On the phase form of a deformation quantization with separation of variables

TL;DR

This paper introduces the phase form of a star product with separation of variables on pseudo-Kaehler manifolds, showing it uniquely characterizes such products and exploring how various transformations affect it.

Contribution

It defines the phase form, proves its bijective correspondence with star products with separation of variables, and analyzes the impact of formal parameter changes.

Findings

Phase forms can be arbitrary and parametrize star products.

The cohomology class of a star product equals that of its phase form.

Transformations affect the star product, Berezin transform, and trace density systematically.

Abstract

Given a star product with separation of variables on a pseudo-Kaehler manifold, we obtain a new formal (1,1)-form from its classifying form and call it the phase form of the star product. The cohomology class of a star product with separation of variables equals the class of its phase form. We show that the phase forms can be arbitrary and they bijectively parametrize the star products with separation of variables. We also describe the action of a change of the formal parameter on a star product with separation of variables, its formal Berezin transform, classifying form, phase form, and canonical trace density.