21,966 research outputs found
Topological Invariants and Fibration Structure of Complete Intersection Calabi-Yau Four-Folds
We investigate the mathematical properties of the class of Calabi-Yau
four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497
configuration matrices which correspond to manifolds that are described as
complete intersections in products of projective spaces. For each manifold in
the list, we compute the full Hodge diamond as well as additional topological
invariants such as Chern classes and intersection numbers. Using this data, we
conclude that there are at least 36,779 topologically distinct manifolds in our
list. We also study the fibration structure of these manifolds and find that
99.95 percent can be described as elliptic fibrations. In total, we find
50,114,908 elliptic fibrations, demonstrating the multitude of ways in which
many manifolds are fibered. A sub-class of 26,088,498 fibrations satisfy
necessary conditions for admitting sections. The complete data set can be
downloaded at
http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/Cicy4folds/index.html .Comment: 25 pages, 7 figures, 1 table. v2: references added and minor changes.
Final version accepted for publicatio
Mirror Symmetry, Mirror Map and Applications to Complete Intersection Calabi-Yau Spaces
We extend the discussion of mirror symmetry, Picard-Fuchs equations,
instanton-corrected Yukawa couplings, and the topological one-loop partition
function to the case of complete intersections with higher-dimensional moduli
spaces. We will develop a new method of obtaining the instanton-corrected
Yukawa couplings through a close study of the solutions of the Picard-Fuchs
equations. This leads to closed formulas for the prepotential for the K\"ahler
moduli fields induced from the ambient space for all complete intersections in
non singular weighted projective spaces. As examples we treat part of the
moduli space of the phenomenologically interesting three-generation models that
are found in this class. We also apply our method to solve the simplest model
in which a topology change was observed and discuss examples of complete
intersections in singular ambient spaces.Comment: 50 page
Black Holes in Type IIA String on Calabi-Yau Threefolds with Affine ADE Geometries and q-Deformed 2d Quiver Gauge Theories
Motivated by studies on 4d black holes and q-deformed 2d Yang Mills theory,
and borrowing ideas from compact geometry of the blowing up of affine ADE
singularities, we build a class of local Calabi-Yau threefolds (CY^{3})
extending the local 2-torus model \mathcal{O}(m)\oplus \mathcal{O}(-m)\to
T^{2\text{}} considered in hep-th/0406058 to test OSV conjecture. We first
study toric realizations of T^{2} and then build a toric representation of
X_{3} using intersections of local Calabi-Yau threefolds \mathcal{O}(m)\oplus
\mathcal{O}(-m-2)\to \mathbb{P}^{1}. We develop the 2d \mathcal{N}=2 linear
\sigma-model for this class of toric CY^{3}s. Then we use these local
backgrounds to study partition function of 4d black holes in type IIA string
theory and the underlying q-deformed 2d quiver gauge theories. We also make
comments on 4d black holes obtained from D-branes wrapping cycles in
\mathcal{O}(\mathbf{m}) \oplus \mathcal{O}(\mathbf{-m-2}%) \to \mathcal{B}_{k}
with \mathbf{m=}(m_{1},...,m_{k}) a k-dim integer vector and \mathcal{B}_{k} a
compact complex one dimension base consisting of the intersection of k
2-spheres S_{i}^{2} with generic intersection matrix I_{ij}. We give as well
the explicit expression of the q-deformed path integral measure of the
partition function of the 2d quiver gauge theory in terms of I_{ij}.Comment: 36 pages, latex, 9 figures. References adde
Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections
We present the most complete list of mirror pairs of Calabi-Yau complete
intersections in toric ambient varieties and develop the methods to solve the
topological string and to calculate higher genus amplitudes on these compact
Calabi-Yau spaces. These symplectic invariants are used to remove redundancies
in examples. The construction of the B-model propagators leads to compatibility
conditions, which constrain multi-parameter mirror maps. For K3 fibered
Calabi-Yau spaces without reducible fibers we find closed formulas for all
genus contributions in the fiber direction from the geometry of the fibration.
If the heterotic dual to this geometry is known, the higher genus invariants
can be identified with the degeneracies of BPS states contributing to
gravitational threshold corrections and all genus checks on string duality in
the perturbative regime are accomplished. We find, however, that the BPS
degeneracies do not uniquely fix the non-perturbative completion of the
heterotic string. For these geometries we can write the topological partition
function in terms of the Donaldson-Thomas invariants and we perform a
non-trivial check of S-duality in topological strings. We further investigate
transitions via collapsing D5 del Pezzo surfaces and the occurrence of free Z2
quotients that lead to a new class of heterotic duals.Comment: 117 pages, 1 Postscript figur
What do Topologists want from Seiberg--Witten theory? (A review of four-dimensional topology for physicists)
In 1983, Donaldson shocked the topology world by using instantons from
physics to prove new theorems about four-dimensional manifolds, and he
developed new topological invariants. In 1988, Witten showed how these
invariants could be obtained by correlation functions for a twisted N=2 SUSY
gauge theory. In 1994, Seiberg and Witten discovered dualities for such
theories, and in particular, developed a new way of looking at four-dimensional
manifolds that turns out to be easier, and is conjectured to be equivalent to,
Donaldson theory.
This review describes the development of this mathematical subject, and shows
how the physics played a pivotal role in the current understanding of this area
of topology.Comment: 51 pages, 10 figures, 8 postscript files. Submitted to International
Journal of Modern Physics A, July 2002 Uses Latex 2e with class file
ws-ijmpa.cls (included in tar file
GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces
We present a detailed study of the generalized hypergeometric system
introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in
the context of toric geometry. GKZ systems arise naturally in the moduli theory
of Calabi-Yau toric varieties, and play an important role in applications of
the mirror symmetry. We find that the Gr\"obner basis for the so-called toric
ideal determines a finite set of differential operators for the local solutions
of the GKZ system. At the special point called the large radius limit, we find
a close relationship between the principal parts of the operators in the GKZ
system and the intersection ring of a toric variety. As applications, we
analyze general three dimensional hypersurfaces of Fermat and non-Fermat types
with Hodge numbers up to . We also find and analyze several non
Landau-Ginzburg models which are related to singular models.Comment: 55 pages, 3 Postscript figures, harvma
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