SU(3)×SU(2)×U(1) vacua in F-theory

The Standard Model group and matter spectrum is obtained in vacua of F-theory, without resorting to an intermediate unification group. The group SU(3)×SU(2)×U(1)Y is the commutant to SU(5)⊥×U(1)Y structure group of a Higgs bundle in E8 and is geometrically realized as a deformation of I5 singularity. Lying along the unification groups of En, our vacua naturally inherit their unification structure. By modding SU(5)⊥ out by Z4 monodromy group, we can distinguish Higgses from lepton doublets by matter parity. Turning on universal G-flux on this part, the spectrum contains three generations of quarks and leptons, as well as vectorlike pairs of electroweak and colored Higgses. Minimal Yukawa couplings is obtained at the renormalizable level.