Observations on Integral and Continuous U-duality Orbits in N=8 Supergravity

TL;DR

This paper investigates the classification of extremal black p-brane solutions in N=8 supergravity across various dimensions, focusing on U-duality orbits and their invariants in the quantum regime using advanced algebraic frameworks.

Contribution

It provides a detailed analysis of discrete U-duality orbits in N=8 supergravity, clarifying known classifications and filling gaps with new insights using Jordan algebras and Freudenthal systems.

Findings

Charge vectors in D=6 and D=5 are classified by unique arithmetic U-duality invariants.

In D=4, black hole orbit structures are complex, with complete classification known only for special subclasses.

E_{7(7)}(Z) acts transitively on charge vectors with fixed entropy for certain black holes.

Abstract

One would often like to know when two a priori distinct extremal black p-brane solutions are in fact U-duality related. In the classical supergravity limit the answer for a large class of theories has been known for some time. However, in the full quantum theory the U-duality group is broken to a discrete subgroup and 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 note is to present and clarify what is currently known about these discrete orbits while at the same time filling in some of the details not yet appearing in the literature. To this end we exploit the mathematical framework of integral Jordan algebras and Freudenthal triple systems. 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. The situation in four dimensions is 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 black holes with a given leading-order entropy.