Attractors in Supergravity and Integrability

Wissam Chemissany, Thomas Van Riet, Pietro Fre, Alexander Sorin, Jan Rosseel

Stationary black holes of massless supergravity theories are described by certain geodesic curves on the target space that is obtained after dimensional reduction over time. When the target space is a symmetric coset space we make use of the group-theoretical structure to prove that the second-order geodesic equations are integrable in the sense of Liouville, by explicitly constructing the correct amount of Hamiltonians in involution. This implies that the Hamilton{Jacobi formalism can be applied, which proves that all such black hole solutions, including non-extremal solutions, possess a description in terms of a (fake) superpotential. Furthermore, we improve the existing integration method by the construction of a Lax integration algorithm that integrates the second-order equations in one step instead of the usual two step procedure. We illustrate this technology with the dilatonic black hole and leave more involved examples for further investigation.