Odd characteristic classes in entire cyclic homology and equivariant loop space homology

TL;DR

This paper constructs an odd Chern character in equivariant cyclic homology for smooth maps on a compact manifold, linking K-theory to homology via a new algebraic framework.

Contribution

It introduces a novel odd Chern character in equivariant cyclic homology and establishes its relation to K^{-1} theory and the Bismut-Chern character.

Findings

Defined a Chern character in the equivariant cyclic complex.

Proved the map induces a group homomorphism from K^{-1}(M) to homology.

Connected the new construction to existing Bismut-Chern character.

Abstract

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$