Background independent geometry and Hopf cyclic cohomology

TL;DR

This paper surveys the development of Hopf cyclic cohomology and its relation to noncommutative geometry, highlighting its potential relevance to background independent approaches in quantum gravity, with added new material for clarity.

Contribution

It provides a detailed overview of Hopf cyclic cohomology's evolution and introduces new insights into its application to transverse index theory and quantum gravity.

Findings

Hopf cyclic cohomology is closely linked to noncommutative index formulas.

The geometric framework is Diff-invariant and background independent.

New material clarifies the structure of Hopf algebras in transverse geometry.

Abstract

This is primarily a survey of the way in which Hopf cyclic cohomology has emerged and evolved, in close relationship with the application of the noncommutative local index formula to transverse index theory on foliations. Being Diff-invariant, the geometric framework that allowed us to treat the `space of leaves' of a general foliation provides a `background independent' set-up for geometry that could be of relevance to the handling of the the background independence problem in quantum gravity. With this potential association in mind, we have added some new material, which complements the original paper and is also meant to facilitate its understanding. Section 2 gives a detailed description of the Hopf algebra that controls the `affine' transverse geometry of codimension $n$ foliations, and Section 5 treats the relative version of Hopf cyclic cohomology in full generality, including the case of Hopf pairs with noncompact isotropy.