We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E_10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.
Keywords: hidden symmetries, duality, Kac-Moody algebras
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Since a Living Reviews in Relativity article may evolve over time, please cite the access <date>, which uniquely identifies the version of the article you are referring to:
Marc Henneaux and Daniel Persson and Philippe Spindel,
"Spacelike Singularities and Hidden Symmetries of Gravity",
Living Rev. Relativity 11, (2008), 1. URL (cited on <date>):
http://www.livingreviews.org/lrr-2008-1
ORIGINAL | http://www.livingreviews.org/lrr-2008-1 |
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Title | Spacelike Singularities and Hidden Symmetries of Gravity |
Author | Marc Henneaux / Daniel Persson / Philippe Spindel |
Date | accepted 12 February 2008, published 24 April 2008 |