Symmetric Spaces and Star representations II : Causal Symmetric Spaces

We construct and identify star representations canonically associated with holonomy reducible simple symplectic symmetric spaces. This leads the a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley type and Hermitian symmetric spaces of tube type.Comment: 13 page

Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces

Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered non-formal quantization. This note illustrates and partially reviews through an example a problematic studied by the author regarding non-formal quantization in presence of large groups of symmetries

Regular Poisson structures on massive non-rotating BTZ black holes

We revisit the non-rotating massive BTZ black hole within a pseudo-Riemannian symmetric space context. Using classical symmetric space techniques we find that every such space intrinsically carries a regular Poisson structure whose symplectic leaves are para-hermitian symmetric surfaces. We also obtain a global expression of the metric yielding a dynamical description of the black hole from its initial to its final singularity.Comment: LaTex, 18 pages, 3 figures, version published in Nucl. Phys.

Global geometry of the 2+1 rotating black hole

The generic rotating BTZ black hole, obtained by identifications in AdS3 space through a discrete subgroup of its isometry group, is investigated within a Lie theoretical context. This space is found to admit a foliation by two-dimensional leaves, orbits of a two-parameter subgroup of SL(2,R) and invariant under the BTZ identification subgroup. A global expression for the metric is derived, allowing a better understanding of the causal structure of the black hole.Comment: 9 pages, 1 figur

The deformation quantizations of the hyperbolic plane

We describe the space of (all) invariant deformation quantizations on the hyperbolic plane as solutions of the evolution of a second order hyperbolic differential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a unified method producing every quantization of the hyperbolic plane, and provides, in the 2-dimensional context, an exact solution to Weinstein's WKB quantization program within geometric terms. The construction reveals the existence of a metric of Lorentz signature canonically attached (or `dual') to the geometry of the hyperbolic plane through the quantization process.Comment: 26 pages, 5 figure

Deformations of quantum field theories on de Sitter spacetime

Quantum field theories on de Sitter spacetime with global U(1) gauge symmetry are deformed using the joint action of the internal symmetry group and a one-parameter group of boosts. The resulting theory turns out to be wedge-local and non-isomorphic to the initial one for a class of theories, including the free charged Dirac field. The properties of deformed models coming from inclusions of CAR-algebras are studied in detail.Comment: 26 pages, no figure