S1-equivariant Chern–Weil constructions on loop space

Abstract

We study the existence of S 1 -equivariant characteristic classes on certain natural infinite rank bundles over the loop space L M of a manifold M . We discuss the different S 1 -equivariant cohomology theories in the literature and clarify their relationships. We attempt to use S 1 -equivariant Chern–Weil techniques to construct S 1 -equivariant characteristic classes. The main result is the construction of a sequence of S 1 -equivariant characteristic classes on the total space of the bundles, but these classes do not descend to the base L M . Nevertheless, we conclude by identifying a class of bundles for which the S 1 -equivariant first Chern class does descend to L M .