# 2+1-dimensional gravity coupled to a dust shell: quantization in terms of global phase space variables

## TL;DR

This paper analyzes a 2+1-dimensional gravity model with a dust shell, identifying global phase space variables with ADS^{2} geometry, leading to non-commutative and discrete coordinates that resolve singularities.

## Contribution

It introduces a global phase space parametrization for the model using ADS^{2} geometry and Euler angles, connecting to Kuchar variables for potential 3+1 dimensional extension.

## Key findings

- Global phase space is an ADS^{2} manifold.
- Coordinate space exhibits non-commutativity and discreteness.
- Singularity is resolved through this geometric and algebraic structure.

## Abstract

We perform canonical analysis of a model in which gravity is coupled to a spherically symmetric dust shell in 2+1 spacetime dimensions. The result is a reduced action depending on a finite number of degrees of freedom. The emphasis is made on finding canonical variables providing the global chart for the entire phase space of the model. It turns out that all the distinct pieces of momentum space could be assembled into a single manifold which has ADS^{2}-geometry, and the global chart for it is provided by the Euler angles. This results in both non-commutativity and discreteness in coordinate space, which allows to resolve the central singularity. We also find the map between ADS^{2} momentum space obtained here and momentum space in Kuchar variables, which could be helpful in extending the present results to 3+1 dimensions.