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 $S^4$.
Highlights
In the matrix models for string and M- theories [1,2], geometry of fundamental objects such as strings and membranes are described in terms of some Hermitian matrices Xμ, which correspond to the quantized version of the embedding functions
We demonstrate that the information metric for these spaces are given by the ordinary round metric, while the Berry connections are given by the configurations of the Wu-Yang monopole and Yang monopole for fuzzy S2 and S4, respectively
We developed the notion of the information metric and Berry connection in the context of the matrix geometry
Summary
In the matrix models for string and M- theories [1,2], geometry of fundamental objects such as strings and membranes are described in terms of some Hermitian matrices Xμ, which correspond to the quantized version of the embedding functions. In this paper, we try to extract the geometric data from a given set of Hermitian matrices Xμ which define a fuzzy space This problem should be important in studying the matrix models, which are formulated completely in the language of matrices. Under the Berezin-Toeplitz quantization map, the defining Hermitian matrices for the fuzzy S2 and S4 can be seen as the images of the embedding functions of S2 and S4, respectively, into the flat target spaces. This result provides a unified viewpoint for fuzzy S2 and fuzzy S4.
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