On the Classification of Reflexive Polyhedra

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.

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

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• Gorenstein toric Fano varieties
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• Canonical toric Fano threefolds
• Three-Family Particle Physics Models from Global F-theory Compactifications
• Heterotic Models from Vector Bundles on Toric Calabi-Yau Manifolds
• Brane Tilings and Specular Duality
• The web of Calabi-Yau hypersurfaces in toric varieties
• Computing symmetry groups of polyhedra
• Toric complete intersections and weighted projective space
• Rational F-Theory GUTs without exotics
• PALP - a User Manual
• The Expanding Zoo of Calabi-Yau Threefolds
• Calabi-Yau Volumes and Reflexive Polytopes
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• Heterotic Bundles on Calabi-Yau Manifolds with Small Picard Number
• All Weight Systems for Calabi-Yau Fourfolds from Reflexive Polyhedra
• Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part
• Fano 3-Folds, Reflexive Polytopes and Brane Brick Models
• Conway Subgroup Symmetric Compactifications Of Heterotic String
• Maximum Likelihood Estimation of Toric Fano Varieties
• Reflexive polytopes of higher index and the number 12
• Polytopes and Machine Learning
• Toric Geometry and String Theory
• On $(2,4)$ complete intersection threefolds that contain an Enriques surface
• Patterns in Calabi-Yau Distributions
• Universal Calabi-Yau Algebra: Towards an Unification of Complex Geometry
• Heterotic String Models on Smooth Calabi-Yau Threefolds
• The making of Calabi-Yau spaces: Beyond toric hypersurfaces
• Combinatorial and Algebraic Mutations of Toric Fano 3-folds and Mass Deformations of 2d (0,2) Quiver Gauge Theories
• Smooth Fano Polytopes With Many Vertices
• Futaki Invariants and Reflexive Polygons
• Strings on Calabi--Yau spaces and Toric Geometry
• Calabi-Yau Four/Five/Six-folds as ${\mathbb{P}}_{\text{w}}^n$ Hypersurfaces: Machine Learning, Approximation, and Generation
• Quiver Gauge Theories: Beyond Reflexivity
• Max Kreuzer's Contributions to the Study of Calabi-Yau Manifolds
• The origin of discrete symmetries in F-theory models
• Birational Transformations and 2d (0,2) Quiver Gauge Theories beyond Toric Fano 3-folds
• Classifying Fano Complexity-One $T$-Varieties via Divisorial Polytopes
• A note on the smooth blow-ups of P(1,1,1,k) in torus-invariant subvarieties
• Beyond Algebraic Superstring Compactification
• Combinatorics in N = 1 Heterotic Vacua