49 research outputs found
Fano hypersurfaces and Calabi-Yau supermanifolds
In this paper, we study the geometrical interpretations associated with
Sethi's proposed general correspondence between N = 2 Landau-Ginzburg orbifolds
with integral \hat{c} and N = 2 nonlinear sigma models. We focus on the
supervarieties associated with \hat{c} = 3 Gepner models. In the process, we
test a conjecture regarding the superdimension of the singular locus of these
supervarieties. The supervarieties are defined by a hypersurface \widetilde{W}
= 0 in a weighted superprojective space and have vanishing super-first Chern
class. Here, \widetilde{W} is the modified superpotential obtained by adding as
necessary to the Gepner superpotential a boson mass term and/or fermion
bilinears so that the superdimension of the supervariety is equal to \hat{c}.
When Sethi's proposal calls for adding fermion bilinears, setting the bosonic
part of \widetilde{W} (denoted by \widetilde{W}_{bos}) equal to zero defines a
Fano hypersurface embedded in a weighted projective space. In this case, if the
Newton polytope of \widetilde{W}_{bos} admits a nef partition, then the
Landau-Ginzburg orbifold can be given a geometrical interpretation as a
nonlinear sigma model on a complete intersection Calabi-Yau manifold. The
complete intersection Calabi-Yau manifold should be equivalent to the
Calabi-Yau supermanifold prescribed by Sethi's proposal.Comment: 24 pages, uses JHEP3.cls; v2: minor corrections, references adde
Deforming, revolving and resolving - New paths in the string theory landscape
In this paper we investigate the properties of series of vacua in the string
theory landscape. In particular, we study minima to the flux potential in type
IIB compactifications on the mirror quintic. Using geometric transitions, we
embed its one dimensional complex structure moduli space in that of another
Calabi-Yau with h^{1,1}=86 and h^{2,1}=2. We then show how to construct
infinite series of continuously connected minima to the mirror quintic
potential by moving into this larger moduli space, applying its monodromies,
and moving back. We provide an example of such series, and discuss their
implications for the string theory landscape.Comment: 41 pages, 5 figures; minor corrections, published versio
Roots of Ehrhart Polynomials of Smooth Fano Polytopes
V. Golyshev conjectured that for any smooth polytope P of dimension at most
five, the roots z\in\C of the Ehrhart polynomial for P have real part equal
to -1/2. An elementary proof is given, and in each dimension the roots are
described explicitly. We also present examples which demonstrate that this
result cannot be extended to dimension six.Comment: 10 page
Motivic Milnor fibre for nondegenerate function germs on toric singularities
We study function germs on toric varieties which are nondegenerate for their
Newton diagram. We express their motivic Milnor fibre in terms of their Newton
diagram. We extend a formula for the motivic nearby fibre to the case of a
toroidal degeneration. We illustrate this by some examples.Comment: 14 page
Linear Toric Fibrations
These notes are based on three lectures given at the 2013 CIME/CIRM summer
school. The purpose of this series of lectures is to introduce the notion of a
toric fibration and to give its geometrical and combinatorial
characterizations. Polarized toric varieties which are birationally equivalent
to projective toric bundles are associated to a class of polytopes called
Cayley polytopes. Their geometry and combinatorics have a fruitful interplay
leading to fundamental insight in both directions. These notes will illustrate
geometrical phenomena, in algebraic geometry and neighboring fields, which are
characterized by a Cayley structure. Examples are projective duality of toric
varieties and polyhedral adjunction theory
Virtual Structure Constants as Intersection Numbers of Moduli Space of Polynomial Maps with Two Marked Points
In this paper, we derive the virtual structure constants used in mirror
computation of degree k hypersurface in CP^{N-1}, by using localization
computation applied to moduli space of polynomial maps from CP^{1} to CP^{N-1}
with two marked points. We also apply this technique to non-nef local geometry
O(1)+O(-3)->CP^{1} and realize mirror computation without using Birkhoff
factorization.Comment: 10 pages, latex, a minor change in Section 4, English is refined,
Some typing errors in Section 3 are correcte
Polynomial Structure of Topological String Partition Functions
We review the polynomial structure of the topological string partition
functions as solutions to the holomorphic anomaly equations. We also explain
the connection between the ring of propagators defined from special K\"ahler
geometry and the ring of almost-holomorphic modular forms defined on modular
curves.Comment: version 2: references fixe
An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts
Even a cursory inspection of the Hodge plot associated with Calabi-Yau
threefolds that are hypersurfaces in toric varieties reveals striking
structures. These patterns correspond to webs of elliptic-K3 fibrations whose
mirror images are also elliptic-K3 fibrations. Such manifolds arise from
reflexive polytopes that can be cut into two parts along slices corresponding
to the K3 fibers. Any two half-polytopes over a given slice can be combined
into a reflexive polytope. This fact, together with a remarkable relation on
the additivity of Hodge numbers, explains much of the structure of the observed
patterns.Comment: 30 pages, 15 colour figure
On the full, strongly exceptional collections on toric varieties with Picard number three
We investigate full strongly exceptional collections on smooth, com- plete
toric varieties. We obtain explicit results for a large family of varieties
with Picard number three, containing many of the families already known. We
also describe the relations between the collections and the split of the push
forward of the trivial line bundle by the toric Frobenius morphism
Lectures on BCOV holomorphic anomaly equations
The present article surveys some mathematical aspects of the BCOV holomorphic
anomaly equations introduced by Bershadsky, Cecotti, Ooguri and Vafa. It grew
from a series of lectures the authors gave at the Fields Institute in the
Thematic Program of Calabi-Yau Varieties in the fall of 2013.Comment: reference added, typos correcte