2,015 research outputs found
Geometry of all supersymmetric type I backgrounds
We find the geometry of all supersymmetric type I backgrounds by solving the
gravitino and dilatino Killing spinor equations, using the spinorial geometry
technique, in all cases. The solutions of the gravitino Killing spinor equation
are characterized by their isotropy group in Spin(9,1), while the solutions of
the dilatino Killing spinor equation are characterized by their isotropy group
in the subgroup Sigma(P) of Spin(9,1) which preserves the space of parallel
spinors P. Given a solution of the gravitino Killing spinor equation with L
parallel spinors, L = 1,2,3,4,5,6,8, the dilatino Killing spinor equation
allows for solutions with N supersymmetries for any 0 < N =< L. Moreover for L
= 16, we confirm that N = 8,10,12,14,16. We find that in most cases the Bianchi
identities and the field equations of type I backgrounds imply a further
reduction of the holonomy of the supercovariant connection. In addition, we
show that in some cases if the holonomy group of the supercovariant connection
is precisely the isotropy group of the parallel spinors, then all parallel
spinors are Killing and so there are no backgrounds with N < L supersymmetries.Comment: 73 pages. v2: minor changes, references adde
The Geometry of D=11 Null Killing Spinors
We determine the necessary and sufficient conditions on the metric and the
four-form for the most general bosonic supersymmetric configurations of D=11
supergravity which admit a null Killing spinor i.e. a Killing spinor which can
be used to construct a null Killing vector. This class covers all
supersymmetric time-dependent configurations and completes the classification
of the most general supersymmetric configurations initiated in hep-th/0212008.Comment: 30 pages, typos corrected, reference added, new solution included in
section 5.1; uses JHEP3.cl
New directions in enumerative chess problems
Normally a chess problem must have a unique solution, and is deemed unsound
even if there are alternatives that differ only in the order in which the same
moves are played. In an enumerative chess problem, the set of moves in the
solution is (usually) unique but the order is not, and the task is to count the
feasible permutations via an isomorphic problem in enumerative combinatorics.
Almost all enumerative chess problems have been ``series-movers'', in which one
side plays an uninterrupted series of moves, unanswered except possibly for one
move by the opponent at the end. This can be convenient for setting up
enumeration problems, but we show that other problem genres also lend
themselves to composing enumerative problems. Some of the resulting
enumerations cannot be shown (or have not yet been shown) in series-movers.
This article is based on a presentation given at the banquet in honor of
Richard Stanley's 60th birthday, and is dedicated to Stanley on this occasion.Comment: 14 pages, including many chess diagrams created with the Tutelaers
font
Topology of Exceptional Orbit Hypersurfaces of Prehomogeneous Spaces
We consider the topology for a class of hypersurfaces with highly nonisolated
singularites which arise as exceptional orbit varieties of a special class of
prehomogeneous vector spaces, which are representations of linear algebraic
groups with open orbits. These hypersurface singularities include both
determinantal hypersurfaces and linear free (and free*) divisors. Although
these hypersurfaces have highly nonisolated singularities, we determine the
topology of their Milnor fibers, complements and links. We do so by using the
action of linear algebraic groups beginning with the complement, instead of
using Morse type arguments on the Milnor fibers. This includes replacing the
local Milnor fiber by a global Milnor fiber which has a complex geometry
resulting from a transitive action of an appropriate algebraic group, yielding
a compact model submanifold for the homotopy type of the Milnor fiber. The
topology includes the (co)homology (in characteristic 0, and 2 torsion in one
family) and homotopy groups, and we deduce the triviality of the monodromy
transformations on rational (or complex) cohomology. The cohomology of the
Milnor fibers and complements are isomorphic as algebras to exterior algebras
or for one family, modules over exterior algebras; and cohomology of the link
is, as a vector space, a truncated and shifted exterior algebra, for which the
cohomology product structure is essentially trivial. We also deduce from Bott's
periodicity theorem, the homotopy groups of the Milnor fibers for determinantal
hypersurfaces in the stable range as the stable homotopy groups of the
associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we
obtain a class of formal linear combinations of exceptional orbit hypersurfaces
which have Milnor fibers which are homotopy equivalent to joins of the compact
model submanifolds.Comment: to appear in the Journal of Topolog
Three-Dimensional Manifolds, Skew-Gorenstein Rings and their Cohomology
Graded skew-commutative rings occur often in practice. Here are two examples:
1) The cohomology ring of a compact three-dimensional manifold. 2) The
cohomology ring of the complement of a hyperplane arrangement (the
Orlik-Solomon algebra). We present some applications of the homological theory
of these graded skew-commutative rings. In particular we find compact oriented
3-manifolds without boundary for which the Hilbert series of the Yoneda
Ext-algebra of the cohomology ring of the fundamental group is an explicit
transcendental function. This is only possible for large first Betti numbers of
the 3-manifold (bigger than -- or maybe equal to -- 11). We give also examples
of 3-manifolds where the Ext-algebra of the cohomology ring of the fundamental
group is not finitely generated.Comment: 21 page
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