Berezin-Toeplitz Quantization and Star Products for Compact Kaehler Manifolds

TL;DR

This paper reviews Berezin-Toeplitz quantization and related star products on compact Kähler manifolds, highlighting their equivalence, asymptotic properties, and various construction methods, including graph-based approaches.

Contribution

It provides a comprehensive review of star products on compact Kähler manifolds, emphasizing their equivalence and detailed constructions, including new insights into graph-based methods.

Findings

Berezin-Toeplitz, Berezin, and geometric star products are equivalent.

The Berezin transform's asymptotic expansion is crucial for understanding star products.

Graph-based methods are introduced for constructing and calculating star products.

Abstract

For compact quantizable K\"ahler manifolds certain naturally defined star products and their constructions are reviewed. The presentation centers around the Berezin-Toeplitz quantization scheme which is explained. As star products the Berezin-Toeplitz, Berezin, and star product of geometric quantization are treated in detail. It is shown that all three are equivalent. A prominent role is played by the Berezin transform and its asymptotic expansion. A few ideas on two general constructions of star products of separation of variables type by Karabegov and by Bordemann--Waldmann respectively are given. Some of the results presented is work of the author partly joint with Martin Bordemann, Eckhard Meinrenken and Alexander Karabegov. At the end some works which make use of graphs in the construction and calculation of these star products