General U(1) x U(1) F-theory compactifications and beyond: geometry of unHiggsings and novel matter structure

We construct the general form of an F-theory compactification with two U(1) factors based on a general elliptically fibered Calabi-Yau manifold with Mordell-Weil group of rank two. This construction produces broad classes of models with diverse matter spectra, including many that are not realized in earlier F-theory constructions with U(1) x U(1) gauge symmetry. Generic U(1) x U(1) models can be related to a Higgsed non-Abelian model with gauge group SU(2) x SU(2) x SU(3), SU(2)(3) x SU(3), or a subgroup thereof. The nonlocal horizontal divisors of the Mordell-Weil group are replaced with local vertical divisors associated with the Cartan generators of non-Abelian gauge groups from Kodaira singularities. We give a global resolution of codimension two singularities of the Abelian model; we identify the full anomaly free matter content, and match it to the unHiggsed non-Abelian model. The non-Abelian Weierstrass model exhibits a new algebraic description of the singularities in the fibration that results in the first explicit construction of matter in the symmetric representation of SU(3). This matter is realized on double point singularities of the discriminant locus. The construction suggests a generalization to U(1)(k) factors with k > 2, which can be studied by Higgsing theories with larger non-Abelian gauge groups.