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:
http://www.livingreviews.org/lrr-2008-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |