$S^1$-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 $LM$ 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 $LM$.

37 pages

References in corpus (5)

• String geometry vs. spin geometry on loop spaces
• Exotic twisted equivariant cohomology of loop spaces, twisted Bismut-Chern character and T-duality
• The differential topology of loop spaces
• The Geometry of Loop Spaces II: Characteristic Classes
• $KR$-theory of compact Lie groups with group anti-involutions