Fourier decompositions of loop bundles

Abstract

Abstract. 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 ofThe 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 byLf : 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.

Finite-dimensional subbundles of loop bundles

“…In [Cohen and Stacey 2004], the bundle ψ is referred to as the underlying (finitedimensional) vector bundle associated to ξ .…”

“…This work began as an elaboration of the work in [Cohen and Stacey 2004]. The author thanks Ralph Cohen for helpful conversations, and also Persi Diaconis and Dan Bump for their help at stages of the work.…”

“…Following [Cohen and Stacey 2004], we define a rank-n loop bundle to be a vector bundle ξ → X with fibre L‫ރ‬ n and structure group LGL n . We will find it convenient to assume a reduction of the structure group to LU n , which is equivalent to choosing a fibrewise inner product on ξ isomorphic in charts to the standard inner product on L‫ރ‬ n .…”

“…The main example of a loop bundle is the tangent bundle of the free loop space of a smooth finite-dimensional manifold M. In [Morava 2001] and [Cohen and Stacey 2004], the question arose as to the existence of a circle equivariant subbundle of TL M nonequivariantly isomorphic to the pull back of T M via an evaluation map. In [Cohen and Stacey 2004], the existence of such a bundle was found to be a strong condition, one of the implications being that the tangent bundle of the based loop space is parallelisable.…”

“…In [Cohen and Stacey 2004], the existence of such a bundle was found to be a strong condition, one of the implications being that the tangent bundle of the based loop space is parallelisable. In this paper we consider a more general setting but the conclusion stays the same.…”