Towards Exotic Matter and Discrete Non-Abelian Symmetries in F-theory

We present a prescription in F-theory for realizing matter in "exotic" representations of product gauge groups. For 6D vacua, bifundamental hypermultiplets are engineered by starting at a singular point in moduli space which includes 6D superconformal field theories coupled to gravity. A deformation in Higgs branch moduli space takes us to a weakly coupled gauge theory description. In the corresponding elliptically fibered Calabi--Yau threefold, the minimal Weierstrass model parameters $(f,g,\Delta)$ vanish at collisions of the discriminant at least to order $(4,6,12)$, but with sufficiently high order of tangency to ensure the existence of T-brane deformations to a weakly coupled gauge theory with exotic bifundamentals. We present explicit examples including bifundamental hypermultiplets of $\mathfrak{e}_7 \times \mathfrak{su}_2$ and $\mathfrak{e}_6 \times \mathfrak{su}_3$, each of which have dual heterotic orbifold descriptions. Geometrically, these matter fields are delocalized across multiple points of an F-theory geometry. Symmetry breaking with such representations can be used to produce high dimension representations of simple gauge groups such as the four-index symmetric representation of $\mathfrak{su}_2$ and the three-index symmetric representation of $\mathfrak{su}_3$, and after further higgsing can yield discrete non-abelian symmetries.