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

• Published in: Journal of noncommutative geometry Vol. 15; no. 2; pp. 615 - 642
• Main Authors: Cacciatori, Sergio L., Güneysu, Batu
• Format: Journal Article
• Language: English
• Published: European Mathematical Society Publishing House 01.01.2021
• Subjects: Homology theory (Mathematics); Integrals; Mappings (Mathematics); Mathematical research; Italy
• ISSN: 1661-6952; 1661-6960
• DOI: 10.4171/jncg/406

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})) .