Encoding Curved Tetrahedra in Face Holonomies: Phase Space of Shapes from Group-Valued Moment Maps

We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi–Civita holonomies around each of the tetrahedron’s faces. This allows the reco...

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Bibliographic Details
Published in:Annales Henri Poincaré 2016-08, Vol.17 (8), p.2001-2048
Main Authors: Haggard, Hal M., Han, Muxin, Riello, Aldo
Format: Article
Language:English
Subjects:
Algebra
Classical and Quantum Gravitation
Cosmological constant
Dynamical Systems and Ergodic Theory
Elementary Particles
Euclidean geometry
Group theory
Mathematical and Computational Physics
Mathematical Methods in Physics
Physics
Physics and Astronomy
Quantum Field Theory
Quantum gravity
Quantum Physics
Reconstruction
Relativity Theory
Tetrahedra
Theoretical
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Summary:We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi–Civita holonomies around each of the tetrahedron’s faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group-valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. This becomes manifest in light of their relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in covariant loop quantum gravity in the presence of a cosmological constant.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-015-0455-4