705 research outputs found
Exceptional Moufang quadrangles and structurable algebras
In 2000, J. Tits and R. Weiss classified all Moufang spherical buildings of
rank two, also known as Moufang polygons. The hardest case in the
classification consists of the Moufang quadrangles. They fall into different
families, each of which can be described by an appropriate algebraic structure.
For the exceptional quadrangles, this description is intricate and involves
many different maps that are defined ad hoc and lack a proper explanation.
In this paper, we relate these algebraic structures to two other classes of
algebraic structures that had already been studied before, namely to
Freudenthal triple systems and to structurable algebras. We show that these
structures give new insight in the understanding of the corresponding Moufang
quadrangles.Comment: 49 page
M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized Spin(7) structures
The requirement of supersymmetry for M-theory backgrounds of the
form of a warped product , where is an eight-manifold
and is three-dimensional Minkowski or AdS space, implies the
existence of a nowhere-vanishing Majorana spinor on . lifts to a
nowhere-vanishing spinor on the auxiliary nine-manifold , where
is a circle of constant radius, implying the reduction of the structure
group of to . In general, however, there is no reduction of the
structure group of itself. This situation can be described in the language
of generalized structures, defined in terms of certain spinors of
. We express the condition for supersymmetry
in terms of differential equations for these spinors. In an equivalent
formulation, working locally in the vicinity of any point in in terms of a
`preferred' structure, we show that the requirement of
supersymmetry amounts to solving for the intrinsic torsion and all irreducible
flux components, except for the one lying in the of , in
terms of the warp factor and a one-form on (not necessarily
nowhere-vanishing) constructed as a bilinear; in addition, is
constrained to satisfy a pair of differential equations. The formalism based on
the group is the most suitable language in which to describe
supersymmetric compactifications on eight-manifolds of structure,
and/or small-flux perturbations around supersymmetric compactifications on
manifolds of holonomy.Comment: 24 pages. V2: introduction slightly extended, typos corrected in the
text, references added. V3: the role of Spin(7) clarified, erroneous
statements thereof corrected. New material on generalized Spin(7) structures
in nine dimensions. To appear in JHE
Exceptional Superconformal Algebras
Reductive W-algebras which are generated by bosonic fields of spin-1, a
single spin-2 field and fermionic fields of spin-3/2 are classified. Three new
cases are found: a `symplectic' family of superconformal algebras which are
extended by , an and an superconformal algebra.
The exceptional cases can be viewed as arising a Drinfeld-Sokolov type
reduction of the exceptional Lie superalgebras and , and have an
octonionic description. The quantum versions of the superconformal algebras are
constructed explicitly in all three cases.Comment: 16 page
AdS solutions with exceptional supersymmetry
Among the possible superalgebras that contain the AdS isometries, two
interesting possibilities are the exceptional and . Their
R-symmetry is respectively SO(7) and , and the amount of supersymmetry
and . We find that there exist two (locally) unique
solutions in type IIA supergravity that realize these superalgebras, and we
provide their analytic expressions. In both cases, the internal space is
obtained by a round six-sphere fibred over an interval, with an O8-plane at one
end. The R-symmetry is the symmetry group of the sphere; in the case, it
is broken to by fluxes. We also find several numerical
solutions with flavor symmetry, with various localized sources, including
O2-planes and O8-planes.Comment: 30 pages, 4 figures; v3: revised appendix, minor correction
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