S1-equivariant Chern-Weil constructions on loop space

We study the existence of S1-equivariant characteristic classes on certain natural infinite rank bundles over the loop space LM of a manifold M. We discuss the different S1-equivariant cohomology theories in the literature and clarify their relationships. We attempt to use S1-equivariant Chern-Weil techniques to construct S1-equivariant characteristic classes. The main result is the construction of a sequence of S1-equivariant characteristic classes on the total space of the bundles, but these classes do not descend to the base LM. Nevertheless, we conclude by identifying a class of bundles for which the S1-equivariant first Chern class does descend to LM.