Description
Three dimensional N=4 gauge theories have enjoyed a rich interplay with geometry and representation theory since the 90's, particularly motivated by the discovery of 3d mirror symmetry. An initial prediction of 3d mirror symmetry was the existence of many pairs of hyperkahler cones "M_H" and "M_C" (Higgs and Coulomb branches), with the rank of the hyperkahler isometry group of M_H equal to the dimension of the resolution space of M_C, and vice versa. This may be seen as an analogue of 2d mirror symmetry relating pairs Calabi-Yau manifolds whose Betti numbers are swapped in an appropriate way.
In the last decade, the interplay between the physics of 3d N=4 gauge theories and mathematics has greatly intensified, in part due to the advent of new methods to access the structure of local and extended operators in these 3d QFT's. A broad goal of my lectures will be to review some of the recent developments, especially those coming from mathematics, while connecting them directly to fundamental ideas in 3d physics. I will organize many of these developments within the framework of topological (and occasionally holomorphic) "twists" of 3d N=4 gauge theories, and the extended TQFT's (topological quantum field theories) that they predict to exist. This leads into a major current research goal: defining and proving a top-level "3d homological mirror symmetry" equivalence of 2-categories, that would encompass all other mirror equivalences.
More specifically, I will touch on: * the physical definition of 3d N=4 gauge theories, and their topological twists * algebraic and geometric structures predicted by these twists -- within the framework of 3d TQFT * what we know (and do not know) about 3d mirror symmetry * constructions of 3d Coulomb branches, e.g. in work of Braverman-Finkelberg-Nakajima * holomorphic twists and elliptic stable envelopes (in brief) * a current frontier: braided tensor categories of line operators and 2-categories of boundary conditions * the appearance and utility of vertex operator algebras in describing line operators * algebraic vs. analytic/symplectic geometry in making mathematical sense of physical structure * relations between 3d mirror symmetry, 2d mirror symmetry, and 4d S-duality
