Spin Foams and Noncommutative Geometry

TL;DR

This paper extends spin foam formalism to include topological data via branched covers, explores convolution algebras related to topology, and proposes integrating matter with noncommutative geometry frameworks.

Contribution

It introduces a novel approach to encode topology in spin foams using branched covers and develops associated convolution algebras, linking topology, quantum gravity, and noncommutative geometry.

Findings

Construction of convolution algebras from spin networks and spin foams.

Identification of dynamical flows and equilibrium states related to topological invariants.

Proposal for integrating matter using spectral triples in noncommutative geometry.

Abstract

We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry.