52,731 research outputs found
Hurwitz numbers and intersections on moduli spaces of curves
This article is an extended version of preprint math.AG/9902104. We find an
explicit formula for the number of topologically different ramified coverings
of a sphere by a genus g surface with only one complicated branching point in
terms of Hodge integrals over the moduli space of genus g curves with marked
points.Comment: 30 pages (AMSTeX). Minor typos are correcte
Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua
We develop a combinatorial approach to the construction of general smooth
compact base surfaces that support elliptic Calabi-Yau threefolds. This extends
previous analyses that have relied on toric or semi-toric structure. The
resulting algorithm is used to construct all classes of such base surfaces
with and all base surfaces over which there is an
elliptically fibered Calabi-Yau threefold with Hodge number . These two sets can be used todescribe all 6D F-theory models that
have fewer than seven tensor multiplets or more than 150 neutral scalar fields
respectively in their maximally Higgsed phase. Technical challenges to
constructing the complete list of base surfaces for all Hodge numbers are
discussed.Comment: 51 pages, 10 figure
A rigid Calabi--Yau 3-fold
The aim of this paper is to analyze some geometric properties of the rigid
Calabi--Yau threefold obtained by a quotient of , where
is a specific elliptic curve. We describe the cohomology of and
give a simple formula for the trilinear form on . We describe
some projective models of and relate these to its generalized
mirror. A smoothing of a singular model is a Calabi--Yau threefold with small
Hodge numbers which was not known before.Comment: 24 pages, Version 2: minor changes, references adde
Quadrature domains and kernel function zipping
It is proved that quadrature domains are ubiquitous in a very strong sense in
the realm of smoothly bounded multiply connected domains in the plane. In fact,
they are so dense that one might as well assume that any given smooth domain
one is dealing with is a quadrature domain, and this allows access to a host of
strong conditions on the classical kernel functions associated to the domain.
Following this string of ideas leads to the discovery that the Bergman kernel
can be zipped down to a strikingly small data set. It is also proved that the
kernel functions associated to a quadrature domain must be algebraic.Comment: 13 pages, to appear in Arkiv for matemati
Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY manifolds)
In this article we discuss the geometry of moduli spaces of (1) flat bundles
over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills
bundles over complex submanifolds in Calabi-Yau manifolds.
These moduli spaces reflect the geometry of the Calabi-Yau itself like a
mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau
manifold is such a moduli space and they argue that the mirror symmetry duality
is a Fourier-Mukai transformation. We review various aspects of the mirror
symmetry conjecture and discuss a geometric approach in proving it.
The existence of rigid Calabi-Yau manifolds poses a serious challenge to the
conjecture. The proposed mirror partners for them are higher dimensional
generalized Calabi-Yau manifolds. For example, the mirror partner for a certain
K3 surface is a cubic fourfold and its Fano variety of lines is birational to
the Hilbert scheme of two points on the K3. This hyperkahler manifold can be
interpreted as the SYZ mirror of the K3 by considering singular special
Lagrangian tori.
We also compare the geometries between a CY and its associated generalized
CY. In particular we present a new construction of Lagrangian submanifolds.Comment: To appear in the proceedings of International Congress of Chinese
Mathematicians 2001, 47 page
Complex Multiplication Symmetry of Black Hole Attractors
We show how Moore's observation, in the context of toroidal compactifications
in type IIB string theory, concerning the complex multiplication structure of
black hole attractor varieties, can be generalized to Calabi-Yau
compactifications with finite fundamental groups. This generalization leads to
an alternative general framework in terms of motives associated to a Calabi-Yau
variety in which it is possible to address the arithmetic nature of the
attractor varieties in a universal way via Deligne's period conjecture.Comment: 28 page
Arithmetic Spacetime Geometry from String Theory
An arithmetic framework to string compactification is described. The approach
is exemplified by formulating a strategy that allows to construct geometric
compactifications from exactly solvable theories at . It is shown that the
conformal field theoretic characters can be derived from the geometry of
spacetime, and that the geometry is uniquely determined by the two-dimensional
field theory on the world sheet. The modular forms that appear in these
constructions admit complex multiplication, and allow an interpretation as
generalized McKay-Thompson series associated to the Mathieu and Conway groups.
This leads to a string motivated notion of arithmetic moonshine.Comment: 36 page
Complex Multiplication of Exactly Solvable Calabi-Yau Varieties
We propose a conceptual framework that leads to an abstract characterization
for the exact solvability of Calabi-Yau varieties in terms of abelian varieties
with complex multiplication. The abelian manifolds are derived from the
cohomology of the Calabi-Yau manifold, and the conformal field theoretic
quantities of the underlying string emerge from the number theoretic structure
induced on the varieties by the complex multiplication symmetry. The geometric
structure that provides a conceptual interpretation of the relation between
geometry and the conformal field theory is discrete, and turns out to be given
by the torsion points on the abelian varieties.Comment: 44 page
Third kind elliptic integrals and 1-motives
In our PH.D. thesis we have showed that the Generalized Grothendieck's
Conjecture of Periods applied to 1-motives, whose underlying semi-abelian
variety is a product of elliptic curves and of tori, is equivalent to a
transcendental conjecture involving elliptic integrals of the first and second
kind, and logarithms of complex numbers. In this paper we investigate the
Generalized Grothendieck's Conjecture of Periods in the case of 1-motives whose
underlying semi-abelian variety is a non trivial extension of a product of
elliptic curves by a torus. This will imply the introduction of elliptic
integrals of the third kind for the computation of the period matrix of M and
therefore the Generalized Grothendieck's Conjecture of Periods applied to M
will be equivalent to a transcendental conjecture involving elliptic integrals
of the first, second and third kind.Comment: paper with an appendix of Michel Waldschmidt and a letter of Yves
Andr\'
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