David Ben-Zvi | Electric-Magnetic Duality for Periods and L-functions

2/24/2021 Colloquium

David Ben-Zvi (U Texas)

Title: Electric-Magnetic Duality for Periods and L-functions

Abstract: I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory.
A fundamental aspect of the Langlands correspondence -- the relative Langlands program -- studies the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index the period integrals (spherical varieties for G) and the L-functions (representations of the dual group G^) don't seem to line up.
We present an approach to this problem via the Kapustin-Witten interpretation of the [geometric] Langlands correspondence as electric-magnetic duality for 4-dimensional supersymmetric Yang-Mills theory. Namely, we rewrite the relative Langlands program as duality in the presence of supersymmetric boundary conditions. As a result, the partial correspondence between periods and L-functions is embedded in a natural duality between Hamiltonian actions of the dual groups.