Comment on 'Gauge networks in noncommutative geometry'

TL;DR

This paper discusses the continuum limit of a lattice-based noncommutative geometric model, showing it converges to pure Yang-Mills theory without a Higgs scalar, clarifying the connection between lattice and continuum gauge theories.

Contribution

It demonstrates that the continuum limit of the spectral action for lattice gauge theories in noncommutative geometry is the pure Yang-Mills action, excluding the Higgs component.

Findings

The lattice spectral action converges to the Yang-Mills functional in the continuum limit.

The Higgs scalar is absent in the continuum limit of the model.

Clarifies the relationship between lattice noncommutative geometry and continuum gauge theories.

Abstract

The article (Gauge networks in noncommutative geometry, J. Geom. Phys. 75 : 71--91, 2014) that motivates this comment provides, in particular, one answer to the following natural question: what is noncommutative geometry on a lattice? In the context of spectral triples, Marcolli and van Suijlekom define in op. cit. a Dirac operator on the lattice and identify the corresponding Spectral Action with the lattice Yang-Mills--Higgs system. In this comment we show that the continuum limit of this theory is the Yang-Mills action functional, without a Higgs scalar. R\'esum\'e: Qu'est-ce que la g\'eom\'etrie non-commutative sur r\'eseau ? \`A cette question l'article ici comment\'e (Gauge networks in noncommutative geometry, J. Geom. Phys. 75 : 71--91, 2014) apporte une des r\'eponses possibles. Marcolli et van Suijlekom, travaillant dans le contexte des triplets spectraux, y construisent un op\'erateur de type Dirac pour le r\'eseau et d\'erivent une th\'eorie sur r\'eseau de type Yang-Mills--Higgs \`a partir de l'Action Spectrale. Ce commentaire montre que la limite continue de ce mod\`ele est la th\'eorie pure de Yang-Mills (sans aucun Higgs).