Hyperbolic Kac Moody Algebras and Einstein Billiards

Abstract

We identify the hyperbolic Kac Moody algebras for which there exists a Lagrangian of gravity, dilatons and $p$-forms which produces a billiard that can be identified with their fundamental Weyl chamber. Because of the invariance of the billiard upon toroidal dimensional reduction, the list of admissible algebras is determined by the existence of a Lagrangian in three space-time dimensions, where a systematic analysis can be carried out since only zero-forms are involved. We provide all highest dimensional parent Lagrangians with their full spectrum of $p$-forms and dilaton couplings. We confirm, in particular, that for the rank 10 hyperbolic algebra, $C{E}_{10}=A_{15}^{(2)\wedge }$, also known as the dual of $B_8^{\wedge \wedge }$, the maximally oxidized Lagrangian is 9 dimensional and involves besides gravity, 2 dilatons, a 2-form, a 1-form and a 0-form.

33 pages

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Cited by in corpus (15)

• Spacelike Singularities and Hidden Symmetries of Gravity
• E10 and SO(9,9) invariant supergravity
• Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits
• E10 Orbifolds
• The classification of almost affine (hyperbolic) Lie superalgebras
• Poincare, Relativity, Billiards and Symmetry
• Quantum billiards in multidimensional models with branes
• Kac-Moody Algebras in M-theory
• Embeddings of hyperbolic Kac-Moody algebras into ${\mathbf{E}}_{10}$}
• On Brane Solutions Related to Non-Singular Kac-Moody Algebras
• The Poincare series of the hyperbolic Coxeter groups with finite volume of fundamental domains
• Infinite affine, hyperbolic and Lorentzian Weyl groups with their associated Calogero models
• Black brane solutions related to non-singular Kac-Moody algebras
• Quantum billiards with branes on product of Einstein spaces
• On a Hidden Supersymmetry of Cosmological Billiards