Locally anti de Sitter spaces and deformation quantization

TL;DR

This paper defines a new class of black holes in anti de Sitter spaces using group orbits, and performs a strict quantization of these black holes, including the computation of their Dirac operators.

Contribution

It introduces a novel geometric construction of black holes in anti de Sitter space and provides a rigorous group-theoretic quantization framework with explicit matrix decompositions.

Findings

Defined BTZ black holes via Iwasawa orbits in anti de Sitter space

Performed strict group action quantization of these black holes

Computed Dirac operators for the quantized black hole models

Abstract

In the first part we define a "BTZ" black hole in anti de Sitter space in any dimension by defining as "singular" the closed orbits of the Iwasawa component of SO(2,n). In the second part, a strict quantization of the black hole by action of group is performed and its Dirac operator is computed. We introduce, in the appendix, most of the notions about homogeneous spaces and Iwasawa decompositions that are needed. Explicit matricial decompositions are given for every Lie algebra that will be used in the thesis: sl(2,R), so(1,n), so(2,n), sl(2,C) and sp(2,R).