Generalized Berezin quantization, Bergman metrics and fuzzy laplacians

We study extended Berezin and Berezin-Toeplitz quantization for compact Kähler manifolds, two related quantization procedures which provide a general framework for approaching the construction of fuzzy compact Kähler 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 Kähler 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 Kähler manifolds. © 2008 SISSA