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


Lara B. Anderson (Department of Physics, Virginia Tech, Blacksburg, Va., U.S.A.)

James Gray (Department of Physics, Virginia Tech, Blacksburg, Va., U.S.A.)

Mohsen Karkheiran (Center for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon, South Korea)

Paul-Konstantin Oehlmann (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden)

Nikhil Raghuram (Department of Physics, Virginia Tech, Blacksburg, Va., U.S.A.)


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.


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