Integrability of supergravity billiards and the generalized Toda lattice equations

We prove that the field equations of supergravity for purely time-dependent backgrounds, which reduce to those of a one-dimensional sigma model, admit a Lax pair representation and are fully integrable. In the case where the effective sigma model is on a maximally split non-compact coset U/H (maximal supergravity or subsectors of lower supersymmetry supergravities) we are also able to construct a completely explicit analytic integration algorithm, adapting a method introduced by Kodama et al. in a recent paper. The properties of the general integral are particularly suggestive. Initial data are represented by a pair C0,h0 where C0 is in the CSA of the Lie algebra of U and h0∈H/W, is in the compact subgroup H modded by the Weyl group of U. At asymptotically early and asymptotically late times the Lax operator is always in the Cartan subalgebra and due to the isospectral property the two limits differ only by the action of some element of the Weyl group. Hence the entire cosmic evolution can be seen as a billiard scattering with quantized angles defined by the Weyl group. The solution algorithm realizes a map from H/W into W.