Formal Symplectic Groupoid of a Deformation Quantization

Abstract

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 Kähler-Poisson manifold.

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

“…morphisms preserving a symplectic structure into a symplectic structure). In such a case, the author of [19] had to work with symplectic groupoids and introduce contravariant connections which modified substantially the Fedosov scheme of quantization.…”

“…One says that certain geometric objects are defined on T M (or T * M ) in N-adapted form [equivalently, in distinguished form, in brief, d-form] if they are given by coefficients defined with respect to frames e α (16) and coframes e α (18) and their tensor products (with respect to frames * e α (17) and coframes * e α (19) and their tensor products). We shall use "boldface" letters in order to emphasize that certain spaces (or geometric objects) are in N-adapted form.…”

“…The aim of this work is to show how Karabegov's approach to Fedosov deformation quantization [19,35,36] can be naturally extended for almost Kähler manifolds endowed with canonical geometric structures generated by semi-Riemannian and/or Einstein metrics and Lagrange-Finsler and Hamilton-Cartan fundamental generating functions. This paper is motivated by the following results: In Refs.…”

Fedosov’s formal symplectic groupoids and contravariant connections

“…Thus, according to [10], there exists a formal symplectic groupoid on (T * M , Z) whose source mapping is S and target mapping is T . We call it Fedosov's formal symplectic groupoid.…”

“…On the one hand, it is known that deformation quantizations with separation of variables (also known as deformation quantizations of the Wick type, see [7] and [1]) are a particular case of Fedosov's deformation quantizations (see [12]). On the other hand, it was shown in [10] that the corresponding formal symplectic groupoids 'with separation of variables' can be naturally extended from Kähler manifolds to Kähler-Poisson manifolds, while it is impossible to extend the star-products with separation of variables to the Kähler-Poisson manifolds in a naive direct way (see [8]). In this paper we show that the construction of the formal symplectic groupoids of Fedosov's deformation quantizations from [9] can be naturally extended to the Poisson manifolds endowed with a torsion-free Poisson contravariant connection.…”