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

TL;DR

This paper links S-duality generating functions with Eisenstein series for Kac-Moody groups, establishing their elliptic properties and providing explicit formulas for special cases and universal functions.

Contribution

It introduces a refined geometric function involving G-bundles with parabolic structures, proves its elliptic behavior, and computes explicit Eisenstein series for affine Kac-Moody groups.

Findings

Proved a functional equation for the geometric function E under the affine Weyl group.

Calculated Eisenstein-Kac-Moody series explicitly for P^1, relating it to Kac-Moody characters.

Derived an explicit formula for the universal blowup function for simply connected 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.