Describing Curved Spaces by Matrices

TL;DR

This paper presents a novel way to describe curved spaces using matrices by mapping covariant derivatives to operators on principal bundles, integrating geometric symmetries into matrix models, and deriving Einstein's equations from matrix model dynamics.

Contribution

It introduces a new interpretation of matrix models where matrices represent operators describing curved spaces, incorporating geometric symmetries and deriving Einstein's equations.

Findings

Matrices can represent covariant derivatives on manifolds.

Diffeomorphism and Lorentz symmetries are included as unitary symmetries.

Einstein's equations emerge from matrix model equations of motion.

Abstract

It is shown that a covariant derivative on any d-dimensional manifold M can be mapped to a set of d operators acting on the space of functions on the principal Spin(d)-bundle over M. In other words, any d-dimensional manifold can be described in terms of d operators acting on an infinite dimensional space. Therefore it is natural to introduce a new interpretation of matrix models in which matrices represent such operators. In this interpretation the diffeomorphism, local Lorentz symmetry and their higher-spin analogues are included in the unitary symmetry of the matrix model. Furthermore the Einstein equation is obtained from the equation of motion, if we take the standard form of the action S=-tr([A_{a},A_{b}][A^{a},A^{b}]).