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

TL;DR

This paper develops a method to quantize anti de Sitter spaces using universal deformation formulas derived from symplectic symmetric spaces, leading to noncommutative geometries relevant for black hole models.

Contribution

It introduces a novel approach to quantize anti de Sitter spaces via universal deformation formulas based on symplectic symmetric spaces, extending noncommutative geometry techniques.

Findings

Constructed a universal deformation formula for Lie subgroups on symplectic symmetric spaces.

Defined Dirac-isospectral noncommutative deformations of anti de Sitter black hole spectral triples.

Linked anti de Sitter geometry to symplectic symmetric spaces through curvature contraction.

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.