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 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 front the minimal accessible value of l(T) to the maximum one or vice versa. (C) 2009 Elsevier B.V. All rights reserved.

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

FRE', Pietro Giuseppe;
2009-01-01

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 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 front the minimal accessible value of l(T) to the maximum one or vice versa. (C) 2009 Elsevier B.V. All rights reserved.
2009
815
430
494
P. Fre;A. S. Sorin
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/101113
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