Coverings of Spectral Triples

TL;DR

This paper generalizes the classical concept of covering spaces of Riemannian manifolds to the noncommutative setting using spectral triples, providing a framework for noncommutative coverings.

Contribution

It introduces a construction of spectral triples on noncommutative covering spaces, extending classical geometric ideas to noncommutative geometry.

Findings

Established a noncommutative analogue of covering space theory

Constructed spectral triples for noncommutative coverings

Bridged classical and noncommutative geometric concepts

Abstract

It is well-known that any covering space of a Riemannian manifold has the natural structure of a Riemannian manifold. This article contains a noncommutative generalization of this fact. Since any Riemannian manifold with a Spin-structure defines a spectral triple, the spectral triple can be regarded as a noncommutative Spin-manifold. Similarly there is an algebraic construction which is a noncommutative generalization of topological covering. This article contains a construction of spectral triple on the "noncommutative covering space".