Spin geometry of the rational noncommutative torus

TL;DR

This paper explores the spin geometry of the rational noncommutative torus, revealing its structure through isomorphisms with spectral triples on bundles and subalgebras, and extends results to curved cases.

Contribution

It demonstrates the twined almost commutative structure of the spectral triple on the rational noncommutative torus and constructs explicit spin structures for all cases.

Findings

Isomorphisms with spectral triples on bundles and subalgebras

Explicit construction of all four spin structures

Extension to curved spectral triples via perturbation

Abstract

The twined almost commutative structure of the standard spectral triple on the noncommutative torus with rational parameter is exhibited, by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of algebras, and a spectral triple on a certain invariant subalgebra of the product algebra. These isomorphisms intertwine also the grading and real structure. This holds for all four inequivalent spin structures, which are explicitly constructed in terms of double coverings of the noncommutative torus (with arbitrary real parameter). These results are extended also to a class of curved (non flat) spectral triples, obtained as a perturbation of the standard one by eight central elements.