From Monge to Higgs: a survey of distance computations in noncommutative geometry

TL;DR

This survey reviews explicit calculations of Connes distance in noncommutative geometry, exploring its applications in physics, connections to other mathematical areas, and its interpretation as a noncommutative optimal transport metric.

Contribution

It provides a comprehensive overview of explicit Connes distance computations across various noncommutative geometries and discusses their physical and mathematical implications.

Findings

Connes distance relates to the Higgs field in physics.

Comparison of Connes distance with quantum spacetime models.

Links between noncommutative geometry and sub-Riemannian geometry.

Abstract

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, and the comparison of Connes distance with the minimal length that emerges in various models of quantum spacetime. Links with other areas of mathematics are studied, in particular the horizontal distance in sub-Riemannian geometry. The interpretation of Connes distance as a noncommutative version of the Monge-Kantorovich metric in optimal transport is also discussed.