The noncommutative Lorentzian cylinder as an isospectral deformation

TL;DR

This paper introduces a noncommutative version of the Euclidean cylinder via isospectral deformation, explores Connes' character formula, and extends the framework to Lorentzian manifolds using Krein spaces, analyzing their symmetries.

Contribution

It provides the first example of a finite-dimensional noncommutative cylinder and extends spectral triple concepts to Lorentzian geometry with Krein spaces.

Findings

Constructed a noncommutative cylinder through isospectral deformation.

Analyzed Connes' character formula for the noncommutative cylinder.

Identified all admissible fundamental symmetries for the noncommutative Lorentzian cylinder.

Abstract

We present a new example of a finite-dimensional noncommutative manifold, namely the noncommutative cylinder. It is obtained by isospectral deformation of the canonical triple associated to the Euclidean cylinder. We discuss Connes' character formula for the cylinder. In the second part, we discuss noncommutative Lorentzian manifolds. Here, the definition of spectral triples involves Krein spaces and operators on Krein spaces. A central role is played by the admissible fundamental symmetries on the Krein space of square integrable sections of a spin bundle over a Lorentzian manifold. Finally, we discuss isospectral deformation of the Lorentzian cylinder and determine all admissible fundamental symmetries of the noncommutative cylinder.