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” [142]. 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:

- First, we exhibit other theories besides pure gravity in four dimensions which also lead to a Coxeter billiard. We stress further how exceptional these theories are in the space of all theories described by the action Equation (2.1).
- Second, we show how to reformulate the dynamics as a motion in the Cartan subalgebra of a Lorentzian Kac–Moody algebra.
- Finally, we connect the Lorentzian Kac–Moody algebra that appears in the BKL-limit to the “hidden” symmetry group in the simplest case when the real Lie algebra of the group is the split real form of the corresponding complexified Lie algebra . (These concepts will be defined below.) The general case will be dealt with in Section 7, after we have recalled the most salient features of the theory of real forms in Section 6.

5.1 More on Coxeter billiards

5.1.1 The Coxeter billiard of pure gravity in spacetime dimensions

5.1.2 The Coxeter billiard for the coupled gravity-3-Form system

Coxeter polyhedra

The Coxeter billiard of eleven-dimensional supergravity

5.2 Dynamics in the Cartan subalgebra

5.2.1 Billiard dynamics in the Cartan subalgebra

Scale factor space and the wall system

Scale factor space and the Cartan subalgebra

5.2.2 The fundamental Weyl chamber and the billiard table

5.2.3 Hyperbolicity implies chaos

Example: Pure gravity in and

5.3 Understanding the emerging Kac–Moody algebra

5.3.1 Invariance under toroidal dimensional reduction

5.3.2 Iwasawa decomposition for split real forms

5.3.3 Starting at the bottom – Overextensions of finite-dimensional Lie algebras

5.4 Models associated with split real forms

5.1.1 The Coxeter billiard of pure gravity in spacetime dimensions

5.1.2 The Coxeter billiard for the coupled gravity-3-Form system

Coxeter polyhedra

The Coxeter billiard of eleven-dimensional supergravity

5.2 Dynamics in the Cartan subalgebra

5.2.1 Billiard dynamics in the Cartan subalgebra

Scale factor space and the wall system

Scale factor space and the Cartan subalgebra

5.2.2 The fundamental Weyl chamber and the billiard table

5.2.3 Hyperbolicity implies chaos

Example: Pure gravity in and

5.3 Understanding the emerging Kac–Moody algebra

5.3.1 Invariance under toroidal dimensional reduction

5.3.2 Iwasawa decomposition for split real forms

5.3.3 Starting at the bottom – Overextensions of finite-dimensional Lie algebras

5.4 Models associated with split real forms

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