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

作者：Cacic Branimir^*

来源：Journal of Geometry and Physics, 2015, 98: 82-109.

DOI：10.1016/j.geomphys.2015.07.028

• Noncommutative geometry
• Spectral triple
• Strict deformation quantisation
• Connes-Landi deformation
• Isospectral deformation
• Toric noncommutative manifold

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.

• 出版日期2015-12