Information metric, Berry connection and Berezin-Toeplitz quantization for matrix geometry

Abstract

We consider the information metric and Berry connection in the context of noncommutative matrix geometry. We propose that these objects give a new method of characterizing the fuzzy geometry of matrices. We first give formal definitions of these geometric objects and then explicitly calculate them for the well-known matrix configurations of fuzzy $S^2$ and fuzzy $S^4$. We find that the information metrics are given by the usual round metrics for both examples, while the Berry connections coincide with the configurations of the Wu-Yang monopole and the Yang monopole for fuzzy $S^2$ and fuzzy $S^4$, respectively. Then, we demonstrate that the matrix configurations of fuzzy $S^n$ $(n=2,4)$ can be understood as images of the embedding functions $S^n\to {\text{R}}^{n+1}$ under the Berezin-Toeplitz quantization map. Based on this result, we also obtain a mapping rule for the Laplacian on fuzzy $S^4$.

arXiv comment33 pages

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Cited by in corpus (11)

• Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model
• Landau-Hall States and Berezin-Toeplitz Quantization of Matrix Algebras
• $SO(5)$ Landau Models and Nested Nambu Matrix Geometry
• The matrix regularization for Riemann surfaces with magnetic fluxes
• $SO(5)$ Landau Model and 4D Quantum Hall Effect in The $SO(4)$ Monopole Background
• Laplacians on Fuzzy Riemann Surfaces
• Topological zero-modes of the spectral localizer of trivial metals
• Category of Quantizations and Inverse Problem
• Diffeomorphisms on Fuzzy Sphere
• Quantum matrix geometry in the lowest Landau level and higher Landau levels
• Deformation of Matrix Geometry via Landau Level Evolution