Black holes admitting a Freudenthal dual

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 ∆(x) determines the lowest order entropy. Here we introduce a Freudenthal duality x →x, for whichx = −x. Although distinct from U-duality it nevertheless leaves ∆(x) invariant. However, the requirement thatx be integer restricts us to the subset of black holes for which ∆(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.

Degeneration of groups of type E 7 and minimal coupling in supergravity

“…The connection of groups of type E 7 to supergravity can be summarized by stating that all 2 N 8-extended supergravities in D = 4 with symmetric scalar manifolds [17,73]. It is intriguing to notice that the first paper on groups of type E 7…”

“…All the cases in which I 4 degenerates to (I 2 ) 2 provide instances of the so-called Freudenthal duality [17,73], whose manifest invariance (by construction, and apart from possible "hidden" symmetries) is given by the U -duality group of the theory obtained after truncation.…”

“…Simple, degenerate groups of type E 7 relevant to D = 4 supergravity (namely, U(1, n) or U(3, n)) share the property that the dyonic charge vector Q (1.7) (element 3 An analysis at the level of quartic invariant polynomial, and dependent on charge configurations, has been considered in [17]. 4 In [5] these groups were called "not of type E7".…”

Creation of matter in the universe and groups of type E7

“…This was about a decade before 2 the discovery of supergravity. In fact, surprisingly, all extended supergravities 2 ≤ N ≤ 8 described by coset spaces G H have G of type E7 [25][26][27], with the exception of N = 2 group G = U(1, n) and N = 3 group G = U(3, n). These G = U(p, n) groups are not of type E7, since they have a primitive quadratic symmetric invariant (in addition to a symplectic bilinear form).…”

“…The first set of all extended supergravities d = 4 N > 1 based on symmetric coset spaces G/H has a remarkable property that almost all groups G are of type E7 [21][22][23][24][25], see tables 1, 2. This universal property will be used to prove that for all extended supergravities d = 4 N > 1 based on symmetric coset space G/H the vector coupling N ΛΣ (φ) must be scalar dependent, with exception of hyper scalars.…”