Complete classification of reflexive polyhedra in four dimensions

TL;DR

The paper presents a complete classification of four-dimensional reflexive polytopes, which encode Calabi–Yau threefolds as hypersurfaces in toric varieties. By developing an algorithm based on r-minimal combined weight systems and IP-polytope properties, the authors bound the search to a manageable finite set and systematically enumerate all reflexive polytopes through lattice refinements. They report 473,800,776 reflexive polytopes (with 236,879,533 dual pairs and 41,710 self-dual cases) and 30,108 distinct Hodge-number pairs for the corresponding Calabi–Yau manifolds, demonstrating that all such vacua are connected via chains of singular transitions. The results, including extensive data and tables, substantially map the landscape of toric Calabi–Yau geometries and advance mirror symmetry and string vacuum studies.

Abstract

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.