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Gauge networks in noncommutative geometry

Marcolli, Matilde and van Suijlekom, Walter D. (2014) Gauge networks in noncommutative geometry. Journal of Geometry and Physics, 75 . pp. 71-91. ISSN 0393-0440. https://resolver.caltech.edu/CaltechAUTHORS:20140109-093610920

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Abstract

We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C^∗-algebra of gauge network correspondences. Given a representation in the category of spectral triples of a quiver embedded in a spin manifold, we define a discretized Dirac operator on the quiver. We compute the spectral action of this Dirac operator on a four-dimensional lattice, and find that it reduces to the Wilson action for lattice gauge theories and a Higgs field lattice system. As such, in the continuum limit it reduces to the Yang–Mills–Higgs system. For the three-dimensional case, we relate the spectral action functional to the Kogut–Susskind Hamiltonian.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.geomphys.2013.09.002 DOIArticle
http://www.sciencedirect.com/science/article/pii/S039304401300168XPublisherArticle
http://arxiv.org/abs/1301.3480arXivDiscussion Paper
Additional Information:© 2013 Elsevier B.V. Received 26 March 2013. Accepted 1 September 2013. Available online 13 September 2013. The first author is partially supported by NSF grants DMS-0901221, DMS-1007207, DMS-1201512, and PHY-1205440. The second author is supported in part by the ESF Research Networking Programme "Low-Dimensional Topology and Geometry with Mathematical Physics (ITGP)". Carlos Perez is acknowledged for a careful reading of the manuscript.
Funders:
Funding AgencyGrant Number
NSFDMS-0901221
NSFDMS-1007207
NSFDMS-1201512
NSFPHY-1205440
ESF Research Networking ProgrammeUNSPECIFIED
Subject Keywords:Noncommutative geometry; Spin networks; Lattice gauge theory
Record Number:CaltechAUTHORS:20140109-093610920
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20140109-093610920
Official Citation:Matilde Marcolli, Walter D. van Suijlekom, Gauge networks in noncommutative geometry, Journal of Geometry and Physics, Volume 75, January 2014, Pages 71-91, ISSN 0393-0440, http://dx.doi.org/10.1016/j.geomphys.2013.09.002. (http://www.sciencedirect.com/science/article/pii/S039304401300168X)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:43288
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Jan 2014 17:54
Last Modified:03 Oct 2019 06:06

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