A reconstruction theorem for Connes–Landi deformations of 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 Pontryagin 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.