Identification of Berezin-Toeplitz deformation quantization

TL;DR

This paper fully characterizes the deformation quantization derived from Berezin-Toeplitz quantization on compact Kähler manifolds, including explicit formulas for the classifying form and characteristic class.

Contribution

It provides a complete identification and explicit calculation of the deformation quantization associated with Berezin-Toeplitz on arbitrary compact Kähler manifolds.

Findings

Explicit classifying form for the deformation quantization.

Characteristic class of the star-product obtained.

Identification of the opposite star-product as a differential deformation with separation of variables.

Abstract

We give a complete identification of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Kaehler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose classifying form is explicitly calculated. Its characteristic class (which classifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szegoe kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjoestrand.