Regularization of the Covariant Derivative on Curved Space by Finite Matrices

TL;DR

This paper extends a matrix-based formulation of covariant derivatives on curved spaces, enabling finite matrix calculations for various coset manifolds and clarifying the embedding of local fields and symmetries.

Contribution

It demonstrates how to perform finite matrix calculations on coset manifolds within the matrix formulation of covariant derivatives, elucidating the embedding of local fields and symmetries.

Findings

Finite matrix representations for covariant derivatives on coset manifolds.

Explicit embedding of local fields and symmetries into matrices.

Method to extract physical degrees of freedom from matrix components.

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.