Generic matter representations in 6D supergravity theories

In six-dimensional supergravity, there is a natural sense in which matter lying in certain representations of the gauge group is "generic," in the sense that other "exotic" matter representations require more fine tuning. From considerations of the dimensionality of the moduli space and anomaly cancellation conditions, we find that the generic sets of matter representations are well-defined for 6D supergravity theories with gauge groups containing arbitrary numbers of nonabelian factors and $\operatorname{U}(1)$ factors. These generic matter representations also match with those that arise in the most generic F-theory constructions, both in 6D and in 4D, with non-generic matter representations requiring more exotic singularity types. The analysis of generic versus exotic matter illuminates long-standing puzzles regarding F-theory models with multiple $\operatorname{U}(1)$ factors and provides a useful framework for analyzing the 6D "swampland" of apparently consistent low-energy theories that cannot be realized through known string constructions. We note also that the matter content of the standard model is generic by the criteria used here only if the global structure is $\operatorname{SU}(3)_\text{c} \times \operatorname{SU}(2)_\text{L} \times \operatorname{U}(1)_Y / \mathbb{Z}_6$.