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 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.

arXiv commentAMS-LaTeX, 41 pp. V3: Corrected the orientability condition in the definition of theta-commutative spectral triple, including a new appendix with a proof of consistency with the relevant commutative case

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Cited by in corpus (3)

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• Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds
• Spin geometry of the rational noncommutative torus