The arrow of time and the Weyl group: all supergravity billiards are integrable

Abstract

In this paper we show that all supergravity billiards corresponding to sigma-models on any U/H non compact-symmetric space and obtained by compactifying supergravity to D=3 are fully integrable. 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 always exists. In this context we establish a remarkable relation between the arrow of time 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 and the direction of time flows is always towards increasing L(T), which plays the unexpected role of an entropy.

arXiv comment73 pages 34 figures. Research paper, not review

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