Non-Higgsable abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces

TL;DR

The paper constructs a broad class of elliptic Calabi–Yau threefolds by fiber products of rational elliptic surfaces with section, extending Schoen’s construction to all Kodaira fiber types. Each resulting threefold carries two elliptic fibrations and typically a nonzero Mordell–Weil rank, giving rise to non-Higgsable abelian gauge sectors in 6D F-theory and revealing intricate singularity structures, including canonical and terminal cases, some of which lack Calabi–Yau resolutions. The authors classify allowed singularity pairings, analyze blowups of the base, and connect the geometry to physics via abelian anomaly coefficients, non-enhanceable $\mathfrak{u}(1)$ factors, and conifold/Higgs transitions, while also establishing an $A$-$B$ duality that exchanges fibrations and leads to T-duality in associated little string theories. They further explore bases with a $\mathbb{C}^{*}$-structure, showing how nonzero Mordell–Weil rank persists and how tuning can or cannot remove abelian factors without introducing pathological singularities. The work provides a concrete bridge between algebraic geometry and 6D/5D/LSFT physics, with implications for classifying F-theory vacua featuring abelian sectors and for understanding dualities of little string theories.

Abstract

We construct a general class of Calabi--Yau threefolds from fiber products of rational elliptic surfaces with section, generalizing a construction of Schoen to include all Kodaira fiber types. The resulting threefolds each have two elliptic fibrations with section over rational elliptic surfaces and blowups thereof. These elliptic fibrations generally have nonzero Mordell--Weil rank. Each of the elliptic fibrations has a physical interpretation in terms of a six-dimensional F-theory model with one or more non-Higgsable abelian gauge fields. Many of the models in this class have mild singularities that do not admit a Calabi--Yau resolution; this does not seem to compromise the physical integrity of the theory and can be associated in some cases with massless hypermultiplets localized at the singular loci. In some of these constructions, however, we find examples of abelian gauge fields that cannot be "unHiggsed" to a nonabelian gauge field without producing unphysical singularities that cannot be resolved. The models studied here can also be used to exhibit T-duality for a class of little string theories.

Figures (10)

• Figure 1: The configuration of intersecting divisors of negative self intersection on $\widehat{B}$ that replaces the $I_0^*$ configuration of $-2$ curves in the surface $B$ when a fiber product space $\widetilde{X}$ is formed from $B$ and a rational elliptic surface $A$ with a type $IV$ singularity over the same point $p$ as the $I_0^*$ singularity in $A$, and $X\to\widetilde{X}$ resolves the singularities, giving an elliptic fibration over the blowup $\widehat{B}$.
• Figure 2: Geometries of the blown up rational curve configurations in $\widehat{B}$ for different B-fiber types. (The B-fiber type is indicated by underlining).
• Figure 3: A classification of divisors in the resolution of an elliptic fibraiton over a rational surface, and their exchange properties under $A$-$B$ duality. The divisors are fibrations, whose bases and generic fibers we have depicted. The fibral divisors are either resolution divisors of $A$-type or coincident singular fibers, while the vertical divisors can be of $MW_B$-type or resolution divisors of either coincident singular fibers or B-type singularities. The sections either are $MW_A$-type divisors, or a global divisor, the blow up along which resolves multiple codimension-two singularities at the coincident loci. The solid arrows with the label "$A$-$B$" denote the exchange under $A$-$B$ duality.
• Figure 4: The behavior of divisors of $X$ (left) and $X^D$ (right) under flops for distinct resolutions of a coincident singular fiber given by $I_0^* \times I_3$.
• Figure 5: Flops relating $X$ (left) and $X^D$ (right).