Black holes admitting a Freudenthal dual

The quantized 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 -> x approximate to, for which x approximate to approximate to=-x. Although distinct from U duality, it nevertheless leaves Delta(x) invariant. However, the requirement that x approximate to be an 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 quantized 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(star), with N(A) necessarily a perfect cube, for which A(star star)=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift.