Black holes admitting a Freudenthal dual

TL;DR

This paper introduces dualities for black hole charges in string theory, based on Freudenthal and Jordan algebra structures, which preserve entropy invariants and relate four- and five-dimensional black holes.

Contribution

It defines new Freudenthal and Jordan dualities for black hole charges, linking algebraic structures to entropy invariants and dimensional lifts.

Findings

Freudenthal duality leaves the quartic norm invariant.

Jordan duality preserves the cubic norm.

Dualities relate 4D and 5D black hole descriptions.

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.