Orbifold modular GUT of flavor

We discuss an $SU\left(5\right)$ grand unified theory based on the $10d$ orbifold $(T^2{)}^3/(Z_4\times Z_2)$ plus three modular $S_4$ groups with moduli at respective fixed points $i,i+2,\omega =e^{2i\pi /3}$. The resulting model has hierarchical quark and charged-lepton mass matrices, arising from a double weighton mechanism, and reproduces the highly predictive littlest seesaw mechanism in the neutrino sector. The down-quark mass matrix has an upper triangular form, contributing to Cabibbo-Kobayashi-Maskawa mixing, while the charged-lepton mass matrix has a lower-triangular form with suppressed contributions to Pontecorvo-Maki-Nakagawa-Sakata mixing. The orbifold yields successful $SU\left(5\right)$ breaking with doublet-triplet splitting of the Higgs multiplets.