Contemporary Mathematics

Volume 450, 2008

Quantized Anti de Sitter spaces and non-formal

deformation quantizations of symplectic symmetric spaces

Pierre Bieliavsky, Laurent Claessens, Daniel Sternheimer,

and Yannick Voglaire

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 os-

cillatory integral kernel on an appropriate symplectic symmetric space. More

precisely we first obtain a UDF for Lie subgroups acting on a symplectic sym-

metric space Min a locally simply transitive manner. Then, observing that

a curvature contraction canonically relates anti de Sitter geometry to the ge-

ometry 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.

1.

Introduction

1.1.

Physical and cosmological motivations. This paper, of independent

interest in itself, can also be seen as a small part in a number of long haul programs

developed by many in the past decades, with a variety of motivations. The refer-

ences that follow are minimal and chosen mostly so as to be a convenient starting

point for further reading, that includes the original articles quoted therein.

An obvious fact (almost a century old) is that anti de Sitter (AdS) space-

time can be obtained from usual Minkowski space-time, deforming it by allowing

a (small) non-zero negative curvature. The Poincare group symmetry of special

relativity is then deformed (in the sense of [Ger64]) to the AdS group 80(2, 3). In

n

+

1 space-time dimensions (n

2:

2) the corresponding AdSn groups are 80(2, n).

Interestingly these are the conformal groups of flat (or AdS)

n

space-times. The

deformation philosophy

(Fla82] makes it then natural, in the spirit of deformation

quantization [DS02], to deform these further [St05, Sto7], i.e. quantizing them,

which many are doing for Minkowski space-time.

2000 Mathematics Subject Classification. 81S10; 53D50, 58B34, 81V25, 83057.

Key words and phroses. Non commutative geometry, BTZ-spaces, Deformation quantization,

Symplectic symmetric spaces.

The first author thanks Victor Gayral for enlightening discussions on noncommutative spec-

tral triples. This research is partially supported by the lAP grant "NOSY'' at UCLouvain.

Laurent Claessens is a FRIA-fellow.

@2008 The Authors

http://dx.doi.org/10.1090/conm/450/08731