Abstract
We study extended Berezin and Berezin-Toeplitz quantization for compact K?hler manifolds, two related quantization procedures which provide a general framework for approaching the construction of fuzzy compact K?hler 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 K?hler 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 K?hler manifolds.
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