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

Abstract

Abstract: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)×U(1) gauge symmetry. Generic U(1)×U(1) models can be related to a Higgsed non-Abelian model with gauge group SU(2)×SU(2)×SU(3), SU(2) 3 ×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.

Matter in transition

“…This behavior parallels that observed in the SU(3) models derived in [80]. There, all of the higher-genus SU(3) models with symmetric matter had non-Tate structures.…”

“…The geometric interpretation of this quantity is conjectured to be that when g > 0, for any representation other than the adjoint, the representation R is realized in F-theory through a Kodaira singularity on a divisor D that is itself singular, where g represents the arithmetic genus contribution of the singularity to the curve D in the 6D case. This correspondence works most simply for the symmetric representation of SU(N ), which has g = 1, and which can be realized on a double point singularity of D as first suggested by Sadov [84], described further in [5], and recently confirmed through an explicit F-theory construction [80]. The explicit F-theory construction of the symmetric matter representation of SU(3) has the unusual feature that the Weierstrass model cannot be built from a standard generic Tate SU(3) construction; rather, the vanishing of the discriminant to order 3 follows from a nontrivial cancellation that involves the explicit algebraic structure of the singular divisor locus carrying the SU(3) gauge group.…”

“…The construction presented here further supports the idea that non-Tate structures are necessary for symmetric matter; indeed, our models can only be expressed in standard forms for exactly those cases without symmetric matter. However, our SU(3) tuning seems to be different from the tuning given in [80]. The connection between these classes of models is left as a question for future investigations.…”

“…Each representation R of a group G can be associated with a genus contribution g, which in F-theory should have the interpretation of an arithmetic genus contribution from a singularity in the divisor supporting the gauge factor G. We have found an explicit example of such a realization in models with the symmetric (6) representation of SU(3). These models are connected through a matter transition to other models with an adjoint representation, but are realized, as in [80], by non-Tate Weierstrass models that exhibit a highly nontrivial cancellation in the vanishing of the discriminant at higher orders. The role of SCFTs in matter transitions that we have uncovered here suggests a resolution of a question regarding under what circumstances a Weierstrass model can exhibit such an exotic higher genus matter representation other than the adjoint: it seems in particular that a model formed by starting with a Tate model for a group G on a smooth divisor D and then deforming the divisor to a singular geometry will not develop an exotic matter representation in the singular limit without moving through a superconformal fixed point to a different branch of the set of Weierstrass models where JHEP04(2016)080 the algebraic structure of the model is changed, modifying the cancellation mechanism in the discriminant.…”

“…Here we explore a few such cases, particularly those related to the SU(N ) Λ 3 matter transitions. These include models with Sp(3) and SO(12) gauge groups, and a class of SU (3) transitions involving matter in the symmetric representation, where there is an intricate structure to the Weierstrass model similar to that found recently in [80]. The existence of distinct families of Sp (3), SO (12), and SU(3) models with varying matter content was recognized in [14]; we explore here the explicit connection of these models through matter transitions and comment on generalizations to other related models.…”