Speaker:  Vasily Krylov (Harvard University)

Abstract: Higgs and Coulomb branches of quiver gauge theories form two important families of Poisson varieties that are expected to be exchanged under so-called 3D mirror symmetry. One important approach to studying modules over quantized Coulomb branches is by analyzing their graded traces. Graded traces generalize the notion of characters and are closely related to the q-characters introduced by Frenkel and Reshetikhin. Any graded trace defines a solution of the D-module of graded traces introduced by Kamnitzer, McBreen, and Proudfoot. In this series of talks, I will discuss various techniques that allow us to explicitly compute characters and graded traces of certain important modules over quantized Coulomb branches, as well as shifted Yangians and affine quantum affine groups. These techniques combine geometry, representation theory, and analytic methods. I will explain how some of these results naturally appear on the Higgs side, leading to an explicit description of the D-module of graded traces for a quantized Coulomb branch via the geometry of the Higgs branch. We will prove these results for ADE quivers and formulate explicit conjectures in the general case. We will also discuss how the picture changes when Coulomb branches are replaced by their "multiplicative" versions.