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

Abstract

In the formulation of (2+1)-dimensional gravity as a Chern-Simons gauge theory, the phase space is the moduli space of flat Poincaré 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 Poincaré 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.

49 pages amslatex, 7 eps figures

References in corpus (3)

• Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
• Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity
• The Phase Space Structure of Multi Particle Models in 2+1 Gravity

Cited by in corpus (44)

• Quantum Spacetime Phenomenology
• The principle of relative locality
• Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
• Spin Foams and Canonical Quantization
• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions
• Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
• Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
• Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
• Symmetries of quantum space-time in 3 dimensions
• Compact phase space, cosmological constant, discrete time
• Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
• The DSR-deformed relativistic symmetries and the relative locality of 3D quantum gravity
• 2+1D Loop Quantum Gravity on the Edge
• Curved momentum spaces from quantum groups with cosmological constant
• Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
• Deformed Carroll particle from 2+1 gravity
• Mapping class group actions in Chern-Simons theory with gauge group $G⋉{\mathfrak{g}}^{\ast }$
• Relativistic compatibility of the interacting $\kappa$-Poincaré model and implications for the relative locality framework
• On the semiduals of local isometry groups in 3d gravity
• Combinatorial quantisation of the Euclidean torus universe
• Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity
• Polygon model from first order gravity
• Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits
• Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant
• The Poincaré group as a Drinfel'd double
• Multi-particle systems in $\kappa$-Poincaré inspired by 2+1D gravity
• $\kappa$-de Sitter and $\kappa$-Poincaré symmetries emerging from Chern-Simons (2+1)D gravity with a cosmological constant
• Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry
• Quantizations of D=3 Lorentz symmetry
• Dual 2+1D Loop Quantum Gravity on the Edge
• A new realization of quantum geometry
• Effective Chern-Simons actions of particles coupled to 3D gravity
• 2+1 gravity with positive cosmological constant in LQG: a proposal for the physical state
• From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes
• Deformed phase spaces with group valued momenta
• Gauge fixing and classical dynamical r-matrices in ISO(2,1)-Chern-Simons theory
• Dual generators of the fundamental group and the moduli space of flat connections
• Towards Non-Commutative Deformations of Relativistic Wave Equations in 2+1 Dimensions
• Drinfel'd double structures for Poincaré and Euclidean groups
• Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
• Spin Networks and Cosmic Strings in 3+1 Dimensions
• Non-commutative waves for gravitational anyons
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes