Origin of the quantum group symmetry in 3d quantum gravity

Abstract

It is well-known that quantum groups are relevant to describe the quantum regime of 3d gravity. They encode a deformation of the gauge symmetries parametrized by the value of the cosmological constant. They appear as a form of regularization either through the quantization of the Chern-Simons formulation or the state sum approach of Turaev-Viro. Such deformations are perplexing from a continuum and classical picture since the action is defined in terms of undeformed gauge invariance. We present here a novel way to derive from first principle and from the classical action such quantum group deformation. The argument relies on two main steps. First we perform a canonical transformation, which deformed the gauge invariance and the boundary symmetries, and makes them depend on the cosmological constant. Second we implement a discretization procedure relying on a truncation of the degrees of freedom from the continuum.

49 pages, 3 figures

References in corpus (55)

• Fault-tolerant quantum computation by anyons
• The Dynamics of General Relativity
• Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy
• A General Definition of "Conserved Quantities" in General Relativity and Other Theories of Gravity
• Local subsystems in gauge theory and gravity
• Entropy from Conformal Field Theory at Killing Horizons
• Remarks on entanglement entropy for gauge fields
• Decomposition of entanglement entropy in lattice gauge theory
• Symplectic structures associated to Lie-Poisson groups
• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Scattering at Planckian Energies
• Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
• Edge modes of gravity -- I: Corner potentials and charges
• Diffeomorphisms and spin foam models
• Geometric entropy and edge modes of the electromagnetic field
• Edge states in Gravity and Black Hole Physics
• The Ponzano-Regge model
• Why Gauge?
• Edge modes of gravity -- II: Corner metric and Lorentz charges
• Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
• Lattice Wess-Zumino-Witten Model and Quantum Groups
• Dressing Symmetries
• Fusion basis for lattice gauge theory and loop quantum gravity
• Turaev-Viro invariants as an extended TQFT
• Lorentz-diffeomorphism edge modes in 3d gravity
• The loop gravity string
• Ponzano-Regge model revisited II: Equivalence with Chern-Simons
• A proposal for 3d quantum gravity and its bulk factorization
• Gravitational edge modes: From Kac-Moody charges to Poincaré networks
• Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
• The Power of Duality -- Exact Results in 4D SUSY Field Theory
• On two approaches to 3-dimensional TQFTs
• Quantum gravity kinematics from extended TQFTs
• Einstein-Cartan Theory
• Towards the Turaev-Viro amplitudes from a Hamiltonian constraint
• Notes on Entanglement in Abelian Gauge Theories
• Turaev-Viro amplitudes from 2+1 Loop Quantum Gravity
• Gauge symmetries in spinfoam gravity: the case for "cellular quantization"
• Comments on Background Independence and Gauge Redundancies
• Edge States and Entanglement Entropy
• 2+1D Loop Quantum Gravity on the Edge
• Discretization of 3d gravity in different polarizations
• Edge States in Canonical Gravity
• Entanglement entropy in lattice gauge theories
• Deformed phase space for 3d loop gravity and hyperbolic discrete geometries
• Non-commutative holonomies in 2+1 LQG and Kauffman's brackets
• On a class of 2-surface observables in general relativity
• Dual loop quantizations of 3d gravity
• Block diagonalisation of four-dimensional metrics
• $q$-deformed 3D Loop Gravity on the Torus
• Deformation Quantization of the Isotropic Rotator
• Dual 2+1D Loop Quantum Gravity on the Edge
• (2-)Drinfel'd Double and (2-)BF Theory
• Canonical transformations generated by the boundary volume: Unimodular and non-Abelian teleparallel gravity
• Discretization of 4d Poincaré BF theory: from groups to 2-groups

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• Edge modes as reference frames and boundary actions from post-selection
• Edge modes as dynamical frames: charges from post-selection in generally covariant theories
• Canonical transformations generated by the boundary volume: Unimodular and non-Abelian teleparallel gravity
• Local Observables in ${\mathrm{SU}}_q(2)$ Lattice Gauge Theory
• Discretization of 4d Poincaré BF theory: from groups to 2-groups
• Entanglement entropy and edge modes in topological string theory: I
• Spinor Representation of the Hamiltonian Constraint in 3D LQG with a Non-zero Cosmological Constant
• 3D Quantum Gravity from Holomorphic Blocks
• Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory
• Polyhedron phase space using 2-groups: kappa-Poincare as a Poisson 2-group