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$USp(32)$ Special Grand Unification

We discuss a grand unified theory (GUT) based on a $USp(32)$ GUT gauge group broken to its subgroups including a special subgroup. A GUT based on an $SO(32)$ GUT gauge group has been discussed on six-dimensional (6D) orbifold space $M^4\times T^2/\mathbb{Z}_2$. It is inspired by the $SO(32)$ string theory behind the $SU(16)$ GUT whose $SU(16)$ is broken to a special subgroup $SO(10)$. Alternative direction is to embed an $SU(16)$ gauge group into a $USp(32)$ GUT gauge group, which is inspired by a non-supersymmetric symplectic-type $USp(32)$ string theory. In a $USp(32)$ GUT, one generation of the SM fermions is embedded into a 6D bulk Weyl fermion in a $USp(32)$ defining representation. For a three generation model, all the 6D and 4D gauge anomalies in the bulk and on the fixed points are canceled out without exotic chiral fermions at low energies. The SM Higgs scalar is embedded into a 6D bulk scalar field in a $USp(32)$ adjoint representation.