Tools for CICYs in F-theory

TL;DR

This paper introduces tools for analyzing the geometry of elliptically fibered Calabi-Yau manifolds in F-theory, emphasizing a total space approach to reveal hidden geometric properties and facilitate the study of fibrations, sections, and related structures.

Contribution

It presents a novel set of geometric tools for analyzing CICYs in F-theory, including methods to identify fibrations, sections, and the Mordell-Weil group directly from the total space.

Findings

Explicit parameterization of Mordell-Weil groups for CICYs.

Identification of non-abelian symmetries in examples.

Analysis of non-flat fibrations in the geometric context.

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.