2014/02/10 by Valentin Bonzom, Bonzom, Valentin, Maïté Dupuis +5 · 1 citation
Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect
paper · pdf · doi:10.48550/arxiv.1402.2323
openalex publication_date 2014/02/10 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28
We revisit the loop gravity space phase for 3D Riemannian gravity by\nalgebraically constructing the phase space\nT^*\SU(2)\∼\ISO(3) as the Heisenberg double of the Lie\ngroup \SO(3) provided with the trivial cocyle. Tackling the issue of\naccounting for a non-vanishing cosmological constraint \Λ \≠ 0 in the\ncanonical framework of 3D loop quantum gravity, \SL(2,\ℂ)\nviewed as the Heisenberg double of \SU(2) provided with a non-trivial\ncocyle is introduced as a phase space. It is a deformation of the flat phase\nspace \ISO(3) and reproduces the latter in a suitable limit. The\n\SL(2,\ℂ) phase space is then used to build a new, deformed\nLQG phase space associated to graphs. It can be equipped with a set of Gauss\nconstraints and flatness constraints, which form a first class system and\nPoisson-generate local 3D rotations and deformed translations. We provide a\ngeometrical interpretation for this lattice phase space with constraints in\nterms of consistently glued hyperbolic triangles, i.e. hyperbolic discrete\ngeometries, thus validating our construction as accounting for a constant\ncurvature \Λ<0. Finally, using ribbon diagrams, we show that our new\nmodel is topological.\n