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

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 $\underline{\mathcal{C}}({\Omega }_{\mathbb{T}}(M\times \mathbb{T}))$ 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 $\underline{\mathcal{C}}({\Omega }_{\mathbb{T}}(M\times \mathbb{T}))$

Growth conditions to the space of differential forms on loop space added; detailed calculations for the equivariant Chen integral map added;