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Tools for CICYs in F-theory

Lara B. Anderson, Xin Gao, James Gray, Seung-Joo Lee

TL;DR

This work develops a direct, geometry-centered toolkit for F-theory compactifications by extracting Weierstrass models, Jacobians, and resolved geometries from genus-one fibrations embedded in complete intersection Calabi–Yau manifolds, with a focus on CICYs. It provides explicit procedures to locate putative and genuine sections, construct global holomorphic (or rational) section maps, and compute the Mordell–Weil group through intersection theory and explicit section arithmetic. The authors apply the framework to a sequence of concrete examples (K3 and threefold CICYs), demonstrating varying Mordell–Weil ranks and non-abelian gauge content, and they validate discriminant data across Weierstrass, Jacobian, and resolved geometries. This methodology clarifies how multiple fibrations arise in known CY geometries and facilitates systematic analysis of F-theory vacua and potential string dualities. The approach is scalable to broader CY datasets and highlights the geometric transparency of F-theory data in resolved Calabi–Yau spaces.

Abstract

We provide a set of tools for analyzing the geometry of elliptically fibered Calabi-Yau manifolds, starting with a description of the total space rather than with a Weierstrass model or a specified type of fiber/base. Such an approach to the subject of F-theory compactification makes certain geometric properties, which are usually hidden, manifest. Specifically, we review how to isolate genus-one fibrations in such geometries and then describe how to find their sections explicitly. This includes a full parameterization of the Mordell-Weil group where non-trivial. We then describe how to analyze the associated Weierstrass models, Jacobians and resolved geometries. We illustrate our discussion with concrete examples which are complete intersections in products of projective spaces (CICYs). The examples presented include cases exhibiting non-abelian symmetries and higher rank Mordell-Weil group. We also make some comments on non-flat fibrations in this context. In a companion paper [1] to this one, these results will be used to analyze the consequences for string dualities of the ubiquity of multiple fibrations in known constructions of Calabi-Yau manifolds.

Tools for CICYs in F-theory

TL;DR

This work develops a direct, geometry-centered toolkit for F-theory compactifications by extracting Weierstrass models, Jacobians, and resolved geometries from genus-one fibrations embedded in complete intersection Calabi–Yau manifolds, with a focus on CICYs. It provides explicit procedures to locate putative and genuine sections, construct global holomorphic (or rational) section maps, and compute the Mordell–Weil group through intersection theory and explicit section arithmetic. The authors apply the framework to a sequence of concrete examples (K3 and threefold CICYs), demonstrating varying Mordell–Weil ranks and non-abelian gauge content, and they validate discriminant data across Weierstrass, Jacobian, and resolved geometries. This methodology clarifies how multiple fibrations arise in known CY geometries and facilitates systematic analysis of F-theory vacua and potential string dualities. The approach is scalable to broader CY datasets and highlights the geometric transparency of F-theory data in resolved Calabi–Yau spaces.

Abstract

We provide a set of tools for analyzing the geometry of elliptically fibered Calabi-Yau manifolds, starting with a description of the total space rather than with a Weierstrass model or a specified type of fiber/base. Such an approach to the subject of F-theory compactification makes certain geometric properties, which are usually hidden, manifest. Specifically, we review how to isolate genus-one fibrations in such geometries and then describe how to find their sections explicitly. This includes a full parameterization of the Mordell-Weil group where non-trivial. We then describe how to analyze the associated Weierstrass models, Jacobians and resolved geometries. We illustrate our discussion with concrete examples which are complete intersections in products of projective spaces (CICYs). The examples presented include cases exhibiting non-abelian symmetries and higher rank Mordell-Weil group. We also make some comments on non-flat fibrations in this context. In a companion paper [1] to this one, these results will be used to analyze the consequences for string dualities of the ubiquity of multiple fibrations in known constructions of Calabi-Yau manifolds.

Paper Structure

This paper contains 40 sections, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Elliptic Fibrations and Sections
  3. Genus-one Fibration Structures in CICYs
  4. Putative Sections to Genus-one Fibrations
  5. Sections to Elliptic Fibrations
  6. Singular Fiber Analysis
  7. Weierstrass Models
  8. Jacobians
  9. Resolved Geometries
  10. The Mordell-Weil Group
  11. Decomposition of the Picard Lattice
  12. Arithmetic of the Sections
  13. Example 1: A K3 example
  14. Section Analysis
  15. Locating the Singular Fibers
  16. ...and 25 more sections

Figures (3)

  • Figure 1: The abundance of CICY threefold configuration matrices, in the standard list, exhibiting a given number of genus-one fibrations which are visible directly in the configuration matrix usscanning.
  • Figure 2: The abundance of CICY fourfold configuration matrices, in the standard list, exhibiting a given number of genus-one fibrations which are visible directly in the configuration matrix Gray:2014fla.
  • Figure 3: Schematic picture of the discriminant locus, $L \subset B = \mathbb P^1_{\bold x_5} \times \mathbb P^1_{\bold x_6}$, where $\Delta(\bold x_5, \bold x_6)$ vanishes. The locus decomposes to four pieces $L^{(I)}$, $I=1, 2, 3, 4$.