Generalized Berezin quantization, Bergman metrics and fuzzy Laplacians

TL;DR

This paper develops a unified framework for Berezin quantization on compact Kähler manifolds, introduces Berezin-Bergman quantization, and proposes a general method for fuzzy scalar field theories with explicit examples.

Contribution

It introduces Berezin-Bergman quantization, linking it to fuzzy Kähler geometries and defining fuzzy Laplacians and scalar field theories within this framework.

Findings

Berezin-Bergman quantization reproduces fuzzy Kähler space constructions.

Explicit examples of fuzzy Laplacians are provided.

A general definition of fuzzy scalar field theory on compact Kähler manifolds is proposed.

Abstract

We study extended Berezin and Berezin-Toeplitz quantization for compact Kaehler manifolds, two related quantization procedures which provide a general framework for approaching the construction of fuzzy compact Kaehler geometries. Using this framework, we show that a particular version of generalized Berezin quantization, which we baptize "Berezin-Bergman quantization", reproduces recent proposals for the construction of fuzzy Kaehler spaces. We also discuss how fuzzy Laplacians can be defined in our general framework and study a few explicit examples. Finally, we use this approach to propose a general explicit definition of fuzzy scalar field theory on compact Kaehler manifolds.