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Published October 21, 2010 | Submitted
Journal Article Open

Spin foams and 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 the 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.

Additional Information

© 2010 IOP Publishing. Received 15 May 2010, in final form 30 July 2010. Published 22 September 2010. This work was inspired by and partially carried out during the workshop 'Noncommutative Geometry and Loop Quantum Gravity' at the Mathematisches Forschungsinstitut Oberwolfach, which the first two authors thank for hospitality and support. The second author is partially supported by NSF grants DMS-0651925, DMS-0901221 and DMS-1007207. The first author was partially supported by a Richter Memorial Fund Summer Undergraduate Research Fellowship from Caltech.

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