Hyperbolic Kac Moody Algebras and Einstein Billiards

TL;DR

This paper classifies hyperbolic Kac-Moody algebras that correspond to gravity billiards, identifying their Lagrangians and spectrum of fields, with a focus on the rank 10 algebra $CE_{10}$ and its maximal oxidation.

Contribution

It systematically determines the hyperbolic Kac-Moody algebras compatible with gravity billiards and provides explicit Lagrangians and field content for these cases.

Findings

Classified hyperbolic Kac-Moody algebras related to gravity billiards.

Identified maximal dimension Lagrangians for these algebras.

Explicit spectrum of fields for the rank 10 algebra $CE_{10}$.

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, $CE_{10} = A_{15}^{(2)\wedge}$, 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.