A cohomological construction of modules over Fedosov deformation quantization algebra

TL;DR

This paper constructs a Fedosov-type star-product on symplectic manifolds that naturally induces module structures on functions over leaves of a polarization, advancing geometric quantization methods.

Contribution

It introduces a cohomological method to build modules over Fedosov deformation algebras, including star-products with separation of variables under specific conditions.

Findings

Constructed a Fedosov-type star-product in neighborhoods of symplectic manifolds.

Established natural module structures on functions over leaves of a polarization.

Demonstrated conditions under which the star-product has separation of variables.

Abstract

In certain neighborhood $U$ of an arbitrary point of a symplectic manifold $M$ we construct a Fedosov-type star-product $\ast_L$ such that for an arbitrary leaf $\wp$ of a given polarization $\mathcal{D}\subset TM$ the algebra $C^\infty (\wp \cap U)[[h]]$ has a natural structure of left module over the deformed algebra $(C^\infty (U)[[h]], \ast_L)$. With certain additional assumptions on $M$, $\ast_L$ becomes a so-called star-product with separation of variables.