## String-Math Seminar: Integrable systems via shifted quantum groups

Seminar | September 9 | 2-3 p.m. | 402 LeConte Hall

Oleksandr Tsymbaliuk, Yale University

Department of Mathematics

In the recent papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of $3d$ $N=4$ quiver gauge theories was proposed, whose quantization is conjecturally described via the so-called shifted Yangians and shifted quantum affine algebras.

The goal of this talk is to explain how both of these shifted algebras provide a new insight towards integrable systems via the RTT realization. In particular, the study of Bethe subalgebras associated to the antidominantly shifted Yangians of $\mathfrak sl(n)$ provides an interesting plethora of integrable systems generalizing the famous Toda and DST systems. As another interesting application, the shifted quantum affine algebras in the simplest case of $\mathfrak sl(2)$ give rise to a new family of $3^{n-2}$ q-Toda systems of $\mathfrak sl(n)$, generalizing the well-known one due to Etingof and Sevostyanov. Time permitted, I will also explain how one can generalize the latter construction to produce exactly $3^{rk(g)-1}$ modified q-Toda systems for any semisimple Lie algebra $\mathfrak g$.

These talks are based on the joint works with M. Finkelberg, R. Gonin and a current project with R. Frassek, V. Pestun.

artamonov@berkeley.edu