Chiral spectrum of the universal tuned $(\text{SU}(3)\times \text{SU}(2)\times \text{U}(1))\mathrm{/}{\mathbb{Z}}_6$ 4D F-theory model

We use the recently developed methods of 2108.07810 to analyze vertical flux backgrounds and associated chiral matter spectra in the 4D universal $(\text{SU}(3)\times \text{SU}(2)\times \text{U}(1))\mathrm{/}{\mathbb{Z}}_6$ model introduced in 1912.10991, which is believed to describe the most general generic family of F-theory vacua with tuned $(\text{SU}(3)\times \text{SU}(2)\times \text{U(}1))\mathrm{/}{\mathbb{Z}}_6$ gauge symmetry. Our analysis focuses on a resolution of a particular presentation of the $(\text{SU}(3)\times \text{SU}(2)\times \text{U}(1))\mathrm{/}{\mathbb{Z}}_6$ model in which the elliptic fiber is realized as a cubic in ${\mathbb{P}}^2$ fibered over an arbitrary smooth threefold base. We show that vertical fluxes can produce nonzero multiplicities for all chiral matter families that satisfy 4D anomaly cancellation, which include as a special case the chiral matter families of the Minimal Supersymmetric Standard Model.