Reflexive polyhedra, weights and toric Calabi-Yau fibrations

We present results from an inductive algebraic approach to the systematic construction and classification of the `lowest-level' CY3 spaces defined as zeroes of polynomial loci associated with reflexive polyhedra, derived from suitable vectors in complex projective spaces. These CY3 spaces may be sorted into `chains' obtained by combining lower-dimensional projective vectors classified previously. We analyze all the 4242 (259, 6, 1) two- (three-, four-, five-) vector chains, w...

We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.

In previous work, we have commenced the task of unpacking the $473,800,776$ reflexive polyhedra by Kreuzer and Skarke into a database of Calabi-Yau threefolds (see http://www.rossealtman.com). In this paper, following a pedagogical introduction, we present a new algorithm to isolate Swiss cheese solutions characterized by "holes," or small 4-cycles, descending from the toric divisors inherent to the original four dimensional reflexive polyhedra. Implementing these methods, we...

These notes contain a brief introduction to the construction of toric Calabi--Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and report on recent results and work in progress, including torsion in cohomology, classification issues and topological transitions.

We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hyper...

We explain how to form a novel dataset of simply connected Calabi-Yau threefolds via the Gross-Siebert algorithm. We expect these to degenerate to Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated) singularities. In particular, we explain how to `smooth the boundary' of a class of $4$-dimensional reflexive polytopes to obtain a polarised tropical manifolds. We compute topological invariants of a compactified torus fibration over each such tropic...

We show that the moduli space of all Calabi-Yau manifolds that can be realized as hypersurfaces described by a transverse polynomial in a four dimensional weighted projective space, is connected. This is achieved by exploiting techniques of toric geometry and the construction of Batyrev that relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the previously known fact that the moduli space of all CICY's is connected, and is moreover connected to the moduli...

Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. ...

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discr...

Using the Kreuzer-Skarke database of 4-dimensional reflexive polytopes, we systematically constructed a new database of orientifold Calabi-Yau threefolds with $h^{1,1}(X) \leq 12$. Our approach involved non-trivial $\mathbb{Z}_2$ involutions, incorporating both divisor exchanges and multi-divisor reflections acting on the Calabi-Yau threefolds. Each proper involution results in an orientifold Calabi-Yau threefolds and we constructed 320,386,067 such examples. We developed a n...