[Paper Review] Reflexive polyhedra, weights and toric Calabi-Yau fibrations

Maximilian Kreuzer, Harald Skarke|ArXiv.org|Jan 19, 2000

Algebraic Geometry and Number Theory33 references4 citations

TL;DR

This paper presents a systematic classification of reflexive polyhedra in 3 and 4 dimensions, using weight systems to encode toric data and identify Calabi–Yau hypersurfaces with fibration structures relevant to string dualities. By leveraging lattice polytopes with a unique interior point and embedding them via multi-linear Diophantine equations, the authors generate a comprehensive database of 184,026 weighted projective space configurations, including 95 K3 weight systems and extensive Hodge data, enabling the study of mirror symmetry and fibration types in toric Calabi–Yau compactifications.

ABSTRACT

During the last years we have generated a large number of data related to Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive polyhedra. We classified all reflexive polyhedra in three dimensions leading to K3 hypersurfaces and have nearly completed the four dimensional case relevant to Calabi-Yau threefolds. In addition, we have analysed for many of the resulting spaces whether they allow fibration structures of the types that are relevant in the context of superstring dualities. In this survey we want to give background information both on how we obtained these data, which can be found at our web site, and on how they may be used. We give a complete exposition of our classification algorithm at a mathematical (rather than algorithmic) level. We also describe how fibration structures manifest themselves in terms of toric diagrams and how we managed to find the respective data. Both for our classification scheme and for simple descriptions of fibration structures the concept of weight systems plays an important role.

Motivation & Objective

• To develop a mathematical framework for classifying reflexive polyhedra in 3 and 4 dimensions using weight systems and combined weight systems.
• To identify and catalog toric Calabi–Yau hypersurfaces with fibration structures relevant to string dualities.
• To provide a complete database of Hodge numbers, reflexive projections, and geometric invariants for these spaces, accessible via a public web resource.
• To establish a constructive method linking Diophantine equations to lattice polytopes with a unique interior point, forming the basis for reflexive polyhedra.

Proposed method

• Use of weight systems defined by non-negative integer solutions to equations of the form ∑n_i^{(j)} a_i = d^{(j)} with d^{(j)} = ∑n_i^{(j)} to embed lattice polytopes into higher codimension spaces.
• Transformation of affine coordinates (a_i) to linear coordinates (x_i = a_i - 1) to express polytopes in subspaces defined by ∑n_i^{(j)}x_i = 0, with the origin as the interior point.
• Application of the reflexive polytope condition: a lattice polytope is reflexive if and only if its dual polytope is also a lattice polytope, with all facets at unit distance from the unique interior point.
• Computation of Hodge numbers h^{1,1} and h^{1,2} from the combinatorial data of the polytope and its dual, using toric geometry and hypersurface construction.
• Identification of fibration structures via reflexive projections onto lower-dimensional faces, particularly facets, and enumeration of such projections.
• Use of the combined weight system formalism to systematically generate and classify all reflexive polyhedra in four dimensions, with a focus on K3 and Calabi–Yau threefold cases.

Experimental results

Research questions

• RQ1 How can reflexive polyhedra in 3 and 4 dimensions be systematically classified using weight systems and Diophantine equations?
• RQ2 Which of the resulting Calabi–Yau hypersurfaces admit fibration structures relevant to string dualities, and how can these be detected from toric data?
• RQ3 What is the distribution of Hodge numbers among the 184,026 weighted projective space configurations, and how symmetric is it with respect to mirror pairs?
• RQ4 How do reflexive projections onto facets relate to the existence of fibration structures in toric Calabi–Yau manifolds?
• RQ5 What is the role of minimality types (r, ls, s, etc.) in classifying weight systems and their associated geometric invariants?

Key findings

• The authors classified 95 K3 weight systems in four dimensions, with Hodge numbers h^{1,1} and h^{1,2} ranging from 11 to 63, and identified 184,026 weighted projective space configurations with transversal or non-transversal polynomials.
• For the 184,026 configurations, the paper reports Hodge numbers h^{1,1} and h^{1,2} ranging from 1 to 491, with the largest example having h^{1,1} = 491 and h^{1,2} = 11.
• The database includes 35,734 configurations with transversal polynomials (T), and 148,292 with non-transversal ones (–), with reflexive projections onto facets counted and reported for each.
• The number of reflexive projections (Π) and their facet projections (F) are computed and listed, with values ranging from 0 to 3, indicating the presence and multiplicity of fibration structures.
• The paper identifies 12 weight systems with unique partitions (indicated in bold), which are minimal and likely correspond to extremal or special fibration types.
• The classification reveals that most configurations are not mirror-symmetric in the Hodge data, but the overall structure supports mirror symmetry through the reflexive polytope duality, with the full database available online for further study.