Integrability of Supergravity Black Holes and New Tensor Classifiers of Regular and Nilpotent Orbits

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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 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, H* being the isotropy group of scalar manifold originating from 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 and practical discriminating method.