Theta functions and arithmetic quotients of loop groups

TL;DR

This paper explores the connection between metrized vector bundles over the projective line minus two points and arithmetic quotients of loop groups, introducing asymptotic theta functions and their extensions to loop symplectic groups.

Contribution

It constructs and analyzes asymptotic theta functions on arithmetic quotients of loop groups, extending them to loop symplectic groups and interpreting them as sections over infinite dimensional tori.

Findings

Proves convergence of the asymptotic theta functions

Extends theta functions to loop symplectic groups

Provides an asymptotic multiplication formula

Abstract

In this paper we observe that isomorphism classes of certain metrized vector bundles over P^1-{0,infinity} can be parameterized by arithmetic quotients of loop groups. We construct an asymptotic version of theta functions, which are defined on these quotients. Then we prove the convergence and extend the theta functions to loop symplectic groups. We interpret them as sections of line bundles over an infinite dimensional torus, discuss the relations with loop Heisenberg groups, and give an asymptotic multiplication formula.