Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity

TL;DR

This paper explores the phase space, Poisson structure, and symmetries in (2+1)-dimensional gravity formulated as a Chern-Simons gauge theory, providing explicit descriptions and physical interpretations.

Contribution

It offers an explicit combinatorial description of the phase space and Poisson structure for general surfaces with punctures, and analyzes the symmetry group actions in this context.

Findings

Explicit phase space description for genus g surfaces with punctures

Decoupling of degrees of freedom for handles and particles

Mapping class group acts via Poisson isomorphisms

Abstract

In the formulation of (2+1)-dimensional gravity as a Chern-Simons gauge theory, the phase space is the moduli space of flat Poincar\'e group connections. Using the combinatorial approach developed by Fock and Rosly, we give an explicit description of the phase space and its Poisson structure for the general case of a genus g oriented surface with punctures representing particles and a boundary playing the role of spatial infinity. We give a physical interpretation and explain how the degrees of freedom associated with each handle and each particle can be decoupled. The symmetry group of the theory combines an action of the mapping class group with asymptotic Poincar\'e transformations in a non-trivial fashion. We derive the conserved quantities associated to the latter and show that the mapping class group of the surface acts on the phase space via Poisson isomorphisms.