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
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
1 space-time dimensions (n
2) the corresponding AdSn groups are 80(2, n).
Interestingly these are the conformal groups of flat (or AdS)
(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