On the Classification of Reflexive Polyhedra

TL;DR

This work presents a structured program to classify reflexive polyhedra in toric geometry, with the goal of enumerating mirror pairs of Calabi–Yau hypersurfaces up to $n=4$. It introduces minimal polytopes $M$ and $ar M$, a pairing-matrix framework $A$ tied to weight systems from weighted projective spaces, and a higher-dimensional lattice formalism that unifies interior/exterior constraints. The authors outline a finite, three-step algorithm: classify minimal-polytope structures, classify interior-point weight systems with span properties, and assemble full vertex-pairing matrices across lattices to realize all dual pairs; they demonstrate the method in the $n=2$ case to recover the classic reflexive-polygon classification. The approach provides a concrete, computable route to complete classifications in the physically relevant $n o 4$ regime, with direct relevance to Calabi–Yau compactifications in string theory.

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.