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

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### Abstract

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.

### Most cited references9

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### Infinite dimensional Lie algebras

(1990)
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### A general solution of the Einstein equations with a time singularity

(2006)
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### Evidence for F-Theory

(1996)
We construct compact examples of D-manifolds for type IIB strings. The construction has a natural interpretation in terms of compactification of a 12 dimensional `F-theory'. We provide evidence for a more natural reformulation of type IIB theory in terms of F-theory. Compactification of M-theory on a manifold $$K$$ which admits elliptic fibration is equivalent to compactification of F-theory on $$K\times S^1$$. A large class of $$N=1$$ theories in 6 dimensions are obtained by compactification of F-theory on Calabi-Yau threefolds. A class of phenomenologically promising compactifications of F-theory is on $$Spin(7)$$ holonomy manifolds down to 4 dimensions. This may provide a concrete realization of Witten's proposal for solving the cosmological constant problem in four dimensions.
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### Author and article information

###### Journal
19 December 2000
2001-04-12
hep-th/0012172 10.1103/PhysRevLett.86.4749