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

### Sergio L. Cacciatori

Università dell’Insubria, Como, Italy; INFN, Milano, Italy### Batu Güneysu

Universität Bonn, Germany

## Abstract

Given a compact manifold $M$ and a smooth map $g\colon M\to U(l\times l;\mathbb{C})$ 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 $\mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ 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 $\mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$.

## Cite this article

Sergio L. Cacciatori, Batu Güneysu, Odd characteristic classes in entire cyclic homology and equivariant loop space homology. J. Noncommut. Geom. 15 (2021), no. 2, pp. 615–642

DOI 10.4171/JNCG/406