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

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

2005-07-31

arxiv.org/abs/gr-qc/0508004v2

Commun.Math.Phys. 266 (2006) 735-775

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

43 pages, 10 .eps figures; minor modifications: 2 figures added, explanations added, typos corrected

Scientific paper

10.1007/s00220-006-0037-x

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 $\theta^2=0$.

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