Background independent geometry and Hopf cyclic cohomology

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.

50 pages

References in corpus (1)

• Very Basic Noncommutative Geometry

Cited by in corpus (7)

• A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
• Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology
• Cup products in Hopf cyclic cohomology via cyclic modules I
• A Short Survey of Cyclic Cohomology
• Lectures on Noncommutative Geometry
• Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1
• Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology