136 research outputs found
Products, coproducts and singular value decomposition
Products and coproducts may be recognized as morphisms in a monoidal tensor
category of vector spaces. To gain invariant data of these morphisms, we can
use singular value decomposition which attaches singular values, ie generalized
eigenvalues, to these maps. We show, for the case of Grassmann and Clifford
products, that twist maps significantly alter these data reducing degeneracies.
Since non group like coproducts give rise to non classical behavior of the
algebra of functions, ie make them noncommutative, we hope to be able to learn
more about such geometries. Remarkably the coproduct for positive singular
values of eigenvectors in yields directly corresponding eigenvectors in
A\otimes A.Comment: 17 pages, three eps-figure
Domesticity in generalized quadrangles
An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and linedomestic. In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4), and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: Either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4
A somewhat gentle introduction to differential graded commutative algebra
Differential graded (DG) commutative algebra provides powerful techniques for
proving theorems about modules over commutative rings. These notes are a
somewhat colloquial introduction to these techniques. In order to provide some
motivation for commutative algebraists who are wondering about the benefits of
learning and using these techniques, we present them in the context of a recent
result of Nasseh and Sather-Wagstaff. These notes were used for the course
"Differential Graded Commutative Algebra" that was part of the Workshop on
Connections Between Algebra and Geometry held at the University of Regina, May
29--June 1, 2012.Comment: 78 page
A scan for new N=1 vacua on twisted tori
We perform a systematic search for N=1 Minkowski vacua of type II string
theories on compact six-dimensional parallelizable nil- and solvmanifolds
(quotients of six-dimensional nilpotent and solvable groups, respectively).
Some of these manifolds have appeared in the construction of string backgrounds
and are typically called twisted tori. We look for vacua directly in ten
dimensions, using the a reformulation of the supersymmetry condition in the
framework of generalized complex geometry. Certain algebraic criteria to
establish compactness of the manifolds involved are also needed. Although the
conditions for preserved N=1 supersymmetry fit nicely in the framework of
generalized complex geometry, they are notoriously hard to solve when coupled
to the Bianchi identities. We find solutions in a large-volume,
constant-dilaton limit. Among these, we identify those that are T-dual to
backgrounds of IIB on a conformal T^6 with self-dual three-form flux, and hence
conceptually not new. For all backgrounds of this type fully localized
solutions can be obtained. The other new solutions need multiple intersecting
sources (either orientifold planes or combinations of O-planes and D-branes) to
satisfy the Bianchi identities; the full list of such new solution is given.
These are so far only smeared solutions, and their localization is yet unknown.
Although valid in a large-volume limit, they are the first examples of
Minkowski vacua in supergravity which are not connected by any duality to a
Calabi-Yau. Finally, we discuss a class of flat solvmanifolds that may lead to
AdS_4 vacua of type IIA strings.Comment: 75 pages, 1 figure. v.2: minor corrections, references added; v3:
several changes and clarification
Augmentations are Sheaves
We show that the set of augmentations of the Chekanov-Eliashberg algebra of a
Legendrian link underlies the structure of a unital A-infinity category. This
differs from the non-unital category constructed in [BC], but is related to it
in the same way that cohomology is related to compactly supported cohomology.
The existence of such a category was predicted by [STZ], who moreover
conjectured its equivalence to a category of sheaves on the front plane with
singular support meeting infinity in the knot. After showing that the
augmentation category forms a sheaf over the x-line, we are able to prove this
conjecture by calculating both categories on thin slices of the front plane. In
particular, we conclude that every augmentation comes from geometry.Comment: 109 pages; v2: added Legendrian mirror example in section 4.4.4,
corrected typos and other minor changes; v3: accepted versio
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