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.

Complete classification of reflexive polyhedra in four dimensions

“…As we showed in [22], V must either be a simplex or contain lower dimensional simplices with the origin in their respective interiors such that any vertex of V belongs to at least one of these simplices. This leads to a coarse classification with respect to the number of vertices and the simplices they belong to: In two dimensions, the only possibilities are the triangle V1V2V3 and the parallelogram ViV^V/V^, where V1V2 and V1V2 are one dimensional simplices (line segments) with the origin in their respective interiors.…”

Brane tilings and reflexive polygons

“…This property led to a systematic study of mirror paired CalabiYau manifolds. The resulting classification [28][29][30][31][32][33] found connections to for instance heterotic string compactifications [34][35][36] or to F-theory backgrounds [37][38][39][40].…”

Patterns in Calabi–Yau Distributions

“…Here are the fit statistics for best fit curve: (A, μ, σ, α) = (4517. 45,10.76, 2.97, −0.031), as shown in Fig. 19.…”

“…To facilitate this approach to a low-energy phenomenology derived from string theory, mathematicians and physicists have constructed large datasets of Calabi-Yau threefolds [7,[9][10][11][12][13][14][15][16][17][18][19][20][21][22] as well as various refined analyses of properties thereof [28][29][30][31][32][33][34][35]. By far the largest database was constructed in a tour de force of algebraic geometry, combinatorics, physics, and computer algorithms by Kreuzer and Skarke based on the theorems of Batyrev and Borisov [9][10][11][12][13][14]36,37].…”

“…By far the largest database was constructed in a tour de force of algebraic geometry, combinatorics, physics, and computer algorithms by Kreuzer and Skarke based on the theorems of Batyrev and Borisov [9][10][11][12][13][14]36,37]. In short, these Calabi-Yau n-manifolds X n are realized as a smooth hypersurface embedded in a toric variety A n+1 of complex dimension n + 1; the Calabi-Yau condition simply translates to the requirement that the polytope defining A n+1 be reflexive.…”