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Spin foams and noncommutative geometry

Denicola, Domenic and Marcolli, Matilde and Zainy al-Yasry, Ahmad (2010) Spin foams and noncommutative geometry. Classical and Quantum Gravity, 27 (20). Art. No. 205025. ISSN 0264-9381. https://resolver.caltech.edu/CaltechAUTHORS:20101012-141717687

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1088/0264-9381/27/20/205025 DOIArticle
http://iopscience.iop.org/0264-9381/27/20/205025/PublisherArticle
http://stacks.iop.org/CQG/27/205025PublisherArticle
http://arxiv.org/abs/1005.1057arXivDiscussion Paper
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.
Funders:
Funding AgencyGrant Number
NSFDMS-0651925
NSFDMS-0901221
NSFDMS-1007207
Caltech Summer Undergraduate Research Fellowship (SURF)UNSPECIFIED
Mathematisches Forschungsinstitut OberwolfachUNSPECIFIED
Richter Memorial FundsUNSPECIFIED
Issue or Number:20
Classification Code:PACS: 02.40.Gh, 04.60.−m. MSC: 18B40; 14A22; 83C45; 83C65.
Record Number:CaltechAUTHORS:20101012-141717687
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20101012-141717687
Official Citation:Domenic Denicola et al 2010 Class. Quantum Grav. 27 205025 doi: 10.1088/0264-9381/27/20/205025
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20410
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Nov 2010 00:35
Last Modified:03 Oct 2019 02:09

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