Generalized Berezin quantization, Bergman metrics and fuzzy Laplacians

Understand

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.