Marcolli, Matilde and van Suijlekom, Walter D. (2014) Gauge networks in noncommutative geometry. Journal of Geometry and Physics, 75 . pp. 7191. ISSN 03930440. http://resolver.caltech.edu/CaltechAUTHORS:20140109093610920

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Abstract
We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almostcommutative 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 wellknown 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 fourdimensional 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 threedimensional case, we relate the spectral action functional to the Kogut–Susskind Hamiltonian.
Item Type:  Article  

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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 DMS0901221, DMS1007207, DMS1201512, and PHY1205440. The second author is supported in part by the ESF Research Networking Programme "LowDimensional Topology and Geometry with Mathematical Physics (ITGP)". Carlos Perez is acknowledged for a careful reading of the manuscript.  
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Subject Keywords:  Noncommutative geometry; Spin networks; Lattice gauge theory  
Record Number:  CaltechAUTHORS:20140109093610920  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20140109093610920  
Official Citation:  Matilde Marcolli, Walter D. van Suijlekom, Gauge networks in noncommutative geometry, Journal of Geometry and Physics, Volume 75, January 2014, Pages 7191, ISSN 03930440, 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:  30 Nov 2015 17:57 
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