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

TL;DR

This paper provides a comprehensive survey of U(1) gauge symmetries in F-theory compactifications, characterizing possible charges and fiber configurations using rational sections, with implications for GUT models and symmetry breaking.

Contribution

It introduces a universal framework for classifying U(1) charges and fiber structures in F-theory using rational sections, applicable to multiple U(1)s and GUT models.

Findings

Classified all possible U(1) charges via codimension two fibers.

Analyzed rational sections and their configurations in Calabi-Yau fibrations.

Proposed a systematic approach to Higgsing U(1)s to discrete symmetries.

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.