Integrability of supergravity black holes and new tensor classifiers of regular and nilpotent orbits

Abstract

In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of $ \mathcal{N} = 2 $ supergravities with symmetric Special Geometry. Our main goal is the classification of these black-holes according to the H⋆-orbits in which the space of possible Lax operators decomposes. By H⋆ one denotes the isotropy group of the coset UD=3/H⋆ which appears in the time-like dimensional reduction of supergravity from D = 4 to D = 3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H⋆-orbits of Lax operators. Our tensor based classification is compared, in the case of the simple one-field model S 3, to the algebraic classification of nilpotent orbits and it is shown to provide a simple discriminating method. In particular we present a detailed analysis of the S 3 model, constructing explicitly its solutions and discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostant-representation of a generic Lie algebra element, we were able to develop an algorithm which produces the necessary number of hamiltonians in involution required by Liouville integrability of generic orbits. The degenerate orbits correspond to extremal black-holes and are nilpotent. We present an in depth discussion of their identification and of the construction of the corresponding supergravity solutions. We dwell on the relation between H⋆ orbits and critical points of the geodesic potential showing that there is correspondence yet not one-to-one.