Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity

We propose a description of open universes in the Chern-Simons formulation of (2+1)-dimensional gravity where spatial infinity is implemented as a puncture. At this puncture, additional variables are introduced which lie in the cotangent bundle of the Poincare group, and coupled minimally to the Chern-Simons gauge field. We apply this description of spatial infinity to open universes of general genus and with an arbitrary number of massive spinning particles. Using results of [9] we give a finite-dimensional description of the phase space and determine its symplectic Structure. In the special case of a genus zero universe with spinless particles, we compare our result to the symplectic structure computed by Matschull in the metric formulation of (2+1)dimensional gravity. We comment on the quantization of the phase space and derive a quantization condition for the total mass and spin of an open universe.