Quark and lepton mass matrices from localization in M-theory on G2 orbifold

M-theory compactified on a G₂ manifold with resolved E₈ singularities realizes four-dimensional N =1 supersymmetric gauge theories coupled to gravity with three families of Standard Model fermions. Beginning with one E₈ singularity, three fermion families emerge when E₈ is broken by geometric engineering deformations to a smaller subgroup with equal rank. In this paper, we use the local geometry of the theory to explain the origin of the three families and their mass hierarchy. We linearize the blowing up of two-cycles associated with resolving E₈ singularities. After imposing explicit constraints on the effectively stabilized moduli, we arrive at Yukawa couplings for the quarks and leptons. We fit the high scale Yukawa couplings approximately which results in the quark masses agreeing reasonably well with the observations, implying that the experimental hierarchy of the masses is achievable within this framework. The hierarchy separation of the top quark from the charm and up is a stringy effect, while the spitting of the charm and up also depends on the Higgs sector. The Higgs sector cannot be reduced to having a single vacuum expectation value (VEV); all three VEVs must be nonzero. Three extra U (1 ) s survive to the low scale but are not massless, so Z states are motivated to occur in the spectrum, but may be massive.