In this section we will begin to explore in more detail the correspondence between Lorentzian Coxeter groups and the limiting behavior of the dynamics of gravitational theories close to a spacelike singularity.
We have seen in Section 2 that in the BKL-limit, the dynamics of gravitational theories is equivalent to a billiard dynamics in a region of hyperbolic space. In the generic case, the billiard region has no particular feature. However, we have seen in Section 3 that in the case of pure gravity in four spacetime dimensions, the billiard region has the remarkable property of being the fundamental domain of the Coxeter group acting on two-dimensional hyperbolic space.
This is not an accident. Indeed, this feature arises for all gravitational theories whose toroidal dimensional reduction to three dimensions exhibits hidden symmetries, in the sense that the reduced theory can be reformulated as three-dimensional gravity coupled to a nonlinear sigma-model based on , where is the maximal compact subgroup of . The “hidden” symmetry group is also called, by a generalization of language, “the U-duality group” . This situation covers the cases of pure gravity in any spacetime dimension, as well as all known supergravity models. In all these cases, the billiard region is the fundamental domain of a Lorentzian Coxeter group (“Coxeter billiard”). Furthermore, the Coxeter group in question is crystallographic and turns out to be the Weyl group of a Lorentzian Kac–Moody algebra. The billiard table is then the fundamental Weyl chamber of a Lorentzian Kac–Moody algebra [45, 46] and the billiard is also called a “Kac–Moody billiard”. This enables one to reformulate the dynamics as a motion in the Cartan subalgebra of the Lorentzian Kac–Moody algebra, hinting at the potential – and still conjectural at this stage – existence of a deeper, infinite-dimensional symmetry of the theory.
The purpose of this section is threefold:
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