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

Abstract

This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kähler 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.

Twist star products and Morita equivalence

“…From a mathematical point of view, fuzzy spaces are strict deformation quantizations of coadjoint orbits of connected compact semisimple Lie groups, obtained via covariant Berezin quantization (see, e.g., [26]). Alternatively, since on any such orbit there is a canonical invariant Kähler structure (see, e.g., [27]), they can also be obtained via Berezin-Toeplitz quantization [28]. It was shown by Schlichenmaier in [29] (see [30] for the original reference in German), using some estimates of [7], that one can associate a natural star product with the Berezin-Toeplitz quantization of any compact Kähler (hence symplectic) manifold, such as CP n−1 .…”