Theta functions and arithmetic quotients of loop groups

In this paper, we observe that isomorphism classes ofcertain meterized vector bundles over$\mb{P}^1_{\mb{Z}}-\{0,\i\}$ can be parameterized byarithmetic quotients of loop groups. We construct anasymptotic version of theta functions, which are defined onthese quotients. Then we prove the convergence and extendthe theta functions to loop symplectic groups. We interpretthem as sections of line bundles over aninfinite-dimensional torus, discuss the relations with loopHeisenberg groups, and give an asymptotic multiplicationformula.

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Published 2 May 2012