On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model

TL;DR

This paper develops a finite-dimensional noncommutative geometric framework to formulate Yang-Mills-Higgs theories as explicit multimatrix models, extending fuzzy geometries and spectral triples to include Higgs fields in a quantized setting.

Contribution

It introduces gauge matrix spectral triples using finite algebras to model Yang-Mills-Higgs theories as multimatrix models within noncommutative geometry.

Findings

Formulation of Yang-Mills-Higgs theory as a multimatrix model.

Extension of fuzzy geometries to include Higgs fields.

Quantization approach via spectral action on fuzzy space.

Abstract

We continue the study of fuzzy geometries inside Connes' spectral formalism and their relation to multimatrix models. In this companion paper to [arXiv 2007:10914, Ann. Henri Poincar\'e] we propose a gauge theory setting based on noncommutative geometry, which -- just as the traditional formulation in terms of almost-commutative manifolds -- has the ability to also accommodate a Higgs field. However, in contrast to "almost-commutative manifolds", the present framework employs only finite dimensional algebras which we call gauge matrix spectral triples. In a path-integral quantization approach to the Spectral Action, this allows to state Yang-Mills--Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here, whose matrix fields mirror those of the Yang-Mills--Higgs theory on a smooth manifold.