Observations on integral and continuous U-duality orbits in N=8 supergravity

One would often like to know when two a priori distinct extremal black p-brane solutions are in fact related by U-duality. In the classical supergravity limit the answer for a large class of theories has been known for some time now. However, in the full quantum theory the U-duality group is broken to a discrete subgroup, a consequence of the Dirac-Zwanziger-Schwinger charge quantization conditions. The question of U-duality orbits in this case is a nuanced matter. In the present work we address this issue in the context of N = 8 supergravity in four, five and six dimensions. The purpose of this paper is to present and clarify what is currently known about these orbits while at the same time filling in some of the details not yet appearing in the literature. For the continuous case we present the cascade of relationships existing between the orbits, generated as one descends from six to four dimensions, together with the corresponding implications for the associated moduli spaces. In addressing the discrete case we exploit the mathematical framework of integral Jordan algebras, the integral Freudenthal triple system and, in particular, the work of Krutelevich. The charge vector of the dyonic black string in D = 6 is SO(5, 5; Z) related to a two-charge reduced canonical form uniquely specified by a set of two arithmetic U-duality invariants. Similarly, the black hole (string) charge vectors in D = 5 are E-6(6)(Z) equivalent to a three-charge canonical form, again uniquely fixed by a set of three arithmetic U-duality invariants. However, the situation in four dimensions is, perhaps predictably, less clear. While black holes preserving more than 1/8 of the supersymmetries may be fully classified by the known arithmetic E-7(7)(Z) invariants, 1/8-BPS and non-BPS black holes yield increasingly subtle orbit structures, which remain to be properly understood. However, for the very special subclass of projective black holes a complete classification is known. All projective black holes are E-7(7)(Z) related to a four- or five-charge canonical form determined uniquely by the set of known arithmetic U-duality invariants. Moreover, E-7(7)(Z) acts transitively on the charge vectors of projective black holes with a given leading-order entropy.