Wick type deformation quantization of Fedosov manifolds

TL;DR

This paper develops a coordinate-free approach to Wick-type deformation quantization on Fedosov manifolds, introducing a bilinear form that characterizes the deviation from Weyl quantization and exploring its geometric and cohomological properties.

Contribution

It provides a novel Fedosov-based, coordinate-free definition of Wick-type symbols and analyzes their geometric structure and cohomology, including superextensions and applications.

Findings

Identified the cohomological class matching the Wick-type star-product

Established a geometric framework for Wick-type deformation quantization

Proposed a superextension of the symbol construction

Abstract

A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The main ingredient of this approach is a bilinear symmetric form defined on the complexified tangent bundle of the symplectic manifold and subject to some set of algebraic and differential conditions. It is precisely the structure which describes a deviation of the Wick-type star-product from the Weyl one in the first order in the deformation parameter. The geometry of the symplectic manifolds equipped by such a bilinear form is explored and a certain analogue of the Newlander-Nirenberg theorem is presented. The 2-form is explicitly identified which cohomological class coincides with the Fedosov class of the Wick-type star-product. For the particular case of K\"ahler manifold this class is shown to be proportional to the Chern class of a complex manifold. We also show that the symbol construction admits canonical superextension, which can be thought of as the Wick-type deformation of the exterior algebra of differential forms on the base (even) manifold. Possible applications of the deformed superalgebra to the noncommutative field theory and strings are discussed.