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

Abstract

Given a compact manifold $M$ and a smooth map $g:M\to U(l\times l;\mathbb{C})$ from $M$ to the Lie group of unitary $l\times l$ matrices with entries in $\mathbb{C}$, we construct a Chern character ${\mathrm{Ch}}^{-}(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized cyclic complex ${\mathcal{N}}_{ϵ}({\Omega }_{\mathbb{T}}(M\times \mathbb{T}))$ 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 ${\mathcal{N}}_{ϵ}({\Omega }_{\mathbb{T}}(M\times \mathbb{T}))$.