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

TL;DR

This paper explores the geometric structures of fuzzy spheres using information metrics and Berry connections, linking matrix configurations to classical geometries via Berezin-Toeplitz quantization.

Contribution

It introduces a novel approach to characterize fuzzy geometries through information metrics and Berry connections, and establishes their relation to classical embeddings via quantization.

Findings

Information metrics match round metrics for fuzzy spheres.

Berry connections correspond to Wu-Yang and Yang monopoles.

Matrix configurations relate to classical embeddings through Berezin-Toeplitz quantization.

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 $S^n\rightarrow \textbf{R}^{n+1}$ under the Berezin-Toeplitz quantization map. Based on this result, we also obtain a mapping rule for the Laplacian on fuzzy $S^4$.