Electric-Magnetic Duality between 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 tool in number theory, the relative Langlands program, is centered on the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index these period integrals (spherical varieties for a reductive group G) and the L-functions (representations of the Langlands dual group G^) don't seem to line up, making the search for integral representations somewhat of an art.

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 gauge theory. Namely, we rewrite the relative Langlands program as duality in the presence of 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.

differential geometrymetric geometry

Audience: researchers in the discipline

Western Hemisphere colloquium on geometry and physics

Series comments: Description: Biweekly colloquium of geometers and physicists

This weekly online colloquium features geometers and physicists presenting current research on a wide range of topics in the interface of the two fields. The talks are aimed at a broad audience. They will take place via Zoom on alternate Mondays at 3pm Eastern, noon Pacific, 4pm BRT. Each session features a 60 minute talk, followed by 15 minutes for questions and discussion. You may join the meeting 15 minutes in advance. Questions and comments may be submitted to the moderator via the chat interface during the talk, or presented in person during the Q&A session. These colloquia will be recorded and will be available (linked from the website) asap after the event.