On the Classification of Reflexive Polyhedra

TL;DR

This paper develops an algorithm to classify dual pairs of reflexive polyhedra in up to four dimensions, aiding the study of Calabi-Yau hypersurfaces in string theory.

Contribution

It introduces a novel method based on non-negative integral matrices and weight systems for classifying reflexive polyhedra and their duals.

Findings

Algorithm for constructing all dual reflexive polyhedra pairs

Efficient enumeration method for up to 4 dimensions

Insights into the geometry of Calabi-Yau hypersurfaces

Abstract

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 corner stone 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.