E_(10), BE_(10) and Arithmetical Chaos in Superstring Cosmology

It is shown that the never ending oscillatory behaviour of the generic solution, near a cosmological singularity, of the massless bosonic sector of superstring theory can be described as a billiard motion within a simplex in 9-dimensional hyperbolic space. The Coxeter group of reflections of this billiard is discrete and is the Weyl group of the hyperbolic Kac-Moody algebra E${}_{10}$ (for type II) or BE${}_{10}$ (for type I or heterotic), which are both arithmetic. These results lead to a proof of the chaotic (Anosov'') nature of the classical cosmological oscillations, and suggest a chaotic quantum billiard’’ scenario of vacuum selection in string theory.