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

TL;DR

It is shown that the neverending oscillatory behavior of the generic solution of the massless bosonic sector of superstring theory can be described as a billiard motion within a simplex in nine-dimensional hyperbolic space, leading to a proof of the chaotic ("Anosov") nature of the classical cosmological oscillations.

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.

Figures (1)

• Figure 1: Dynkin diagrams defined (for each $n = 2,1,0$) by the ten wall forms $w_i^{[n]}(\beta^{\mu}), i = 1,\ldots,10$ that determine the billiard dynamics, near a cosmological singularity, of the three blocks of theories ${\cal B}_2 = \{$M, IIA, IIB$\}$, ${\cal B}_1 = \{$I, HO, HE$\}$ and ${\cal B}_0 = \{D = 10$ closed bosonic$\}$. The node labels $1,\ldots,10$ correspond to the form label $i$ used in the text.