Spin Foams and Noncommutative Geometry

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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.
(Based on joint work with Domenic Denicola and Ahmad Zainy al-Yasry)