# Canonical Chern-Simons Gravity

## TL;DR

This paper provides a canonical analysis of axisymmetric vacuum solutions in 2+1 dimensional gravity formulated as a Chern-Simons gauge theory, revealing simplified constraints and a trivial Wheeler-DeWitt equation.

## Contribution

It introduces a canonical framework for 2+1D Chern-Simons gravity with axisymmetry, deriving explicit variables and constraints, and analyzing the quantum structure.

## Key findings

- Constraints imply vanishing momenta conjugate to Killing time and curvature radius.
- Remaining variables are the total mass and angular momentum.
- Wheeler-DeWitt equation is trivial, indicating time independence.

## Abstract

We study the canonical description of the axisymmetric vacuum in 2+1 dimensional gravity, treating Einstein's gravity as a Chern Simons gauge theory on a manifold with the restriction that the dreibein is invertible. Our treatment is in the spirit of Kucha\v r's description of the Schwarzschild black hole in 3+1 dimensions, where the mass and angular momentum are expressed in terms of the canonical variables and a series of canonical transformations are performed that turn the curvature coordinates and their conjugate momenta into new canonical variables. In their final form, the constraints are seen to require that the momenta conjugate to the Killing time and curvature radius vanish and what remains are the mass, the angular momentum and their conjugate momenta, which we derive. The Wheeler-DeWitt equation is trivial and describes time independent systems with wave functions described only by the total mass and total angular momentum.