Closure constraints for hyperbolic tetrahedra

Abstract

We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R^3. One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Lambda, and in particular that deforming the loop gravity phase space with real parameter q>0 would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in SU(2) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in SB(2,C)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.

24 pages

References in corpus (10)

• Polyhedra in 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
• Observables in Loop Quantum Gravity with a cosmological constant
• Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
• Towards the Turaev-Viro amplitudes from a Hamiltonian constraint
• Turaev-Viro amplitudes from 2+1 Loop Quantum Gravity
• Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries
• Deformed phase space for 3d loop gravity and hyperbolic discrete geometries
• 6j symbols for U_q(sl_2) and non-Euclidean tetrahedra

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