Three-Index Symmetric Matter Representations of SU(2) in F-Theory from Non-Tate Form Weierstrass Models

TL;DR

This paper constructs explicit F-theory models with three-index symmetric matter representations of SU(2), realized through non-Tate Weierstrass models, and explores their unHiggsing to more complex gauge groups.

Contribution

It provides the first explicit realization of matter in F-theory with genus contributions greater than one, using non-Tate Weierstrass models and unHiggsing techniques.

Findings

Realization of SU(2) matter in a three-index symmetric representation.

Construction of models with singular divisors and non-Tate Weierstrass forms.

UnHiggsing to G_2xSU(2) and SU(2)^3 models with diverse matter content.

Abstract

We give an explicit construction of a class of F-theory models with matter in the three-index symmetric (4) representation of SU(2). This matter is realized at codimension two loci in the F-theory base where the divisor carrying the gauge group is singular; the associated Weierstrass model does not have the form associated with a generic SU(2) Tate model. For 6D theories, the matter is localized at a triple point singularity of arithmetic genus g=3 in the curve supporting the SU(2) group. This is the first explicit realization of matter in F-theory in a representation corresponding to a genus contribution greater than one. The construction is realized by "unHiggsing" a model with a U(1) gauge factor under which there is matter with charge q=3. The resulting SU(2) models can be further unHiggsed to realize non-Abelian G_2xSU(2) models with more conventional matter content or SU(2)^3 models with trifundamental matter. The U(1) models used as the basis for this construction do not seem to have a Weierstrass realization in the general form found by Morrison-Park, suggesting that a generalization of that form may be needed to incorporate models with arbitrary matter representations and gauge groups localized on singular divisors.