Fourier Decompositions of Loop Bundles

Abstract

In this paper we investigate bundles whose structure group is the loop group LU(n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle $\xi$. This is essentially a decomposition of $\xi$ as $\zeta \otimes L\mathbb C$, where $\zeta$ is a finite dimensional subbundle of $\xi$ and $L\mathbb C$ is the loop space of the complex numbers. The criterion is a reduction of the structure group to the finite rank unitary group U(n) viewed as the subgroup of LU(n) consisting of constant loops. Next we study the case where $\xi$ is the loop space of an $n$ dimensional bundle $\zeta \to M$. The tangent bundle of $LM$ is such a bundle. We then show how to twist such a bundle by elements of the automorphism group of the pull back of $\zeta$ over $LM$ via the map $LM \to M$ that evaluates a loop at a basepoint. Given a connection on $\zeta$, we view the associated parallel transport operator as an element of this gauge group and show that twisting the loop bundle by such an operator satisfies the criterion and admits a Fourier decomposition.