Communications in Number Theory and Physics

Volume 9 (2015)

Number 3

$D_5$ elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory

Pages: 583 – 642



Mboyo Esole (Department of Mathematics and Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts, U.S.A.)

James Fullwood (Institute of Mathematical Research, The University of Hong Kong, Pokfulam, Hong Kong)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


A $D_5$ elliptic fibration is a fibration whose generic fiber is modeled by the complete intersection of two quadric surfaces in $\mathbb{P}^3$. They provide simple examples of elliptic fibrations admitting a rich spectrum of singular fibers (not all on the list of Kodaira) without introducing singularities in the total space of the fibration, thus avoiding a discussion of their resolutions. We study systematically the fiber geometry of such fibrations using Segre symbols and compute several topological invariants.

We present for the first time Sen’s (orientifold) limits for $D_5$ elliptic fibrations. These orientifolds limits describe different weak coupling limits of F-theory to type IIB string theory giving a system of three brane-image-brane pairs in presence of a $\mathbb{Z}_2$ orientifold. The orientifold theory is mathematically described by the double cover the base of the elliptic fibration. Such orientifold theories are characterized by a transition from a semi-stable singular fiber to an unstable one. In this paper, we describe the first example of a weak coupling limit in F-theory characterized by a transition to a non-Kodaira (and non-ADE) fiber. Inspired by string dualities, we obtain non-trivial topological relations connecting the elliptic fibration and the different loci that appear in its weak coupling limit. Mathematically, these are very surprising relations which relate the total Chern class of the $D_5$ elliptic fibration and those of different loci that naturally appear in the weak coupling limit. We work over bases of arbitrary dimension and our results are independent of any Calabi-Yau hypothesis.

Published 11 September 2015