Calabi-Yau 4-folds and toric fibrations

TL;DR

This paper develops a systematic method to identify fibrations in Calabi-Yau 4-folds using toric geometry, providing extensive data on weights and Hodge numbers, and clarifying the geometric structure of fibrations.

Contribution

It introduces a general scheme for detecting fibrations in toric Calabi-Yau 4-folds and computes their Hodge numbers using toric and conformal field theory methods.

Findings

Identified over 900,000 weights with reflexive Newton polyhedra.

Computed Hodge numbers for large classes of Calabi-Yau 4-folds.

Clarified the geometric relationship between fibrations and polyhedral structures.

Abstract

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi--Yau 4-folds. We find 914,164 weights with degree $d\le150$ whose maximal Newton polyhedra are reflexive and 525,572 weights with degree $d\le4000$ that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's fomulas (obtained by counting of Ramond ground states in N=2 LG models) for the latter class, checking their consistency for the 109,308 weights in the overlap. Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold.