Regularization of the Covariant Derivative on Curved Space by Finite Matrices

Abstract

In a previous paper [M.~Hanada, H.~Kawai and Y.~Kimura, Prog. Theor. Phys. 114 (2005), 1295] it is shown that a covariant derivative on any n-dimensional Riemannian manifold can be expressed in terms of a set of n matrices, and a new interpretation of IIB matrix model, in which the diffeomorphism, the local Lorentz symmetry and their higher spin analogues are embedded in the unitary symmetry, is proposed. In this article we investigate several coset manifolds in this formulation and show that on these backgrounds, it is possible to carry out calculations at the level of finite matrices by using the properties of the Lie algebras. We show how the local fields and the symmetries are embedded as components of matrices and how to extract the physical degrees of freedom satisfying the constraint proposed in the previous paper.