3,078 research outputs found
Fibered geometries
AbstractOur aim is to initiate the study of fibered geometries, in particular fibered projective planes and fibered generalized polygons. In fact, we apply the theory of fuzzy sets in a particular way on incidence geometry. Combinatorial and geometric questions arise. But also classical objects are recognized by this alternative view: for instance apartments arise naturally in the theory of “contagious values”
Geometric constraints in dual F-theory and heterotic string compactifications
We systematically analyze a broad class of dual heterotic and F-theory models
that give four-dimensional supergravity theories, and compare the geometric
constraints on the two sides of the duality. Specifically, we give a complete
classification of models where the heterotic theory is compactified on a smooth
Calabi-Yau threefold that is elliptically fibered with a single section and
carries smooth irreducible vector bundles, and the dual F-theory model has a
corresponding threefold base that has the form of a P^1 bundle. We formulate
simple conditions for the geometry on the F-theory side to support an
elliptically fibered Calabi-Yau fourfold. We match these conditions with
conditions for the existence of stable vector bundles on the heterotic side,
and show that F-theory gives new insight into the conditions under which such
bundles can be constructed. In particular, we find that many allowed F-theory
models correspond to vector bundles on the heterotic side with exceptional
structure groups, and determine a topological condition that is only satisfied
for bundles of this type. We show that in many cases the F-theory geometry
imposes a constraint on the extent to which the gauge group can be enhanced,
corresponding to limits on the way in which the heterotic bundle can decompose.
We explicitly construct all (4962) F-theory threefold bases for dual
F-theory/heterotic constructions in the subset of models where the common
twofold base surface is toric, and give both toric and non-toric examples of
the general results.Comment: 81 pages, 2 figures; v2, v3: references added, minor corrections; v4:
minor errors, Table 5 correcte
On Isosystolic Inequalities for T^n, RP^n, and M^3
If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology
has the maximal cup-length, then for any riemannian metric g on M, we show that
the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g)
are related by the following isosystolic inequality: Sys(M,g)^n \leq n!
Vol(M,g). The inequality can be regarded as a generalization of Burago and
Hebda's inequality for closed essential surfaces and as a refinement of Guth's
inequality for closed n-manifolds whose Z/2Z-cohomology has the maximal
cup-length. We also establish the same inequality in the context of possibly
non-compact manifolds under a similar cohomological condition. The inequality
applies to (i) T^n and all other compact euclidean space forms, (ii) RP^n and
many other spherical space forms including the Poincar\'e dodecahedral space,
and (iii) most closed essential 3-manifolds including all closed aspherical
3-manifolds.Comment: 34 pages, 0 figures. v2 contains expository revisions and some
additional reference
Warped Entanglement Entropy
We study the applicability of the covariant holographic entanglement entropy
proposal to asymptotically warped AdS spacetimes with an SL(2,R) x U(1)
isometry. We begin by applying the proposal to locally AdS backgrounds
which are written as a real-line fibration over AdS. We then perturb away
from this geometry by considering a warping parameter to get an
asymptotically warped AdS spacetime and compute the dual entanglement
entropy perturbatively in . We find that for large separation in the
fiber coordinate, the entanglement entropy can be computed to all orders in
and takes the universal form appropriate for two-dimensional CFTs. The
warping-dependent central charge thus identified exactly agrees with previous
calculations in the literature. Performing the same perturbative calculations
for the warped BTZ black hole again gives universal two-dimensional CFT
answers, with the left-moving and right-moving temperatures appearing
appropriately in the result.Comment: 25 pages plus appendices; v2 references added, discussions clarified
and equations sharpene
On three-manifolds dominated by circle bundles
We determine which three-manifolds are dominated by products. The result is
that a closed, oriented, connected three-manifold is dominated by a product if
and only if it is finitely covered either by a product or by a connected sum of
copies of the product of the two-sphere and the circle. This characterization
can also be formulated in terms of Thurston geometries, or in terms of purely
algebraic properties of the fundamental group. We also determine which
three-manifolds are dominated by non-trivial circle bundles, and which
three-manifold groups are presentable by products.Comment: 12 pages; to appear in Math. Zeitschrift; ISSN 1103-467
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