JournalsjncgVol. 15, No. 2pp. 615–642

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

Given a compact manifold MM and a smooth map g ⁣:MU(l×l;C)g\colon M\to U(l\times l;\mathbb{C}) from MM to the Lie group of unitary l×ll\times l matrices with entries in C\mathbb{C}, we construct a Chern character Ch(g)\mathrm{Ch}^-(g) which lives in the odd part of the equivariant (entire) cyclic Chen-normalized cyclic complex Nϵ(ΩT(M×T))\mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T})) of MM, and which is mapped to the odd Bismut–Chern character under the equivariant Chen integral map. It is also shown that the assignment gCh(g)g\mapsto \mathrm{Ch}^-(g) induces a well-defined group homomorphism from the K1K^{-1} theory of MM to the odd homology group of Nϵ(ΩT(M×T))\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