On a self-dual phase space for 3+1 lattice Yang-Mills theory
arXiv:1706.07811 · doi:10.1103/PhysRevD.97.025003
Abstract
I propose a self-dual deformation of the classical phase space of lattice Yang--Mills theory, in which both the electric and magnetic fluxes take value in the gauge Lie group. A local construction of the deformed phase space requires the machinery of "quasi-Hamiltonian spaces" by Alekseev et al., which is here reviewed. The results is a full-fledged finite-dimensional and gauge-invariant phase space, whose self-duality properties are largely enhanced in (3+1) spacetime dimensions. This enhancement is due to a correspondence with the moduli space of an auxiliary non-commutative flat connection living on a Riemann surface defined from the lattice itself, which in turn equips the duality between electric and magnetic fluxes with a neat geometrical interpretation in terms of a Heegaard splitting of the space manifold. Finally, I discuss the consequences of the proposed deformation on the quantization of the phase space, its quantum gravitational interpretation, as well as its relevance for the construction of (3+1) dimensional topological field theories with defects.
arXiv commentReferences, a footnote, and a final comment added; typos corrected in the appendix; a few minor corrections; about 30 pages + 3 of appendix, many figures
References in corpus (42)
- Real Ashtekar Variables for Lorentzian Signature Space-times
- Two Dimensional Gauge Theories Revisited
- Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
- Twisted geometries: A geometric parametrisation of SU(2) phase space
- Symplectic structures associated to Lie-Poisson groups
- Diffeomorphisms and spin foam models
- Twisted Gauge Theory Model of Topological Phases in Three Dimensions
- From twistors to twisted geometries
- A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
- Spinor Representation for Loop Quantum Gravity
- Quantum deformation of two four-dimensional spin foam models
- Fusion basis for lattice gauge theory and loop quantum gravity
- SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
- A new vacuum for Loop Quantum Gravity
- Cosmological Deformation of Lorentzian Spin Foam Models
- Cosmological Constant in LQG Vertex Amplitude
- Observables in the Turaev-Viro and Crane-Yetter models
- Four-dimensional Quantum Gravity with a Cosmological Constant from Three-dimensional Holomorphic Blocks
- Poisson structure on moduli of flat connections on Riemann surfaces and -matrix
- Observables in Loop Quantum Gravity with a cosmological constant
- Quantum gravity kinematics from extended TQFTs
- Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps
- A categorical construction of 4D TQFTs
- Towards the Turaev-Viro amplitudes from a Hamiltonian constraint
- Kitaev lattice models as a Hopf algebra gauge theory
- Flux formulation of loop quantum gravity: Classical framework
- The Fock Space of Loopy Spin Networks for Quantum Gravity
- Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries
- Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double
- Hopf algebra gauge theory on a ribbon graph
- (3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces
- Symplectic Forms on Moduli Spaces of Flat Connections on 2-manifolds
- SL(2,C) Chern-Simons Theory, Flat Connections, and Four-dimensional Quantum Geometry
- Closure constraints for hyperbolic tetrahedra
- Dichromatic state sum models for four-manifolds from pivotal functors
- From 3D topological quantum field theories to 4D models with defects
- Gauge-invariant Hamiltonian formulation of lattice Yang-Mills theory and the Heisenberg double
- The closure constraint for the hyperbolic tetrahedron as a Bianchi identity
- SU(2) Flat Connection on Riemann Surface and Twisted Geometry with Cosmological Constant
- Quasi-Poisson Manifolds
- Hamiltonian lattice gauge models and the Heisenberg double
- Physical phase space of lattice Yang-Mills theory and the moduli space of flat connections on a Riemann surface
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- Quasi-local holographic dualities in non-perturbative 3d quantum gravity I - Convergence of multiple approaches and examples of Ponzano-Regge statistical duals
- Towards the map of quantum gravity
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- Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back
- Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases
- Quantum edge modes in 3d gravity and 2+1d topological phases of matter
- Squeezing of light from Planck-scale physics
- Quantum fluctuations of the compact phase space cosmology
- Polyhedron phase space using 2-groups: kappa-Poincare as a Poisson 2-group