14619 2007.12635 10.1007/JHEP01(2021)100 100 100 Edge modes of gravity. Part III. Corner simplicity constraints 01 Mathematical Sciences 0101 Pure Mathematics Regular Article - Theoretical Physics 1 64 2021 1 19 2020 7 31 2020 12 2 2020 12 10 2021 1 19 The Author(s) 2021 Open Access. This article is distributed under the terms of the Creative Commons Attribution License ( CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 13130 2021 2021 1 1 Laurent Freidel lfreidel@perimeterinstitute.ca Marc Geiller marc.geiller@ens-lyon.fr Daniele Pranzetti dpranzetti@perimeterinstitute.ca grid.420198.6 0000 0000 8658 0851 Perimeter Institute for Theoretical Physics 31 Caroline Street North Waterloo ON N2L 2Y5 Canada École Normale Superieure (ENS) de Lyon 5 parvis René Descartes — BP 7000 F-69342 Lyon Cedex 07 France grid.5390.f 0000 0001 2113 062X Università degli Studi di Udine via Palladio 8 I-33100 Udine Italy Abstract In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincaré and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints. This amounts to fixing the Heisenberg frame with a choice of position and spin operators. The simplicity constraints then further reduce the Poincaré symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a particle-like description of (quantum) geometry: the internal normal plays the role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the flux that of a relativistic position, and the frame that of a spin harmonic oscillator. Moreover, we show that the corner area element corresponds to the Poincaré spin Casimir. We achieve this central result by properly splitting, in the continuum, the corner simplicity constraints into first and second class parts. We construct the complete set of Dirac observables, which includes the generators of the local sl 2 ℂ $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) $$ subalgebra of Poincaré, and the components of the tangential corner metric satisfying an sl 2 ℝ $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$ algebra. We then present a preliminary analysis of the covariant and continuous irreducible representations of the infinite-dimensional corner algebra. Moreover, as an alternative path to quantization, we also introduce a regularization of the corner algebra and interpret this discrete setting in terms of an extended notion of twisted geometries. Keywords Classical Theories of Gravity Models of Quantum Gravity Space-Time Symmetries Gauge Symmetry ArXiv ePrint: 2007.12635