On the Classification of Reflexive Polyhedra

Two of my collaborations with Max Kreuzer involved classification problems related to string vacua. In 1992 we found all 10,839 classes of polynomials that lead to Landau-Ginzburg models with c=9 (Klemm and Schimmrigk also did this); 7,555 of them are related to Calabi-Yau hypersurfaces. Later we found all 473,800,776 reflexive polytopes in four dimensions; these give rise to Calabi-Yau hypersurfaces in toric varieties. The missing piece - toric constructions that need not be...

We study various geometrical quantities for Calabi-Yau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the Sasaki-Einstein base of the corresponding Calabi-Yau cone are calculated. By doing so, we conjecture new bounds for the Sasaki-Einstein volume with respect to various topological quantities of the corr...

The problem of counting points is revisited from the perspective of reflexive 4-dimensional polytopes. As an application, the Hilbert series of the 473,800,776 reflexive polytopes (equivalently, their Calabi-Yau hypersurfaces) are computed.

We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.

Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, i...

For a particular toric variety, I explore to what extent the SYZ conjecture applied to the orbits of the torus action gives the mirror manifold, in the sense of Batyrev's mirror construction using reflexive polytopes.

We use the notions of reflexivity and of reflexive dimensions in order to introduce probability measures for lattice polytopes and initiate the investigation of their statistical properties. Examples of applications to discrete geometry include a study of randomness of self-duality of reflexive polytopes and implications for expectation values of the numbers of such polytopes in higher dimensions. We also discuss enumeration problems and related algorithms and point out inter...

We introduce a special class of convex rational polyhedral cones which allows to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$, where $r$ is a positive integer and d is the dimension of critical string vacua with central chatge $c = 3d$. It is conjectured that the natural combinatorial duality satisfies by these cones corresponds to the mirror involution. Using the theory of toric varieties, we show that our conjecture includes as special cases all al...

We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.

The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index r extends the well-known polar duality for reflexive polytopes (case r=1). We consider the Borisov duality between two nef-par...