360 research outputs found
On commuting -operators in Jordan algebras
Recently J.A.Anquela, T.Cort\'es, and H.Petersson proved that for elements
in a non-degenerate Jordan algebra , the relation
implies that the -operators of and commute: . We
show that the result may be not true without the assumption on non-degeneracity
of . We give also a more simple proof of the mentioned result in the case of
linear Jordan algebras, that is, when
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
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.
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
Entropy on Spin Factors
Recently it has been demonstrated that the Shannon entropy or the von Neuman
entropy are the only entropy functions that generate a local Bregman
divergences as long as the state space has rank 3 or higher. In this paper we
will study the properties of Bregman divergences for convex bodies of rank 2.
The two most important convex bodies of rank 2 can be identified with the bit
and the qubit. We demonstrate that if a convex body of rank 2 has a Bregman
divergence that satisfies sufficiency then the convex body is spectral and if
the Bregman divergence is monotone then the convex body has the shape of a
ball. A ball can be represented as the state space of a spin factor, which is
the most simple type of Jordan algebra. We also study the existence of recovery
maps for Bregman divergences on spin factors. In general the convex bodies of
rank 2 appear as faces of state spaces of higher rank. Therefore our results
give strong restrictions on which convex bodies could be the state space of a
physical system with a well-behaved entropy function.Comment: 30 pages, 6 figure
Three fermions with six single particle states can be entangled in two inequivalent ways
Using a generalization of Cayley's hyperdeterminant as a new measure of
tripartite fermionic entanglement we obtain the SLOCC classification of
three-fermion systems with six single particle states. A special subclass of
such three-fermion systems is shown to have the same properties as the
well-known three-qubit ones. Our results can be presented in a unified way
using Freudenthal triple systems based on cubic Jordan algebras. For systems
with an arbitrary number of fermions and single particle states we propose the
Pl\"ucker relations as a sufficient and necessary condition of separability.Comment: 23 pages LATE
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
Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups
We study the symmetries of generalized spacetimes and corresponding phase
spaces defined by Jordan algebras of degree three. The generic Jordan family of
formally real Jordan algebras of degree three describe extensions of the
Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation,
Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and
SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple
Jordan algebras of degree three correspond to extensions of Minkowskian
spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra
(2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal
triple systems defined over these Jordan algebras describe conformally
covariant phase spaces. Following hep-th/0008063, we give a unified geometric
realization of the quasiconformal groups that act on their conformal phase
spaces extended by an extra "cocycle" coordinate. For the generic Jordan family
the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are
given. The minimal unitary representations of the quasiconformal groups F_4(4),
E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our
earlier work hep-th/0409272.Comment: A typo in equation (37) corrected and missing titles of some
references added. Version to be published in JHEP. 38 pages, latex fil
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