Fourier Decompositions of Loop Bundles

In this paper we investigate bundles whose structure group is the loop group LU(n). These bundles are classified by maps to the loop space of the classifying space, LBU(n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle ξ. This is essentially a decomposition of ξ as ζ⊗LC, where ζ is a finite dimensional subbundle of ξ and LC is the space of functions, C∞(S1,C). 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 one starts with an n dimensional bundle ζ → M classified by a map f : M → BU(n) from which one constructs a loop bundle Lζ → LM classified by Lf : LM → LBU(n). 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 ζ over LM via the map LM → M that evaluates a loop at a basepoint. Given a connection on ζ, 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.