Black holes admitting a Freudenthal dual

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.Comment: 32 pages revtex, 10 tables; minor corrections, references adde

Contractions of low-dimensional nilpotent Jordan algebras

In this paper we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among them. In particular, we prove that J2 and J3 are irreducible and that J4 is the union of the Zariski closures of two rigid Jordan algebras.Comment: 12 pages, 3 figure

Small Orbits

We study both the "large" and "small" U-duality charge orbits of extremal black holes appearing in D = 5 and D = 4 Maxwell-Einstein supergravity theories with symmetric scalar manifolds. We exploit a formalism based on cubic Jordan algebras and their associated Freudenthal triple systems, in order to derive the minimal charge representatives, their stabilizers and the associated "moduli spaces". After recalling N = 8 maximal supergravity, we consider N = 2 and N = 4 theories coupled to an arbitrary number of vector multiplets, as well as N = 2 magic, STU, ST^2 and T^3 models. While the STU model may be considered as part of the general N = 2 sequence, albeit with an additional triality symmetry, the ST^2 and T^3 models demand a separate treatment, since their representative Jordan algebras are Euclidean or only admit non-zero elements of rank 3, respectively. Finally, we also consider minimally coupled N = 2, matter coupled N = 3, and "pure" N = 5 theories.Comment: 40 pages, 9 tables. References added. Expanded comments added to sections III. C. 1. and III. F.

Response of selected plant and insect species to simulated solid rocket exhaust mixtures and to exhaust components from solid rocket fuels

The effects of solid rocket fuel (SRF) exhaust on selected plant and and insect species in the Merritt Island, Florida area was investigated in order to determine if the exhaust clouds generated by shuttle launches would adversely affect the native, plants of the Merritt Island Wildlife Refuge, the citrus production, or the beekeeping industry of the island. Conditions were simulated in greenhouse exposure chambers and field chambers constructed to model the ideal continuous stirred tank reactor. A plant exposure system was developed for dispensing and monitoring the two major chemicals in SRF exhaust, HCl and Al203, and for dispensing and monitoring SRF exhaust (controlled fuel burns). Plants native to Merritt Island, Florida were grown and used as test species. Dose-response relationships were determined for short term exposure of selected plant species to HCl, Al203, and mixtures of the two to SRF exhaust

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

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.Comment: 43 pages, 8 tables; minor corrections, references added; version to appear in Class. Quantum Gra

Jordan Pairs, E6 and U-Duality in Five Dimensions

By exploiting the Jordan pair structure of U-duality Lie algebras in D = 3 and the relation to the super-Ehlers symmetry in D = 5, we elucidate the massless multiplet structure of the spectrum of a broad class of D = 5 supergravity theories. Both simple and semi-simple, Euclidean rank-3 Jordan algebras are considered. Theories sharing the same bosonic sector but with different supersymmetrizations are also analyzed.Comment: 1+41 pages, 1 Table; v2 : a Ref. and some comments adde

Unified N=2 Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions

We study unified N=2 Maxwell-Einstein supergravity theories (MESGTs) and unified Yang-Mills Einstein supergravity theories (YMESGTs) in four dimensions. As their defining property, these theories admit the action of a global or local symmetry group that is (i) simple, and (ii) acts irreducibly on all the vector fields of the theory, including the ``graviphoton''. Restricting ourselves to the theories that originate from five dimensions via dimensional reduction, we find that the generic Jordan family of MESGTs with the scalar manifolds [SU(1,1)/U(1)] X [SO(2,n)/SO(2)X SO(n)] are all unified in four dimensions with the unifying global symmetry group SO(2,n). Of these theories only one can be gauged so as to obtain a unified YMESGT with the gauge group SO(2,1). Three of the four magical supergravity theories defined by simple Euclidean Jordan algebras of degree 3 are unified MESGTs in four dimensions. Two of these can furthermore be gauged so as to obtain 4D unified YMESGTs with gauge groups SO(3,2) and SO(6,2), respectively. The generic non-Jordan family and the theories whose scalar manifolds are homogeneous but not symmetric do not lead to unified MESGTs in four dimensions. The three infinite families of unified five-dimensional MESGTs defined by simple Lorentzian Jordan algebras, whose scalar manifolds are non-homogeneous, do not lead directly to unified MESGTs in four dimensions under dimensional reduction. However, since their manifolds are non-homogeneous we are not able to completely rule out the existence of symplectic sections in which these theories become unified in four dimensions.Comment: 47 pages; latex fil

Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N=2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N=2 MESGTs are the quasiconformal groups QConf(J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D, N=2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results.Comment: 55 pages; Latex fil