QFT for Mathematicians 2022

I will discuss a close parallel between Gaiotto and Witten's S-duality for supersymmetric boundary conditions in 4d N=4 SYM and the relative Langlands program, an enhancement of the Langlands program that was developed to provide a framework for the theory of integral representations of L-functions. A special and conjecturally self-dual class of boundary conditions is provided by quantizations of "small" or "multiplicity-free" hamiltonian spaces called hyperspherical varieties. I'll explain how a hyperspherical variety produces objects of interest in all the different settings of the Langlands program (local / global, geometric / arithmetic) and a collection of conjectures providing S-dual descriptions of these objects. The talk is based on forthcoming joint work with Yiannis Sakellaridis and Akshay Venkatesh.

We present a series of (partly proven) conjectures
describing geometric realizations of
categories of (finite-dimensional) representations of quantum
super-groups U_q(g) corresponding
to Lie super-algebras g with reductive even part and a non-degenerate
invariant form.
We shall also discuss the meaning of these conjectures from the point
of view of local geometric Langlands correspondence as well as a
connection to the work of Ben-Zvi, Sakellaridis and Venkatesh.
Based on joint works with M.Finkelberg, V.Ginzburg and R.Travkin as
well as the work of R.Travkin and R.Yang.