A bivariant Chern character for families of spectral triples

TL;DR

This paper develops a bivariant Chern character for families of spectral triples, extending the JLO cocycle using superconnections and heat kernel techniques within the framework of bornological algebras.

Contribution

It introduces a new bivariant Chern character for spectral triples families, utilizing Quillen's cochains and superconnections, generalizing existing cocycles.

Findings

Constructed a bivariant Chern character for spectral triples families.

Extended the JLO cocycle to a bivariant setting.

Provided a heat kernel regularization approach for traces.

Abstract

In this paper we construct a bivariant Chern character defined on ``families of spectral triples''. Such families should be viewed as a version of unbounded Kasparov bimodules adapted to the category of bornological algebras. The Chern character then takes its values in the bivariant entire cyclic cohomology of Meyer. The basic idea is to work within Quillen's algebra cochains formalism, and construct the Chern character from the exponential of the curvature of a superconnection, leading to a heat kernel regularization of traces. The obtained formula is a bivariant generalization of the JLO cocycle.