$\mathbb{P}^1$-fibrations in F-theory and String Dualities

TL;DR

This paper investigates $P^1$-fibered Calabi-Yau manifolds in F-theory, revealing new phenomena like base transitions and their dual heterotic implications, and provides tools for analyzing such compactifications.

Contribution

It offers a comprehensive study of $P^1$-fibered Calabi-Yau geometries in F-theory, including general formulations, degenerations, and duality insights, with new phenomena like base jumping.

Findings

Discovered base transition phenomena in $P^1$-fibered Calabi-Yau manifolds.

Linked degenerations to heterotic 5-branes and monodromy effects.

Provided new formulae and tools for F-theory compactifications.

Abstract

In this work we study F-theory compactifications on elliptically fibered Calabi-Yau n-folds which have $\mathbb{P}^1$-fibered base manifolds. Such geometries, which we study in both 4- and 6-dimensions, are both ubiquitous within the set of Calabi-Yau manifolds and play a crucial role in heterotic/F-theory duality. We discuss the most general formulation of $\mathbb{P}^1$-bundles of this type, as well as fibrations which degenerate at higher codimension loci. In the course of this study, we find a number of new phenomena. For example, in both 4- and 6-dimensions we find transitions whereby the base of a $\mathbb{P}^1$-bundle can change nature, or "jump", at certain loci in complex structure moduli space. We discuss the implications of this jumping for the associated heterotic duals. We argue that $\mathbb{P}^1$-bundles with only rational sections lead to heterotic duals where the Calabi-Yau manifold is elliptically fibered over the section of the $\mathbb{P}^1$- bundle, and not its base. As expected, we see that degenerations of the $\mathbb{P}^1$-fibration of the F-theory base correspond to 5-branes in the dual heterotic physics, with the exception of cases in which the fiber degenerations exhibit monodromy. Along the way, we discuss a set of useful formulae and tools for describing F-theory compactifications on this class of Calabi-Yau manifolds.