Three-index symmetric matter representations of SU(2) in F-theory from non-Tate form Weierstrass models

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 codimen-sion 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 2 × su(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 be needed to incorporate models with arbitrary matter representations and gauge groups localized on singular divisors.