A Fedosov Star Product of Wick Type for K\"ahler Manifolds

TL;DR

This paper introduces a Fedosov star product of Wick type tailored for K"ahler manifolds, extending the concept from complex Euclidean spaces and providing an existence proof for these structures.

Contribution

It generalizes the Wick type star product to all K"ahler manifolds using a Fedosov type construction, expanding the applicability of deformation quantization.

Findings

Defined a star product of Wick type on K"ahler manifolds.

Proved the existence of such star products for any K"ahler manifold.

Analyzed properties like symmetry and differentiation order of the Fedosov star product.

Abstract

In this letter we compute some elementary properties of the Fedosov star product of Weyl type, such as symmetry and order of differentiation. Moreover, we define the notion of a star product of Wick type on every K\"ahler manifold by a straight forward generalization of the corresponding star product in $\mathbb C^n$: the corresponding sequence of bidifferential operators differentiates its first argument in holomorphic directions and its second argument in antiholomorphic directions. By a Fedosov type procedure we give an existence proof of such star products for any K\"ahler manifold.