Abstract: 

Mirror symmetry for 3d N=4 SUSY QFTs has recently received much attention in geometry and representation theory. Theories within this class give rise to interesting moduli spaces of vacua, whose most relevant components are called the Higgs and Coulomb branches. Nakajima initiated the mathematical study of Higgs branches in the 90s; since then, their geometry has been pivotal in diverse areas of enumerative geometry and geometric representation theory. On the other hand, a mathematically precise definition of the Coulomb branch has only recently been proposed, and its study has started. 

Physically, 3d mirror symmetry is understood as a duality for pairs of theories whose Higgs and Coulomb branches are interchanged. Mathematically, it descends to a number of statements relating invariants attached to the dual sides. One of its key predictions is the identification of dual pairs of elliptic stable envelopes, which are certain topological classes intimately related to elliptic quantum groups. In this talk, I will first explain how to use brane diagrams and Cherkis bow varieties to describe both branches of a type A gauge theory. Then, I will discuss the main ideas behind the proof of mirror symmetry of sable envelopes for bow varieties (joint with Richard Rimanyi). A crucial step of our proof involves the process of ”resolving” large charge branes into multiple smaller ones. This phenomenon turns out to be the geometric counterpart of the algebraic fusion for R-matrices (and its dual). Time permitting, I will also hint at recent work in progress to extend this result to affine type A (with Rimanyi) and prove its enumerative counterpart, known as mirror symmetry of vertex functions (with Dinkins and Rimanyi).