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# Pure and Applied Mathematics Quarterly

## Volume 18 (2022)

### Number 4

### Special issue celebrating the work of Herb Clemens

Guest Editor: Ron Donagi

### $\mathbb{P}^1$-fibrations in F-theory and string dualities

Pages: 1264 – 1354

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a2

#### Authors

#### 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.

#### Keywords

heterotic string theory, F-theory, duality, Calabi–Yau geometry, rational fibrations

The work of M.K. was supported by IBS under the project code, IBS-R018-D1. The work of P.K.O. is supported by a grant of the Carl Trygger Foundation for Scientific Research. The work of L.A. and J.G. is supported in part by NSF grant PHY-2014086.

Received 23 September 2021

Received revised 29 March 2022

Accepted 16 April 2022

Published 25 October 2022