We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.

Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties

http://cds.cern.ch/record/994225/files/0610327.pdf

Four-dimensional string compactifications with D-branes, orientifolds and fluxes

We review recent work in which compactifications of string and M theory are constructed in which all scalar fields (moduli) are massive, and supersymmetry is broken with a small positive cosmological constant, features needed to reproduce real world physics We explain how this work implies that there is a ``landscape'' of string/M theory vacua, perhaps containing many candidates for describing real world physics, and present the arguments for and against this idea We discuss statistical surveys of the landscape, and the prospects for testable consequences of this picture, such as observable effects of moduli, constraints on early cosmology, and predictions for the scale of supersymmetry breaking

Flux Compactification

Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory. We describe how we obtained all 473,800,776 reflexive polyhedra that exist in four dimensions and the 30,108 distinct pairs of Hodge numbers of the resulting Calabi-Yau manifolds. As a by-product we show that all these spaces (and hence the corresponding string vacua) are connected via a chain of singular transitions.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2000/0004/0006/ATMP-2000-0004-0006-a002.pdf

Complete classification of reflexive polyhedra in four-dimensions

Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi–Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of these matrices is the existence of a pair of weight systems, indicating a relation to weighted projective spaces. This is the cornerstone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi–Yau compactifications in string theory.

On the Classification of Reflexive Polyhedra

The Web of Calabi-Yau hypersurfaces in toric varieties

According to a recently proposed scheme for the classification of reflexive polyhedra, weight systems of a certain type play a prominent role. These weight systems are classified for the cases n = 3 and n = 4, corresponding to toric varieties with K3 and Calabi-Yau hypersurfaces, respectively. For n = 3 we find the well-known 95 weight systems corresponding to weighted ℙ3’s that allow transverse polynomials, whereas for n = 4 there are 184,026 weight systems, including the 7555 weight systems for weighted ℙ4’s. It is proven (without computer) that the Newton polyhedra corresponding to all these weight systems are reflexive.

Weight systems for toric calabi-yau varieties and reflexivity of newton polyhedra

Abelian Landau-Ginzburg orbifolds and mirror symmetry☆

The Einstein equations for an inhomogeneous irrotational dust universe are analyzed. A set of mild assumptions, all of which are shared by the standard Friedmann-Lemaitre-Robertson-Walker\char21{}type scenarios, results in a model that depends only on the distribution of scalar spatial curvature. If the shape of this distribution is made to fit the structure of the present Universe, with most of the matter in galaxy clusters and very little in the voids that will eventually dominate the volume, then there is a period of accelerated expansion after cluster formation, even in the absence of a cosmological constant.

Harald Skarke

Papers

On the Classification of Reflexive Polyhedra

The Web of Calabi-Yau hypersurfaces in toric varieties

Weight systems for toric calabi-yau varieties and reflexivity of newton polyhedra

Abelian Landau-Ginzburg orbifolds and mirror symmetry☆

Inhomogeneity implies accelerated expansion