The aim of this talk is to give a mathematical definition of the superconformal index of 3d supersymmetric gauge theories. This can be computed exactly using supersymmetric localisation, leading to an explicit contour integral formula involving infinite q-Pochammer symbols. I will explain how this may be understood as the Witten index of a supersymmetric quantum mechanics, or index of a twisted Dirac operator on a certain infinite-dimensional space closely related to one introduced by Braverman-Finkelberg-Nakajima.

I will discuss connections between supersymmetric gauge theories in three dimensions and an exciting development in representation theory known as symplectic duality. I will focus on the simplest example of this phenomenon, which arises from a U(1) gauge theory with N hypermultiplets.