Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant

TL;DR

This paper explores the relationship between grafting, Poisson structures, and phase space in (2+1)-dimensional gravity with zero cosmological constant, using the Chern-Simons formulation and geometric transformations like Dehn twists.

Contribution

It explicitly connects grafting along geodesics to Poisson brackets in the Chern-Simons framework and interprets grafting as a Dehn twist with a formal parameter.

Findings

Derived explicit grafting actions on holonomies.

Proved grafting is generated by gauge-invariant observables.

Linked grafting to Dehn twists with a nilpotent parameter.

Abstract

We relate the geometrical construction of (2+1)-spacetimes via grafting to phase space and Poisson structure in the Chern-Simons formulation of (2+1)-dimensional gravity with vanishing cosmological constant on manifolds of topology $R\times S_g$, where $S_g$ is an orientable two-surface of genus $g>1$. We show how grafting along simple closed geodesics \lambda is implemented in the Chern-Simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on S_g. We prove that this action is generated via the Poisson bracket by a gauge invariant observable associated to the holonomy of $\lambda$. We deduce a symmetry relation between the Poisson brackets of observables associated to the Lorentz and translational components of the holonomies of general closed curves on S_g and discuss its physical interpretation. Finally, we relate the action of grafting on the phase space to the action of Dehn twists and show that grafting can be viewed as a Dehn twist with a formal parameter $\theta$ satisfying $\theta^2=0$.