$S^1$-equivariant Chern-Weil constructions on loop space

TL;DR

This paper investigates the construction of $S^1$-equivariant characteristic classes on infinite rank bundles over loop spaces, clarifies related cohomology theories, and identifies cases where classes descend to the base space.

Contribution

It introduces a sequence of $S^1$-equivariant characteristic classes on certain bundles over loop space and identifies conditions under which these classes descend to the base.

Findings

Constructed $S^1$-equivariant characteristic classes that do not descend to the base.

Clarified relationships among different $S^1$-equivariant cohomology theories.

Identified bundles for which the $S^1$-equivariant first Chern class descends to $LM$.

Abstract

We study the existence of $S^1$-equivariant characteristic classes on certain natural infinite rank bundles over the loop space $LM$ 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 $LM$. Nevertheless, we conclude by identifying a class of bundles for which the $S^1$-equivariant first Chern class does descend to $LM$.