Wick quantisation of a symplectic manifold

TL;DR

This paper introduces a covariant Wick star-product for symplectic manifolds with two transverse polarisations, providing an explicit Fedosov-based construction and exploring conditions for equivalence with other star-products.

Contribution

It extends Wick quantisation to general symplectic manifolds with a covariant approach and identifies conditions for equivalence with Fedosov and Weyl star-products.

Findings

Wick star-product constructed explicitly on symplectic manifolds

Cohomological obstruction identified for equivalence with Fedosov star-product

In Kähler case, Wick and Weyl star-products are equivalent iff the manifold is Calabi-Yau

Abstract

The notion of the Wick star-product is covariantly introduced for a general symplectic manifold equipped with two transverse polarisations. Along the lines of Fedosov method, the explicit procedure is given to construct the Wick symbols on the manifold. The cohomological obstruction is identified to the equivalence between the Wick star-product and the Fedosov one. In particular in the K\"ahler case, the Wick star-product is shown to be equivalent the Weyl one, iff the manifold is a Calabi-Yau one.