December 3: Sasha Tsymbaliuk

Title: Coulomb branches, shifted quantum algebras and modified q-Toda systems.

Abstract:

In the recent series of papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d N=4 quiver gauge theories was proposed (they are symplectic dual to the corresponding well-understood Higgs branches). They can be also realized as slices in the affine Grassmannian and therefore admit a multiplication.

In this talk, we shall discuss the quantizations of these Coulomb branches and their K-theoretic analogues, and the (conjectural) down-to-earth realization of these quantizations via shifted Yangians and shifted quantum affine algebras. Those admit a coproduct quantizing the aforementioned multiplication of slices. In type A, they also act on equivariant cohomology/K-theory of parabolic Laumon spaces.

As another interesting application, the shifted quantum affine algebras in the simplest case of sl(2) give rise to a new family of 3^{n-2} q-Toda systems of sl(n), generalizing the well-known one due to Etingof and Sevostyanov. If time permits, we shall explain how to obtain 3^{rk(g)-1} modified q-Toda systems for any simple Lie algebra g.

This talk is based on the joint works with M. Finkelberg and R. Gonin.