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

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

Jul 2009

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2009nuphb.815..430f&link_type=abstract

Nuclear Physics B, Volume 815, Issue 3, p. 430-494.

Physics

Nuclear Physics

2

Scientific paper

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, the number of reflections with respect to the simple roots. The direction of time flows is always from the minimal accessible value of ℓ to the maximum one or vice versa.

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