Spectral triples and differential calculi related to the Kronecker foliation

TL;DR

This paper constructs two spectral triples associated with the Kronecker foliation, exploring their differential calculi and relating them to signature operators, thereby advancing noncommutative geometric analysis.

Contribution

It introduces two novel spectral triples linked to the Kronecker foliation and analyzes their differential calculi, expanding the understanding of noncommutative geometric structures.

Findings

Different differential calculi for the two spectral triples

Relation of Dirac operators to signature operators

Description of calculus on the noncommutative torus

Abstract

Following ideas of Connes and Moscovici, we describe two spectral triples related to the Kronecker foliation, whose generalized Dirac operators are related to first and second order signature operators. We also consider the corresponding differential calculi $\Omega_D$, which are drastically different in the two cases. As a side-remark, we give a description of a known calculus on the two-dimensional noncommutative torus in terms of generators and relations.