The Torus Universe in the Polygon Approach to 2+1-Dimensional Gravity

TL;DR

This paper investigates the structure of matter-free toroidal spacetime in 2+1-dimensional gravity using 't Hooft's polygon approach, revealing the necessity of multiple polygons and confirming the closure of the constraint algebra.

Contribution

It demonstrates that at least two polygons are required to describe the torus, correcting previous assumptions, and shows the constraint algebra closes in this framework.

Findings

A single polygon is insufficient to describe the torus.

The constraint algebra of the polygons closes.

Multiple polygons are necessary for an accurate description.

Abstract

In this paper we describe the matter-free toroidal spacetime in 't Hooft's polygon approach to 2+1-dimensional gravity (i.e. we consider the case without any particles present). Contrary to earlier results in the literature we find that it is not possible to describe the torus by just one polygon but we need at least two polygons. We also show that the constraint algebra of the polygons closes.