Describing Curved Spaces by Matrices

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 defined 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([Aa, Ab][Aa, Ab]).