F -theory GUTs with U ( 1 ) symmetries: Generalities and survey

We study the structure of $SU(5)$ $F$-theory grand unified theory (GUT) models that engineer additional $U(1)$ symmetries. These are highly constrained by a set of relations observed by Dudas and Palti (DP) that originate from the physics of four-dimensional anomaly cancellation. Using the DP relations, we describe a general tension between unification and the suppression of dimension 5 proton decay when one or more $U(1)$'s are Peccei-Quinn (PQ) symmetries and hypercharge flux is used to break the $SU(5)$ GUT group. We then specialize to spectral cover models, whose global completions in $F$ theory we know how to construct. In that setting, we provide a technical derivation of the DP relations, construct spectral covers that yield all possible solutions to them, and provide a complete survey of spectral cover models for $SU(5)$ GUTs that exhibit two $U(1)$ symmetries.