Polarized deformation quantization

Abstract

Let $A$ be a star product on a symplectic manifold $(M,{\omega }_0)$, $\frac{1}{t}[\omega ]$ its Fedosov class, where $\omega$ is a deformation of ${\omega }_0$. We prove that for a complex polarization of $\omega$ there exists a commutative subalgebra, $O$, in $A$ that is isomorphic to the algebra of functions constant along the polarization. Let $F(A)$ consists of elements of $A$ whose commutator with $O$ belongs to $O$. Then, $F(A)$ is a Lie algebra which is an $O$-extension of the Lie algebra of derivations of $O$. We prove a formula which relates the class of this extension, the Fedosov class, and the Chern class of $P$.

Latex2e, 23 pp