The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups

Abstract

We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for Kac-Moody groups. For a pair consisting of a surface and a curve on it, we consider a refined geometric function E (involving G-bundles with parabolic structures along the curve) which depends both on elliptic and modular variables. We prove a functional equation for E with respect to the affine Weyl group, thus establishing the elliptic behavior. When the curve is P^1, we calculate the Eisenstein-Kac-Moody series explicitly and it turns out to be a certain deformation of an irreducible Kac-Moody character, more precisely, an analog of the Hall-Littlewood polynomial for the affine root system. We also get an explicit formula for the universal blowup function for any simply connected structure group.