Generalized Berezin quantization, Bergman metrics and fuzzy Laplacians

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.

65+1 pages, JHEP style, version to appear in JHEP

References in corpus (3)

• Calabi-Yau Metrics for Quotients and Complete Intersections
• Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds
• Fuzzy Toric Geometries

Cited by in corpus (13)

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• Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras
• The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP^n
• Groupoid Quantization of Loop Spaces
• Calabi-Yau CFTs and Random Matrices
• The Phase Diagram of Scalar Field Theory on the Fuzzy Disc
• Generalized Berezin-Toeplitz quantization of Kaehler supermanifolds
• Eigenvalues and eigenforms on Calabi-Yau threefolds
• Berezin quantization of noncommutative projective varieties
• Magnetic Domains
• Fuzzy Scalar Field Theory as Matrix Quantum Mechanics
• Branes, Quantization and Fuzzy Spheres