ \mathcal{N} $$ N = 4 gauge theories on S 1 × Σ where Σ 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 solutions to generalised vortex equations on Σ, 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 $$ \mathcal{N} $$ 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.