Spacelike Singularities and Hidden Symmetries of Gravity.
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 algebraE10, 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.
Physique Théorique et Mathématique, Université Libre de Bruxelles & International Solvay Institutes, Boulevard du Triomphe, ULB - C.P. 231, B-1050 Bruxelles, Belgium.
Marc
M
Henneaux
Physique Théorique et Mathématique, Université Libre de Bruxelles & International Solvay Institutes, Boulevard du Triomphe, ULB - C.P. 231, B-1050 Bruxelles, Belgium.
Daniel
D
Persson
Service de Mécanique et Gravitation, Université de Mons-Hainaut, Académie Wallonie-Bruxelles, Avenue du Champ de Mars 6, B-7000 Mons, Belgium.
Philippe
P
Spindel