Hyperbolic Kac-Moody Algebras and Chaos in Kaluza-Klein Models

Some time ago, it was found that the never-ending oscillatory chaotic behaviour discovered by Belinsky, Khalatnikov and Lifshitz (BKL) for the generic solution of the vacuum Einstein equations in the vicinity of a spacelike ("cosmological") singularity disappears in spacetime dimensions $D= d+1>10$. Recently, a study of the generalization of the BKL chaotic behaviour to the superstring effective Lagrangians has revealed that this chaos is rooted in the structure of the fundamental Weyl chamber of some underlying hyperbolic Kac-Moody algebra. In this letter, we show that the same connection applies to pure gravity in any spacetime dimension $\geq 4$, where the relevant algebras are $AE_d$. In this way the disappearance of chaos in pure gravity models in $D > 10$ dimensions becomes linked to the fact that the Kac-Moody algebras $AE_d$ are no longer hyperbolic for $d > 9$.