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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, CE10=A15(2)∧, also known as the dual of B8∧∧, the maximally oxidized Lagrangian is nine-dimensional and involves besides gravity, 2 dilatons, a 2-form, a 1-form, and a 0-form. [ABSTRACT FROM AUTHOR]

A new geometrical framework for tetrad-affine formulation of gravity, pure or coupled with Yang–Mills fields, is proposed. After analyzing the geometrical properties of the new mathematical setting, field equations are deduced from a variational principle in the Poincaré–Cartan formalism. A generalized Noether Theorem is stated and classical relationship between symmetries and conserved quantities are recovered in the newer scheme. Some explicit examples are given. [ABSTRACT FROM AUTHOR]