Unoriented Spectral Triples

TL;DR

This paper introduces a framework for defining spectral triples on unoriented Riemannian manifolds by leveraging their oriented double covers and extends this concept to noncommutative geometries.

Contribution

It proposes a novel notion of unoriented spectral triples, generalizing the concept to noncommutative spaces via their oriented coverings.

Findings

Unoriented spectral triples can be constructed from oriented double covers.

The approach extends to noncommutative finite-fold coverings.

Provides a new perspective on noncommutative geometry for unoriented spaces.

Abstract

Any oriented Riemannian manifold with a Spin-structure defines a spectral triple, so the spectral triple can be regarded as a noncommutative Spin-manifold. Otherwise for any unoriented Riemannian manifold there is the two-fold covering by oriented Riemannian manifold. Moreover there are noncommutative generalizations of finite-fold coverings. This circumstances yield a notion of unoriented spectral triple which is covered by oriented one.