Infinitesimal deformations of a formal symplectic groupoid

Abstract

Given a formal symplectic groupoid $G$ over a Poisson manifold $(M,{\pi }_0)$, we define a new object, an infinitesimal deformation of $G$, which can be thought of as a formal symplectic groupoid over the manifold $M$ equipped with an infinitesimal deformation ${\pi }_0+\epsilon {\pi }_1$ of the Poisson bivector field ${\pi }_0$. The source and target mappings of a deformation of $G$ are deformations of the source and target mappings of $G$. To any pair of natural star products $(\ast ,\tilde{\ast })$ having the same formal symplectic groupoid $G$ we relate an infinitesimal deformation of $G$. We call it the deformation groupoid of the pair $(\ast ,\tilde{\ast })$. We give explicit formulas for the source and target mappings of the deformation groupoid of a pair of star products with separation of variables on a Kaehler- Poisson manifold. Finally, we give an algorithm for calculating the principal symbols of the components of the logarithm of a formal Berezin transform of a star product with separation of variables. This algorithm is based upon some deformation groupoid.

22 pages, the paper is reworked, new proofs are added

References in corpus (5)

• Natural star products on symplectic manifolds and quantum moment maps
• Formal symplectic groupoid
• Formal symplectic groupoid of a deformation quantization
• On Dequantization of Fedosov's Deformation Quantization
• Deformation quantization of a Kaehler-Poisson structure vanishing on a Levi nondegenerate hypersurface

Cited by in corpus (1)

• Deformations of symplectic groupoids