Algebra cochains, the bivariant JLO cocycle and the Mathai-Quillen form

TL;DR

This paper explores the connections between superconnection formalism, cyclic cocycles, and KK-theory, extending the Mathai-Quillen Thom form through a bivariant JLO cocycle to unify various mathematical frameworks.

Contribution

It introduces a novel extension of the Mathai-Quillen Thom form using a bivariant JLO cocycle, linking superconnections, cyclic cohomology, and KK-theory.

Findings

Extended Mathai-Quillen Thom form via bivariant JLO cocycle

Unified superconnection formalism with cyclic cocycles and KK-theory

Proposed perspective of KK-cycles as superconnection forms

Abstract

This is a first investigation by the author of the similarity between Quillen's superconnection formalism, his constructions of (periodic) cyclic cocycles via algebra cochains on a bar construction, and Kasparov bimodules for KK-theory. In this article, we do so by deriving a slight extension of the Mathai-Quillen Thom form via a bivariant JLO cocycle. The main idea (which is in fact not really new) is that KK-cycles should be thought of as superconnection forms; these methods will be applied to other contexts elsewhere.