[For zoom link please email s.nagyATqmul.ac.uk There will be a pre-seminar for students at 13:30] I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. I will show that the twisted index reproduces the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface. I will also discuss 3d N = 4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to the moduli space of vortices. Finally, I will comment on level structures and a wall-crossing formula of the twisted indices derived from the gauge theory point of view.

I discuss the dependence of supersymmetric partition functions on continuous parameters for the flavour symmetry group. In the presence of the 't Hooft anomalies, the supersymmetric Ward identities imply that the partition function computed in the Wess-Zumino gauge has a non-holomorphic dependence on the flavour parameters. I show this explicitly for a large class of 4d N=1 partition functions on half-BPS four manifolds. I propose a new expression for the partition functions on M3 x S1, which differs from earlier holomorphic results by a non-holomorphic Casimir pre-factor.