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 \(\mathbb{R} \times S_g\), where S g is an orientable two-surface of genus g>1. We show how grafting along simple closed geodesics λ 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 λ. 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 θ satisfying θ2 = 0.
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Communicated by G.W. Gibbons
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Meusburger, C. Grafting and Poisson Structure in (2+1)-Gravity with Vanishing Cosmological Constant. Commun. Math. Phys. 266, 735–775 (2006). https://doi.org/10.1007/s00220-006-0037-x
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DOI: https://doi.org/10.1007/s00220-006-0037-x