Berezin-Toeplitz quantization for compact Kaehler manifolds. A Review of Results

TL;DR

This review paper summarizes the key concepts, results, and mathematical structures of Berezin-Toeplitz quantization on compact Kähler manifolds, emphasizing semi-classical limits and deformation quantization.

Contribution

It provides a comprehensive overview of Berezin-Toeplitz operators, deformation quantization, and related symbols, consolidating foundational and recent results in the field.

Findings

Clarification of semi-classical limit behavior

Uniqueness of Berezin-Toeplitz deformation quantization

Discussion of covariant and contravariant Berezin symbols

Abstract

This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kaehler manifolds. The basic objects, concepts, and results are given. This concerns the correct semi-classical limit behaviour of the operator quantization, the unique Berezin-Toeplitz deformation quantization (star product), covariant and contravariant Berezin symbols, and Berezin transform. Other related objects and constructions are also discussed.