Edge modes of gravity. Part II. Corner metric and Lorentz charges
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Received: July 31, 2020 Accepted: October 2, 2020 Published: November 9, 2020
Edge modes of gravity. Part II. Corner metric and Lorentz charges
a
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada b Ecole Normale Superieure (ENS) de Lyon, 5 parvis René Descartes — BP 7000, F-69342 Lyon Cedex 07, France c Università degli Studi di Udine, via Palladio 8, I-33100 Udine, Italy
E-mail: [email protected], [email protected], [email protected] Abstract: In this second paper of the series we continue to spell out a new program for quantum gravity, grounded in the notion of corner symmetry algebra and its representations. Here we focus on tetrad gravity and its corner symplectic potential. We start by performing a detailed decomposition of the various geometrical quantities appearing in BF theory and tetrad gravity. This provides a new decomposition of the symplectic potential of BF theory and the simplicity constraints. We then show that the dynamical variables of the tetrad gravity corner phase space are the internal normal to the spacetime foliation, which is conjugated to the boost generator, and the corner coframe field. This allows us to derive several key results. First, we construct the corner Lorentz charges. In addition to sphere diffeomorphisms, common to all formulations of gravity, these charges add a local sl(2, C) component to the corner symmetry algebra of tetrad gravity. Second, we also reveal that the corner metric satisfies a local sl(2, R) algebra, whose Casimir corresponds to the corner area element. Due to the space-like nature of the corner metric, this Casimir belongs to the unitary discrete series, and its spectrum is therefore quantized. This result, which reconciles discreteness of the area spectrum with Lorentz invariance, is proven in the continuum and without resorting to a bulk connection. Third, we show that the corner phase space explains why the simplicity constraints become non-commutative on the corner. This fact requires a reconciliation between the bulk and corner symplectic structures, already in the classical continuum theory. Understanding this leads inevitably to the introduction of edge modes. Keywords: Classical Theories of Gravity, Models of Quantum Gravity, Space-Time Symmetries, Gauge Symmetry ArXiv ePrint: 2007.03563
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)027
JHEP11(2020)027
Laurent Freidel,a Marc Geillerb and Daniele Pranzettia,c
Contents 1 Introduction 1.1 Motivations 1.2 Questions 1.3 Summary of the results theory and tetrad formulation of gravity BF theory Einstein-Cartan-Holst gravity A new look at canonical analysis
8 9 9 11
3 3 + 1 decompositions 3.1 Normal/tangential decomposition 3.1.1 Decomposition of Lorentz tensors 3.1.2 Decomposition of connections 3.1.3 Boost/rotation decomposition of the Gauss constraint 3.2 Horizontal/vertical decomposition 3.3 BF coframes 3.4 Equations of motion 3.5 Bulk simplicity constra
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