Gauge networks in noncommutative geometry

TL;DR

This paper introduces gauge networks as a unifying framework extending spin networks and lattice gauge theories within noncommutative geometry, providing new models, Hamiltonians, and connections to known physical theories.

Contribution

It develops the concept of gauge networks in noncommutative geometry, explores their properties, and links spectral actions to lattice gauge theories and the Yang-Mills-Higgs system.

Findings

Gauge networks form an orthonormal basis in a Hilbert space.

Spectral action reduces to Wilson action and Higgs lattice system.

In 3D, spectral action relates to Kogut-Susskind Hamiltonian.

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.