15 research outputs found
Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity
In the formulation of (2+1)-dimensional gravity as a Chern-Simons gauge theory, the phase space is the moduli space of flat Poincar\'e 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 Poincar\'e 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
Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
We propose a description of open universes in the Chern-Simons formulation of
(2+1)-dimensional gravity where spatial infinity is implemented as a puncture.
At this puncture, additional variables are introduced which lie in the
cotangent bundle of the Poincar\'e group, and coupled minimally to the
Chern-Simons gauge field. We apply this description of spatial infinity to open
universes of general genus and with an arbitrary number of massive spinning
particles. Using results of [9] we give a finite dimensional description of the
phase space and determine its symplectic structure. In the special case of a
genus zero universe with spinless particles, we compare our result to the
symplectic structure computed by Matschull in the metric formulation of
(2+1)-dimensional gravity. We comment on the quantisation of the phase space
and derive a quantisation condition for the total mass and spin of an open
universe.Comment: 44 pages, 3 eps figure
Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
Each of the local isometry groups arising in 3d gravity can be viewed as the
group of unit (split) quaternions over a ring which depends on the cosmological
constant. In this paper we explain and prove this statement, and use it as a
unifying framework for studying Poisson structures associated with the local
isometry groups. We show that, in all cases except for Euclidean signature with
positive cosmological constant, the local isometry groups are equipped with the
Poisson-Lie structure of a classical double. We calculate the dressing action
of the factor groups on each other and find, amongst others, a simple and
unified description of the symplectic leaves of SU(2) and SL(2,R). We also
compute the Poisson structure on the dual Poisson-Lie groups of the local
isometry groups and on their Heisenberg doubles; together, they determine the
Poisson structure of the phase space of 3d gravity in the so-called
combinatorial description.Comment: 34 pages, minor corrections, references adde
Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
We define a theory of Galilean gravity in 2+1 dimensions with cosmological
constant as a Chern-Simons gauge theory of the doubly-extended Newton-Hooke
group, extending our previous study of classical and quantum gravity in 2+1
dimensions in the Galilean limit. We exhibit an r-matrix which is compatible
with our Chern-Simons action (in a sense to be defined) and show that the
associated bi-algebra structure of the Newton-Hooke Lie algebra is that of the
classical double of the extended Heisenberg algebra. We deduce that, in the
quantisation of the theory according to the combinatorial quantisation
programme, much of the quantum theory is determined by the quantum double of
the extended q-deformed Heisenberg algebra.Comment: 22 page
3d gravity and quantum deformations: a Drinfel'd double approach
Loops 11: Non-Perturbative / Background Independent Quantum Gravity
23–28 May 2011, Madrid, SpainThe constant curvature spacetimes of 3d gravity and their associated symmetry algebras are shown to arise from the 6d Drinfel'd double that underlies the two-parametric 'hybrid' quantum deformation of the fraktur sfraktur l(2, Script R) algebra. Moreover, the quantum deformation supplies the additional structures (star structure and pairing) that enter in the Chern-Simons formulation of the theory, thus establishing a direct link between quantum fraktur sfraktur l(2, Script R) algebras and 3d gravity models. In this approach the flat spacetimes and Newtonian models arise as Lie algebra contractions that are governed by two dimensionful fraktur sfraktur l(2, Script R) deformation parameters, which are directly related to the cosmological constant and to the speed of light
Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
We show how to properly gauge fix all the symmetries of the Ponzano-Regge
model for 3D quantum gravity. This amounts to do explicit finite computations
for transition amplitudes. We give the construction of the transition
amplitudes in the presence of interacting quantum spinning particles. We
introduce a notion of operators whose expectation value gives rise to either
gauge fixing, introduction of time, or insertion of particles, according to the
choice. We give the link between the spin foam quantization and the hamiltonian
quantization. We finally show the link between Ponzano-Regge model and the
quantization of Chern-Simons theory based on the double quantum group of SU(2)Comment: 48 pages, 15 figure
HWM-03-20
hep-th/0310218 The quantisation of Poisson structures arising in Chern-Simons theory with gauge group G ⋉ g