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- Spin Foams and Noncommutative Geometry

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11020110

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)

©2012 Perimeter Institute for Theoretical Physics