Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces

Abstract

We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Dirac-isospectral noncommutative deformations of the spectral triples of locally anti de Sitter black holes. The study is motivated by physical and cosmological considerations.

24 pages, to appear in Contemporary Mathematics (AMS) in the volume of the proceedings of the conference Poisson 2006 held at Keio Univ (Japan)

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Cited by in corpus (6)

• Exploring the Causal Structures of Almost Commutative Geometries
• Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces
• Locally anti de Sitter spaces and deformation quantization
• The horizon of the BTZ black hole
• Transitive Subgroups of Transvections Acting on Some Symplectic Symmetric Spaces of Ricci Type
• BTZ black hole from the structure of the algebra so(2,n)