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

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 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 lT , the number of reflections with respect to the simple roots and the direction of time flows is always towards increasing lT , which plays the unexpected role of an entropy. † This work is supported in part by the European Union RTN contract MRTN-CT-2004-005104 and by the Italian Ministry of University (MIUR) under contracts PRIN 2005-024045 and PRIN 2005-023102. Furthermore the work of A.S. was partially supported by the RFBR Grant No. 06-01-00627-a, RFBR-DFG Grant No. 06-02-04012-a, DFG Grant 436 RUS 113/669-3, the Program for Supporting Leading Scientific Schools (Grant No. NSh-5332.2006.2), and the Heisenberg-Landau Program.