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

TL;DR

This paper develops a new framework for describing open universes in (2+1)-dimensional gravity using Chern-Simons theory, introducing boundary conditions at spatial infinity as punctures and analyzing the resulting phase space and symplectic structure.

Contribution

It introduces a novel description of spatial infinity as a puncture with additional variables, providing a finite-dimensional phase space and analyzing its symplectic structure in the Chern-Simons formulation.

Findings

Finite-dimensional phase space description for open universes.

Explicit symplectic structure for the phase space.

Quantisation conditions for total mass and spin.

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 Poincar\'e 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 quantisation of the phase space and derive a quantisation condition for the total mass and spin of an open universe.