Reflexive polyhedra, weights and toric Calabi-Yau fibrations

TL;DR

The paper addresses the problem of systematically classifying Calabi–Yau hypersurfaces in toric varieties via reflexive polyhedra and understanding their fibration structures in the context of string dualities. It delivers a mathematical account of the reflexive-polyhedron classification, reports complete 3D results (leading to K3 surfaces) and near-complete 4D results for Calabi–Yau threefolds, and connects these data to fibration structures through toric diagrams. A key contribution is the emphasis on weight systems as organizing constructs and the provision of data and methods accessible to the community, including a detailed explanation of the underlying algorithm. The work advances mirror-symmetry studies and toric-CY constructions by offering a rigorous framework and a publicly available dataset for further exploration.

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.