Compact $\kappa$-deformation and spectral triples

TL;DR

This paper develops discrete models of $$-Minkowski space within noncommutative geometry, using spectral triples linked to discrete groups, achieving finite summability despite previous obstructions.

Contribution

It introduces a novel construction of finitely summable spectral triples for discrete $$-Minkowski spaces using group actions, bypassing known limitations.

Findings

Successfully constructs finitely summable spectral triples for discrete $$-Minkowski models.

Utilizes Baumslag--Solitar groups to achieve spectral triples with arbitrary dimension.

Shows the spectral triple dimension is independent of the number of space coordinates.

Abstract

We construct discrete versions of $\kappa$-Minkowski space related to a certain compactness of the time coordinate. We show that these models fit into the framework of noncommutative geometry in the sense of spectral triples. The dynamical system of the underlying discrete groups (which include some Baumslag--Solitar groups) is heavily used in order to construct \emph{finitely summable} spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation. The dimension of these spectral triples is unrelated to the number of coordinates defining the $\kappa$-deformed Minkowski spaces.