Fibrations in CICY Threefolds

TL;DR

The authors perform a comprehensive, topology-grounded survey of fibrations in the CICY threefold dataset, leveraging Kollár–Oguiso–Wilson criteria alongside a rich analysis of topology, Kähler/Mori cones, and configuration-matrix splittings. They produce two parallel enumeration tracks—obvious fibrations from the algebraic form and Kollár divisors from topological data—finding near-universal genus-one fibrations in Kahler-favorable geometries and revealing substantial undercounting by OGFs in non-favorable cases. A large-scale splitting program yields 2,946 new favorable CICY descriptions and a complete fibration classification for 4,957 Kahler-favorable geometries, culminating in 377,559 genus-one fibrations across the dataset, plus an explicit infinite family on the Schoen manifold. The work provides extensive publicly available data, strengthens the link between geometry and string dualities, and opens avenues for applying these methods to broader CY constructions and F-/heterotic-duality landscapes.

Abstract

In this work we systematically enumerate genus one fibrations in the class of 7,890 Calabi-Yau manifolds defined as complete intersections in products of projective spaces, the so-called CICY threefolds. This survey is independent of the description of the manifolds and improves upon past approaches that probed only a particular algebraic form of the threefolds (i.e. searches for "obvious" genus one fibrations as in [1,2]). We also study K3-fibrations and nested fibration structures. That is, K3 fibrations with potentially many distinct elliptic fibrations. To accomplish this survey a number of new geometric tools are developed including a determination of the full topology of all CICY threefolds, including triple intersection numbers. In 2,946 cases this involves finding a new "favorable" description of the manifold in which all divisors descend from a simple ambient space. Our results consist of a survey of obvious fibrations for all CICY threefolds and a complete classification of all genus one fibrations for 4,957 "Kahler favorable" CICYs whose Kahler cones descend from a simple ambient space. Within the CICY dataset, we find 139,597 obvious genus one fibrations, 30,974 obvious K3 fibrations and 208,987 nested combinations. For the Kahler favorable geometries we find a complete classification of 377,559 genus one fibrations. For one manifold with Hodge numbers (19,19) we find an explicit description of an infinite number of distinct genus-one fibrations extending previous results for this particular geometry that have appeared in the literature. The data associated to this scan is available at http://www1.phys.vt.edu/cicydata .

Figures (4)

• Figure 1: Distribution of obvious torus fibration abundance in the CICY threefold list (excluding product manifolds). The values lie in the range 0 - 93. We find $139,597$ fibrations in total and on average each CICY threefold configuration is elliptically fibered in $17.7$ different ways.
• Figure 2: Distribution of obvious K3 fibration abundance in the CICY threefold list (excluding product manifolds). The values lie in the range 0 - 9. We find $30,974$ fibrations in total and on average each CICY threefold configuration is obviously K3 fibered in $3.9$ different ways.
• Figure 3: Distribution of the abundance of obvious torus fibrations nested inside obvious K3 fibrations in the CICY threefold list (excluding product manifolds). The values lie in the range 0 - 174. We find $208,987$ such nested fibrations in total and on average each CICY threefold configuration admits $26.6$ different nested fibrations of this type.
• Figure 4: An example entry in the new maximally favorable CICY list.

Theorems & Definitions (2)

• Lemma 2.1
• Lemma 2.2