F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds

TL;DR

This paper analyzes the abelian sector of F-theory in six dimensions by focusing on elliptically fibered Calabi–Yau threefolds with Mordell-Weil rank one over ${\mathbb{P}}^2$. It shows that abelian anomaly coefficients are governed by the Néron–Tate height pairing of Mordell-Weil generators via the Shioda map, enabling a geometric classification of U(1) theories with charges limited to 1 and 2. The authors construct rank-one examples $X_n$ yielding theories $\mathcal{T}_n$ for $n=0$–$6$ (with $b=2(n+3)$), verify these via field-theory Higgsing from $SU(2)$ and explicit intersection computations, and count two types of fibral curves that realize charges 1 and 2. They discuss the remaining models $\mathcal{T}_7$ and $\mathcal{T}_8$, potential extensions to higher charges, and a possible generalized Kodaira bound for the abelian sector, illuminating how the Mordell-Weil group shapes the landscape of six-dimensional F-theory vacua and their phenomenology.

Abstract

The Mordell-Weil group of an elliptically fibered Calabi-Yau threefold X contains information about the abelian sector of the six-dimensional theory obtained by compactifying F-theory on X. After examining features of the abelian anomaly coefficient matrix and U(1) charge quantization conditions of general F-theory vacua, we study Calabi-Yau threefolds with Mordell-Weil rank-one as a first step towards understanding the features of the Mordell-Weil group of threefolds in more detail. In particular, we generate an interesting class of F-theory models with U(1) gauge symmetry that have matter with both charges 1 and 2. The anomaly equations --- which relate the Neron-Tate height of a section to intersection numbers between the section and fibral rational curves of the manifold --- serve as an important tool in our analysis.

Figures (5)

• Figure 1: The intersection between the section ${\hat{S}}$ and fiber components. ${\hat{S}}$ intersects a generic fiber at a point. There are 108 loci in the base above which the fiber becomes reducible --- in fact, an $I_2$ fiber. The $I_2$ fiber consists of two rational curves $c_+$ and $c_-$ intersecting at two points. The section ${\hat{S}}$ intersects the $I_2$ fibers at a point on the $c_-$ component.
• Figure 2: The section ${\hat{S}}_{-1}$. ${\hat{S}}_{-1}$ intersects a generic fiber at a point while is resolved into the $c_-$ component at the $I_2$ loci.
• Figure 3: The section ${\hat{S}}_{2}$. ${\hat{S}}_{2}$ intersects a generic fiber at a point while is resolved into the $c_+$ component at the $I_2$ loci.
• Figure 4: The intersection between the section ${\hat{S}}$ and fiber components in ${\hat{X}}_n$. ${\hat{S}}$ intersects a generic fiber at a point. There are $n(n+3)$ charge-two loci and $4(n+3)(9-n)$ charge-one loci in the base above which the fiber degenerates into an $I_2$ fiber. The fibral curves $\chi_-^\iota$ localized above the charge-two loci intersect the section ${\hat{S}}$ twice, while the fibral curves $c_-^I$ at the charge-one loci intersect ${\hat{S}}$ once.
• Figure 5: Toric data for ambient spaces of some elliptic curve embeddings. These are among the 16 reflexive toric surfaces MR1463052 (see also arXiv:1201.0930).