Einstein billiards and overextensions of finite-dimensional simple Lie algebras

TL;DR

This paper explores the connection between gravitational theories near singularities and billiard dynamics, revealing that their symmetry properties relate to the overextensions of finite-dimensional Lie algebras, independent of spacetime dimension.

Contribution

It demonstrates that billiards in theories with gravity correspond to the fundamental Weyl chambers of Kac-Moody overextended algebras, generalizing previous results to broader classes of theories.

Findings

Billiards are linked to overextended Kac-Moody algebras.

The billiard properties are independent of spacetime dimension.

The symmetry algebra depends on the dimension, but the billiard shape does not.

Abstract

In recent papers, it has been shown that (i) the dynamics of theories involving gravity can be described, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space bounded by hyperplanes; and (ii) that the relevant billiard has remarkable symmetry properties in the case of pure gravity in $d+1$ spacetime dimensions, or supergravity theories in 10 or 11 spacetime dimensions, for which it turns out to be the fundamental Weyl chamber of the Kac-Moody algebras $AE_d$, $E_{10}$, $BE_{10}$ or $DE_{10}$ (depending on the model). We analyse in this paper the billiards associated to other theories containing gravity, whose toroidal reduction to three dimensions involves coset models $G/H$ (with $G$ maximally non compact). We show that in each case, the billiard is the fundamental Weyl chamber of the (indefinite) Kac-Moody ``overextension'' (or ``canonical Lorentzian extension'') of the finite-dimensional Lie algebra that appears in the toroidal compactification to 3 spacetime dimensions. A remarkable feature of the billiard properties, however, is that they do not depend on the spacetime dimension in which the theory is analyzed and hence are rather robust, while the symmetry algebra that emerges in the toroidal dimensional reduction is dimension-dependent.