Spacelike Singularities and Hidden Symmetries of Gravity

Abstract

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 E10, 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.

228 pages. Typos corrected. References added. Subject index added. Published version

References in corpus (21)

• E(11) and the Embedding Tensor
• Hidden symmetries and the fermionic sector of eleven-dimensional supergravity
• Extended E8 Invariance of 11-Dimensional Supergravity
• Dual fields and E_{11}
• Monopoles for Gravitation and for Higher Spin Fields
• K(E10), Supergravity and Fermions
• Describing general cosmological singularities in Iwasawa variables
• Enhanced Coset Symmetries and Higher Derivative Corrections
• Curvature corrections and Kac-Moody compatibility conditions
• Duality Groups, Automorphic Forms and Higher Derivative Corrections
• The arrow of time and the Weyl group: all supergravity billiards are integrable
• K(E9) from K(E10)
• Geometric Configurations, Regular Subalgebras of E10 and M-Theory Cosmology
• Quantization of Wilson loops in Wess-Zumino-Witten models
• Eleven dimensional supergravity and the E10/KE10 sigma-model at low A9 levels
• Pure type I supergravity and DE(10)
• On the relation between mathematical and numerical relativity
• Higher Derivative Corrections, Dimensional Reduction and Ehlers Duality
• Hidden Borcherds symmetries in Z_n orbifolds of M-theory and magnetized D-branes in type 0' orientifolds
• A Special Class of Rank 10 and 11 Coxeter Groups
• Kac-Moody Algebras and Controlled Chaos

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