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

Abstract

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 Poincaré 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.