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

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 λis 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 ${\theta }^2=0$.

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

References in corpus (3)

• Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
• Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity
• Mapping class group actions in Chern-Simons theory with gauge group $G⋉{\mathfrak{g}}^{\ast }$

Cited by in corpus (8)

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• Combinatorial quantisation of the Euclidean torus universe
• Dual generators of the fundamental group and the moduli space of flat connections
• In Search of Fundamental Discreteness in 2+1 Dimensional Quantum Gravity
• A quantization of moduli spaces of 3-dimensional gravity
• Geometry and observables in (2+1)-gravity