Black holes admitting a Freudenthal dual

TL;DR

The paper develops two dualities, Freudenthal duality in four dimensions and Jordan duality in five dimensions, acting on integer black-hole charges within Freudenthal triple systems and Jordan algebras. It establishes that these dualities preserve the leading entropy (via $\Delta(x)$ in 4D and $N(A)$ in 5D) and commute with U-duality in a controlled way, though not all discrete U-duality invariants are invariant under the dualities. Integrality conditions demand $|\Delta(x)|$ to be a perfect square (4D) or $N(A)$ to be a perfect cube (5D), restricting to a subset of charges; the authors derive explicit duality maps, their effect on gcd-based invariants, and their behavior under the 4D/5D lift, including detailed canonical-form analyses and STU/model examples. Open questions remain about higher-order corrections and the full relationship between Freudenthal/Jordan duals and U-duality outside the projective sector, motivating further exploration of exact dyon degeneracies and orbit classifications.

Abstract

The quantised charges x of four dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U-duality and whose U-invariant quartic norm Delta(x) determines the lowest order entropy. Here we introduce a Freudenthal duality x -> \tilde{x}, for which \tilde{\tilde{x}}=-x. Although distinct from U-duality it nevertheless leaves Delta(x) invariant. However, the requirement that \tilde{x} be integer restricts us to the subset of black holes for which Delta(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantised charges A of five dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N(A) determines the lowest order entropy. We introduce an analogous Jordan dual A*, with N(A) necessarily a perfect cube, for which A**=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift.