Polarized deformation quantization

TL;DR

This paper establishes a link between polarized deformation quantization on symplectic manifolds and algebraic structures, providing formulas connecting Fedosov classes, Chern classes, and Lie algebra extensions.

Contribution

It introduces a construction of a commutative subalgebra associated with a polarization and relates the algebraic extension class to geometric invariants.

Findings

Existence of a commutative subalgebra isomorphic to functions constant along polarization.

The algebra of elements with specific commutator properties forms a Lie algebra extension.

Derived a formula connecting the Fedosov class, Chern class, and the extension class.

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 $P$.