On the Classification of Reflexive Polyhedra

Details On the Classification of Reflexive Polyhedra On the Classification of Reflexive Polyhedra

1995-12-27

arxiv.org/abs/hep-th/9512204v2

Commun.Math.Phys. 185 (1997) 495-508

Physics

High Energy Physics

High Energy Physics - Theory

13 pages, LaTeX (updated to agree with shortened published version)

Scientific paper

10.1007/s002200050100

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.

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