Abstract: I will discuss some ongoing work (with Mat Bullimore, Davide Gaiotto, and Justin Hilburn) on 3d gauge theories with half-maximal supersymmetry ("N=4"). These theories are labelled by a compact group G and a quaternionic representation R. From this data one obtains many geometric objects. Among them are the "Higgs branch" of vacua (a hyperkahler quotient R///G), the "Coulomb branch" of vacua (a modification of the cotangent bundle of the dual complex torus T*(T')), deformation quantizations of these spaces, and Fukaya-like categories of modules for the quantizations. These various objects and their relations generalize many constructions in geometric representation theory, including a phenomenon called symplectic duality (studied by Braden-Licata-Proudfoot-Webster, generalizing the Koszul duality of Beilinson-Ginzburg-Soergel) and a finite version of the AGT correspondence (by Braverman-Feigin-Finkelberg-Rybnikov).

I will give an introduction to some of these ideas.