616 research outputs found
Octonions, E6, and Particle Physics
In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan
identity, for a sensible theory of quantum mechanics. All but one of the
algebras that satisfy this condition can be described by Hermitian matrices
over the complexes or quaternions. The remaining, exceptional Jordan algebra
can be described by 3x3 Hermitian matrices over the octonions.
We first review properties of the octonions and the exceptional Jordan
algebra, including our previous work on the octonionic Jordan eigenvalue
problem. We then examine a particular real, noncompact form of the Lie group
E6, which preserves determinants in the exceptional Jordan algebra.
Finally, we describe a possible symmetry-breaking scenario within E6: first
choose one of the octonionic directions to be special, then choose one of the
2x2 submatrices inside the 3x3 matrices to be special. Making only these two
choices, we are able to describe many properties of leptons in a natural way.
We further speculate on the ways in which quarks might be similarly encoded.Comment: 13 pages; 6 figures; TonyFest plenary talk (York 2008
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.
Geometric scaling in high-energy QCD at nonzero momentum transfer
We show how one can obtain geometric scaling properties from the
Balitsky-Kovchegov (BK) equation. We start by explaining how, this property
arises for the b-independent BK equation. We show that it is possible to extend
this model to the full BK equation including momentum transfer. The saturation
scale behaves like max(q,Q_T) where q is the momentum transfer and Q_T a
typical scale of the target.Comment: 4 pages, 2 figures. Talk given by G. Soyez at the "Rencontres de
Moriond", 12-19 March 2005, La Thuile, Ital
Common Representation of Information Flows for Dynamic Coalitions
We propose a formal foundation for reasoning about access control policies
within a Dynamic Coalition, defining an abstraction over existing access
control models and providing mechanisms for translation of those models into
information-flow domain. The abstracted information-flow domain model, called a
Common Representation, can then be used for defining a way to control the
evolution of Dynamic Coalitions with respect to information flow
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
Report and preliminary results of POSEIDON Cruise POS 272 [POS272], Las Palmas - Las Palmas; 01.04.2001 - 14.04.2001
Educating Health Professionals about Disability: A Review of Interventions
Health professionals need to understand the human rights and health needs of disabled people. This review of evidence on interventions demonstrates that a range of often innovative approaches have been trialled. Lectures by faculty are less effective in changing attitudes than contact with disabled people themselves. Existing examples of good practice need to be scaled up, and better and more long-term evaluations of impact are required
Topological wave functions and heat equations
It is generally known that the holomorphic anomaly equations in topological
string theory reflect the quantum mechanical nature of the topological string
partition function. We present two new results which make this assertion more
precise: (i) we give a new, purely holomorphic version of the holomorphic
anomaly equations, clarifying their relation to the heat equation satisfied by
the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian
symmetric tube domain , we show that the general solution of the anomaly
equations is a matrix element \IP{\Psi | g | \Omega} of the
Schr\"odinger-Weil representation of a Heisenberg extension of , between an
arbitrary state and a particular vacuum state .
Based on these results, we speculate on the existence of a one-parameter
generalization of the usual topological amplitude, which in symmetric cases
transforms in the smallest unitary representation of the duality group in
three dimensions, and on its relations to hypermultiplet couplings, nonabelian
Donaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic
changes, published version; v4: typos fixed, small clarification adde
The Seven-sphere and its Kac-Moody Algebra
We investigate the seven-sphere as a group-like manifold and its extension to
a Kac-Moody-like algebra. Covariance properties and tensorial composition of
spinors under are defined. The relation to Malcev algebras is
established. The consequences for octonionic projective spaces are examined.
Current algebras are formulated and their anomalies are derived, and shown to
be unique (even regarding numerical coefficients) up to redefinitions of the
currents. Nilpotency of the BRST operator is consistent with one particular
expression in the class of (field-dependent) anomalies. A Sugawara construction
is given.Comment: 22 pages. Macropackages used: phyzzx, epsf. Three epsf figure files
appende
Co-operative learning and adaptive instruction in a mathematics curriculum
The AGO 12 to 16 Project (the acronym AGO stands for the Dutch equivalent of 'Adaptive Instruction and Co-operative Learning') seeks to develop and evaluate a mathematics curriculum which is suitable for mixed-ability groups in secondary education. The research questions we will address here are, first, whether this curriculum is feasible and effective, and, second, what effects, if any, the context variables time and mean cognitive level of the class have on learning. Many mathematics programmes make insufficient allowance for the differences in intellectual ability that exist in mixed-ability classes. In order to change this situation we developed a mathematics curriculum with adaptive qualities. The evaluation of the experimental curriculum was carried out in two stages. During the first stage the curriculum was used at two schools with the aim of investigating the feasibility of the programme. Experience with the implementation of the programme led to some improvements in the experimental materials. By and large the AGO model appeared to be feasible in secondary classrooms. In the second stage, which was on a large scale, the focus was on the effectiveness of the programme. Six hundred students, 13 teachers and six schools were involved in the research. Teachers in the experimental group were trained in AGO methods and in implementing the new AGO curriculum. Teachers in the control groups worked with the existing programme following their usual methods of teaching. The main conclusion of the study is positive. The AGO model as a whole proved to be practical and effective in learning mathematics. The AGO model has a positive effect on the intercept, which means that the mean scores of AGO classes are higher than the mean scores of non-AGO classes. It may be concluded that, on the average, students benefit from learning in AGO classes as compared with non-AGO classes. AGO does not increase or decrease the differences between students in the same class. As expected, positive effects of two context variables were found: (1) the total amount of time spent in class covering the mathematical content and (2) class composition as indicated by the mean pretest score (aptitude) of the class
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