These connections appear for the cases at hand because in the BKL-limit, not only can the equations of motion be reformulated as dynamical equations for billiard motion in a region of hyperbolic space, but also this region possesses unique features: It is the fundamental Weyl chamber of some Kac–Moody algebra. The dynamical motion in the BKL-limit is then a succession of reflections in the walls bounding the fundamental Weyl chamber and defines “words” in the Weyl group of the Kac–Moody algebra.
Which billiard region of hyperbolic space actually emerges – and hence which Kac–Moody algebra is relevant – depends on the theory at hand, i.e., on the spacetime dimension, the menu of matter fields, and the dilaton couplings. The most celebrated case is eleven-dimensional supergravity, for which the billiard region is the fundamental region of , one of the four hyperbolic Kac–Moody algebras of highest rank 10. The root lattice of is furthermore one of the few even, Lorentzian, self-dual lattices – actually the only one in 10 dimensions – a fact that could play a key role in our ultimate understanding of M-theory.
Other gravitational theories lead to other billiards characterized by different algebras. These algebras are closely connected to the hidden duality groups that emerge upon dimensional reduction to three dimensions [41, 95].
That one can associate a regular billiard and an infinite discrete reflection group (Coxeter group) to spacelike singularities of a given gravitational theory in the BKL-limit is a robust fact (even though the BKL-limit itself is yet to be fully understood), which, in our opinion, will survive future developments. The mathematics necessary to appreciate the billiard structure and its connection to the duality groups in three dimensions involve hyperbolic Coxeter groups, Kac–Moody algebras and real forms of Lie algebras.
The appearance of infinite Coxeter groups related to Lorentzian Kac–Moody algebras has triggered fascinating conjectures on the existence of huge symmetry structures underlying gravity . Similar conjectures based on different considerations had been made earlier in the pioneering works [113, 167]. The status of these conjectures, however, is still somewhat unclear since, in particular, it is not known how exactly the symmetry would act.
The main purpose of this article is to explain the emergence of infinite discrete reflection groups in gravity in a self-contained manner, including giving the detailed mathematical background needed to follow the discussion. We shall avoid, however, duplicating already existing reviews on BKL billiards.
Contrary to the main core of the review, devoted to an explanation of the billiard Weyl groups, which is indeed rather complete, we shall also discuss some paths that have been taken towards revealing the conjectured infinite-dimensional Kac–Moody symmetry. Our goal here will only be to give a flavor of some of the work that has been done along these lines, emphasizing its dynamical relevance. Because we feel that it would be premature to fully review this second subject, which is still in its infancy, we shall neither try to be exhaustive nor give detailed treatments.
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