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.
Highlights
In many approaches to compactifications of F-theory, the identification of the fiber and base of the internal manifold is built in from the start
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
We illustrate our discussion with concrete examples which are complete intersections in products of projective spaces (CICYs)
Summary
In many approaches to compactifications of F-theory, the identification of the fiber and base of the internal manifold is built in from the start. Such a construction has many advantages, including the fact that one is guaranteed that the associated manifold is genus-one fibered and is suitable for use in Ftheory This methodology has the drawback that, instead of using the large data sets of Calabi-Yau manifolds that have already been constructed In cases where a section is present we describe how to obtain an explicit form for it in terms of the original description of the manifold This methodology is closely related to descriptions of holomorphic functions used in recent constructions of “generalized complete intersection CY manifolds” (gCICYs) [25]. In addition to utilizing pre-existing data sets of Calabi-Yau manifolds, this approach to analyzing global F-theory compactifications makes evident some features of fibrations that are not as obvious in more standard methodologies.
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