Canonical Chern-Simons gravity

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.