Black Holes and the Orbits of E_{7(7)}(Z)

In supergravity one would often like to know when two a priori distinct
extremal black p-brane charge configurations are in fact related by U-duality,
the symmetries of the theory. We address this issue in the context of N=8
supergravity in 6, 5 and 4 dimensions where the U-dualities are given by the
discrete groups: SO(5,5; Z), E6(Z) and E7(Z), respectively. The first
challenge is to determine what we mean by E7(Z) precisely. To this end 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 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 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 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.