On Dequantization of Fedosov's Deformation Quantization

To each natural deformation quantization on a Poisson manifold M we associate a Poisson morphism from the formal neighborhood of the zero section of the cotangent bundle to M to the formal neighborhood of the diagonal of the product M x M~, where M~ is a copy of M with the opposite Poisson structure. We call it dequantization of the natural deformation quantization. Then we "dequantize" Fedosov's quantization.Comment: 16 pages, latex; references, terminology, notation, and several typos corrected; to appear in "Letters in Math. Phys.

Formal symplectic groupoid of a deformation quantization

We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid `with separation of variables' over an arbitrary Kaehler-Poisson manifold.Comment: 41 page, Lemma 13, several typos and notations correcte

Fedosov's formal symplectic groupoids and contravariant connections

Using Fedosov's approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kaehler-Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.Comment: 29 page

On Fedosov's approach to Deformation Quantization with Separation of Variables

It was shown in our earlier paper that the deformation quantizations with separation of variables on a Kaehler manifold are parametrized by the formal deformations of the Kaehler form. The Fedosov star-product of Wick type constructed by M. Bordemann and S. Waldmann is proven to coincide with the star-product with separation of variables which corresponds to the trivial deformation of the Kaehler form. To this end a formal Fock bundle on a Kaehler manifold is introduced and an associative multiplication on its sections is defined

Almost Kähler Deformation Quantization

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kähler manifold which respects the almost-Kähler structure to construct a Fedosov-type deformation quantization on this manifold