Some approaches to 2+1-dimensional gravity coupled to point-particles

TL;DR

This paper reviews various methods for analyzing 2+1-dimensional gravity coupled with point-particles, including exact solutions, the polygon approach, Chern-Simons formulation, and Riemann-Hilbert mapping.

Contribution

It compares multiple approaches to 2+1 gravity with point-particles, highlighting their methodologies and connections.

Findings

Exact static and stationary solutions identified

Polygon approach elucidated for 2+1 gravity

Chern-Simons formulation applied to the problem

Abstract

In these notes we will review some approaches to 2+1 dimensional gravity and the way it is coupled to point-particles. First we look into some exact static and stationary solutions with and without cosmological constant. Next we study the polygon approach invented by 't Hooft. The third section treats the Chern-Simonons formulation of 2+1-gravity. In the last part we map the problem of finding the gravitational field around point-particles to the Riemann-Hilbert problem.