F-theory and All Things Rational: Surveying U(1) Symmetries with Rational Sections

TL;DR

This work delivers a universal framework for classifying abelian U(1) symmetries in F-theory by examining rational sections of elliptic CY three- and four-folds. By combining box-graph–driven Coulomb-phase data with rigorous normal-bundle constraints for rational curves, the authors enumerate all codimension-two fiber configurations that carry U(1) charges, first for SU(5) with fundamental and antisymmetric matter and then in the broader SU(n) context. They provide explicit charge spectra for one and multiple U(1)s, analyze fiber flops and codimension-three Yukawas, and study U(1)-charged singlets and their potential to Higgs the abelian sector to discrete symmetries. The results extend known explicit models by revealing a larger landscape of possible charges and fiber types, with systematic implications for model building and the realization of discrete gauge symmetries in F-theory.

Abstract

We study elliptic fibrations for F-theory compactifications realizing 4d and 6d supersymmetric gauge theories with abelian gauge factors. In the fibration these U(1) symmetries are realized in terms of additional rational sections. We obtain a universal characterization of all the possible U(1) charges of matter fields by determining the corresponding codimension two fibers with rational sections. In view of modelling supersymmetric Grand Unified Theories, one of the main examples that we analyze are U(1) symmetries for SU(5) gauge theories with \bar{5} and 10 matter. We use a combination of constraints on the normal bundle of rational curves in Calabi-Yau three- and four-folds, as well as the splitting of rational curves in the fibers in codimension two, to determine the possible configurations of smooth rational sections. This analysis straightforwardly generalizes to multiple U(1)s. We study the flops of such fibers, as well as some of the Yukawa couplings in codimension three. Furthermore, we carry out a universal study of the U(1)-charged GUT singlets, including their KK-charges, and determine all realizations of singlet fibers. By giving vacuum expectation values to these singlets, we propose a systematic way to analyze the Higgsing of U(1)s to discrete gauge symmetries in F-theory.

Figures (19)

• Figure 1: The ${\bf 5}$ and ${\bf 10}$ representation of $SU(5)$. Each box represents a weight $L_i$ ($L_i+ L_j$) of the fundamental (anti-symmetric) representation and the walls inbetween each box correspond to the action of the simple roots $\alpha_k =L_k - L_{k+1}$ on the weights as indicated by the arrows. The direction of the arrow indicates the addition of the corresponding simple root.
• Figure 2: Box graphs for $\mathfrak{u}(5)$ phases with ${\bf 5}$ matter. On the left are the splittings that occur over matter loci for the corresponding phase.
• Figure 3: Three types of codimension one $I_5$ fibers with sections $\sigma_0$ (blue) and $\sigma_1$ (red) distributed as $I_5^{(01)}$, $I_5^{(0|1)}$ and $I_5^{(0||1)}$, respectively.
• Figure 4: Box graphs and codimension two fibers where the $F_j$ that split into $C^\pm$ in codimension two are shown with dashed lines, for the $\mathfrak{su}(5)\oplus \mathfrak{u}(1)$ theory with matter in the fundamental representation.
• Figure 6: Codimension two fibers and charges for ${\overline{\bf 5}}$ matter for $I_5^{(01)}$ models. For details see (\ref{['figlab']}).

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Corollary 3.4
• Theorem 3.5
• Theorem 3.6
• Theorem 3.7
• Theorem 3.8