Supergravity Black Holes and Billiards and Liouville integrable structure of dual Borel algebras

Abstract

In this paper we show that the supergravity equations describing both cosmic billiards and a large class of black-holes are, generically, both Liouville integrable as a consequence of the same universal mechanism. This latter is provided by the Liouville integrable Poissonian structure existing on the dual Borel algebra B_N of the simple Lie algebra A_{N-1}. As a by product we derive the explicit integration algorithm associated with all symmetric spaces U/H^{*} relevant to the description of time-like and space-like p-branes. The most important consequence of our approach is the explicit construction of a complete set of conserved involutive hamiltonians h_α that are responsible for integrability and provide a new tool to classify flows and orbits. We believe that these will prove a very important new tool in the analysis of supergravity black holes and billiards.

48 pages, 7 figures, LaTex; V1: misprints corrected, two references added

References in corpus (6)

• Non-Supersymmetric Attractor Flow in Symmetric Spaces
• Universal BPS structure of stationary supergravity solutions
• Generating Geodesic Flows and Supergravity Solutions
• Quantum Attractor Flows
• The full integration of black hole solutions to symmetric supergravity theories
• Tits-Satake projections of homogeneous special geometries

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• The Integration Algorithm of Lax equation for both Generic Lax matrices and Generic Initial Conditions
• Extremal Multicenter Black Holes: Nilpotent Orbits and Tits Satake Universality Classes
• Black holes as generalised Toda molecules
• The $c$-map, Tits Satake subalgebras and the search for $\mathcal{N}=2$ inflaton potentials
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• Brane solutions and integrability: a status report