Super-connections and non-commutative geometry

TL;DR

This paper extends Quillen's superconnection formalism to a broader class of operators, removing spectral gap conditions, and provides new proofs linking superconnection Chern characters with classical definitions.

Contribution

It generalizes the superconnection approach to arbitrary operators with functional calculus, utilizing non-commutative geometry techniques.

Findings

Extended superconnection formalism to all operators with functional calculus.

Removed spectral gap restrictions in index theory.

Provided a new proof of the equivalence of superconnection and classical Chern characters.

Abstract

We show that Quillen's formalism for computing the Chern character of the index using superconnections extends to arbitrary operators with functional calculus. We thus remove the condition that the operators have, up to homotopy, a gap in the spectrum. This is proved using differential graded algebras and non-commutative differential forms. Our results also give a new proof of the coincidence of the Chern character of a difference bundle defined using super-connections with the classical definition.