Lin Feng

The Standard Model (SM) is an SU(3)xSU(2)xU(1) gauge theory with the assumption of field representations and renormalizability. The fermion masses are not predicted by the SM, and the Yukawa coupling constants had to be introduced, as representations of the SM gauge group. The motivation is the fact that string compactification on tori or orbifolds can lead to a modular-invariant Lagrangian, where, in the Minimal Supersymmetric Standard Model (MSSM), the superpotential and therefore Yukawa coupling constants are modular forms. It is thus natural to extend the SM gauge group with some subgroups of the modular group SL(2,Z). The Higgs mechanism for the extended gauge group now gives the fermion masses, from engineering the gauge group and vacuum selection.