Regularization of matrices in the covariant derivative interpretation of matrix models

TL;DR

This paper develops a regularization method for matrices representing covariant derivatives in matrix models, enabling nonperturbative calculations of curved spacetimes like spheres and tori.

Contribution

It introduces a Berezin-Toeplitz quantization-based regularization for covariant derivatives in two dimensions within matrix models.

Findings

Regularization applied to $S^2$ and $T^2$ cases.

Facilitates numerical simulations of curved spacetimes.

Provides a finite matrix approximation for differential operators.

Abstract

We study regularization of matrices in the covariant derivative interpretation of matrix models, a typical example of which is the type IIB matrix model. The covariant derivative interpretation provides a possible way in which curved spacetimes are described by matrices, which are viewed as differential operators. One needs to regularize the operators as matrices with finite size in order to apply the interpretation to nonperturbative calculations such as numerical simulations. We develop a regularization of the covariant derivatives in two dimensions by using the Berezin-Toeplitz quantization. As examples, we examine the cases of $S^2$ and $T^2$ in details.