Integrability of Supergravity Billiards and the generalized Toda lattice equation

TL;DR

This paper demonstrates that supergravity field equations for time-dependent backgrounds are integrable via a Lax pair, allowing explicit solutions that interpret cosmic evolution as a Weyl group billiard scattering.

Contribution

It establishes the integrability of supergravity billiards and constructs an explicit analytic algorithm for solutions in maximally split cosets, linking cosmic evolution to Weyl group symmetries.

Findings

Supergravity equations admit a Lax pair representation and are fully integrable.

Explicit solution algorithm maps initial data to Weyl group elements.

Cosmic evolution corresponds to a billiard scattering process with quantized angles.

Abstract

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 C_0, h_0 where C_0 is in the CSA of the Lie algebra of U and h_0 in 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 iso-spectral 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.