Curvatures and discrete Gauss-Codazzi equation in (2+1)-dimensional loop quantum gravity

TL;DR

This paper derives the Gauss-Codazzi equation in discrete (2+1)-dimensional loop quantum gravity, enabling the representation of curvatures as operators on spin network states within a triangulated manifold.

Contribution

It introduces a discrete Gauss-Codazzi equation in holonomy and plane-angle forms for (2+1)-dimensional LQG, linking curvature operators to spin network states.

Findings

Derived the Gauss-Codazzi equation in holonomy and plane-angle representations.

Formulated operators for intrinsic and extrinsic curvatures acting on spin networks.

Applied the framework to a triangulated manifold with isosceles tetrahedra.

Abstract

We derive the Gauss-Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss-Codazzi equation for a discrete (2+1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2+1)-dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures.