D5 elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory

TL;DR

<3-5 sentence high-level summary>We study D5 elliptic fibrations, whose generic fiber is the complete intersection of two quadrics in $\mathbb{P}^3$, revealing a rich spectrum of singular fibers beyond Kodaira via Segre symbols and presenting the first Sen-type orientifold limits for these fibrations. The paper develops a systematic classification of singular fibers, including a unique non-Kodaira fiber $I^{*-}_0$, and constructs birational equivalents to $E_6$ and $E_7$ models, illustrating how non-Kodaira fibers arise without total-space singularities. It extends Sethi–Vafa–Witten formulas and Euler-characteristic relations to $D_5$ fibrations in arbitrary base dimension and without a Calabi–Yau assumption, deriving universal tadpole relations that match F-theory to Type IIB weak coupling limits. Finally, it presents explicit weak coupling limits, including a non-Kodaira fiber case with three brane–image-brane pairs and verifies the corresponding Chern-class tadpole identities, highlighting the global geometric and physical implications for F-theory compactifications beyond Calabi–Yau settings.

Abstract

A D5 elliptic fibration is a fibration whose generic fiber is modeled by the complete intersection of two quadric surfaces in P3. 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 and therefore 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 D5 elliptic fibrations. These orientifolds limit 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 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 surprising relations relating the total Chern class of the D5 elliptic fibration and those of different loci that naturally appear in the weak coupling limit. We work in arbitrary dimension and our results don't assume the Calabi-Yau condition.

Figures (3)

• Figure 1: A torus seen as the quotient $\mathbb{C}/ (\mathbb{Z}+\tau \mathbb{Z})$.
• Figure 2: Singular fibers of a $D_5$ elliptic fibration with four sections. There are a total of 8 singular fibers. This includes all the Kodaira fibers with at most 4 components and the fiber $I^{*-}_0$ which is not on Kodaira's list. Down arrows represent an increase in the number of components while up arrows indicate a specialization from a semi-stable to an unstable fiber while preserving the number of components of the fiber.
• Figure 3: One dimensional non-Kodaira fibers appearing in $D_5$ elliptic fibrations.

Theorems & Definitions (31)

• Proposition 1.1: Fiber geometry of $E_8$, $E_7$ and $E_6$ elliptic fibrations
• Proposition 1.2: Fiber geometry of a $D_5$ elliptic fibration
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 3.1: Pencil of quadrics
• Lemma 3.2: Characterization of singularities of the complete intersection of two quadrics
• Definition 3.3: $s$-Cones
• Definition 3.4: $s$-minors
• Lemma 3.5: $s$-cones in a pencil of quadrics