An Asymmetric Noncommutative Torus

TL;DR

This paper introduces a new family of spectral triples for the curved noncommutative two-torus, demonstrating that the Gauss-Bonnet theorem holds in this setting, extending previous results.

Contribution

It constructs a novel family of Dirac operators for the curved noncommutative two-torus and verifies the Gauss-Bonnet theorem in this context.

Findings

Computed the dressed scalar curvature for the new spectral triples

Established the validity of the Gauss-Bonnet theorem in the curved noncommutative setting

Extended the class of noncommutative geometries where classical geometric theorems hold

Abstract

We introduce a family of spectral triples that describe the curved noncommutative two-torus. The relevant family of new Dirac operators is given by rescaling one of two terms in the flat Dirac operator. We compute the dressed scalar curvature and show that the Gauss-Bonnet theorem holds (which is not covered by the general result of Connes and Moscovici).