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 $S^4$.

33 pages