A reconstruction theorem for Connes-Landi deformations of commutative spectral triples

TL;DR

This paper extends Connes's reconstruction theorem to Connes-Landi deformations of commutative spectral triples, showing their stability under deformation and their near-commutative nature for rational parameters.

Contribution

It introduces an abstract framework for Connes-Landi deformations and proves their well-behaved nature and near-commutative properties under certain conditions.

Findings

Deformation parameter in second group cohomology governs noncommutativity.

Spectral triples are stable under further Connes-Landi deformation.

Rational deformations yield almost-commutative spectral triples.

Abstract

We formulate and prove an extension of Connes's reconstruction theorem for commutative spectral triples to so-called Connes-Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group $G$, also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontrjagin dual of $G$, and then show that such spectral triples are well-behaved under further Connes-Landi deformation, thereby allowing for both quantisation from and dequantisation to $G$-equivariant abstract commutative spectral triples. We then use a refinement of the Connes-Dubois-Violette splitting homomorphism to conclude that suitable Connes-Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense.