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

Abstract

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. A concrete realization of this is provided by the 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 3+1 dimensional covariant loop quantum gravity in the presence of a cosmological constant.

38 pages and 9 figures

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• Asymptotic analysis of the EPRL model with timelike tetrahedra
• Einstein Equation from Covariant Loop Quantum Gravity in Semiclassical Continuum Limit
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• (3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces
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• The closure constraint for the hyperbolic tetrahedron as a Bianchi identity
• SU(2) Flat Connection on Riemann Surface and Twisted Geometry with Cosmological Constant
• A new realization of quantum geometry
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• Quantum fluctuations of the compact phase space cosmology
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• Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory
• Hessian in the spinfoam models with cosmological constant