  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.
