Complete classification of reflexive polyhedra in four dimensions

TL;DR

This paper exhaustively classifies all 4D reflexive polyhedra, revealing the structure of Calabi-Yau threefolds relevant for string theory and demonstrating their interconnectedness through singular transitions.

Contribution

It provides the complete enumeration of four-dimensional reflexive polyhedra and their associated Calabi-Yau manifolds, a significant advancement in the field.

Findings

All 473,800,776 reflexive polyhedra in four dimensions are catalogued.

Identified 30,108 distinct pairs of Hodge numbers for these Calabi-Yau manifolds.

Showed that all these spaces are connected via chains of singular transitions.

Abstract

Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory. We describe how we obtained all 473,800,776 reflexive polyhedra that exist in four dimensions and the 30,108 distinct pairs of Hodge numbers of the resulting Calabi-Yau manifolds. As a by-product we show that all these spaces (and hence the corresponding string vacua) are connected via a chain of singular transitions.