  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.
