The Weyl group and asymptotics: All supergravity billiards have a closed form general integral

Abstract

In this paper we show that all supergravity billiards corresponding to σ-models on any U / H non-compact-symmetric space and obtained by compactifying supergravity to D = 3 admit a closed form general integral depending analytically on a complete set of integration constants. The key point in establishing the integration algorithm is provided by an upper triangular embedding of the solvable Lie algebra associated with U / H into sl ( N , R ) which is guaranteed to exist for all non-compact symmetric spaces and also for homogeneous special geometries non-corresponding to symmetric spaces. In this context we establish a remarkable relation between the end-points of the time-flow and the properties of the Weyl group. The asymptotic states of the developing Universe are in one-to-one correspondence with the elements of the Weyl group which is a property of the Tits–Satake universality classes and not of their single representatives. Furthermore the Weyl group admits a natural ordering in terms of ℓ T , the number of reflections with respect to the simple roots. The direction of time flows is always from the minimal accessible value of ℓ T to the maximum one or vice versa.