Twisted indices of 3d supersymmetric gauge theories and enumerative geometry of quasi-maps

I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. In the first part of the talk, I will show that the twisted index computes the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface, which can be understood algebraically as quasi-maps to the Higgs branch. I will explain 3d N=4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to these moduli spaces. Finally, I will present a wall-crossing formula for these invariants derived from the gauge theory point of view.