Proton decay, Yukawa couplings and underlying gauge symmetry in string theory

In string theory, massless particles often originate from a symmetry breaking of a large gauge symmetry G to its subgroup H  . The absence of dimension-4 proton decay in supersymmetric theories suggests that (D¯,L) are different from H¯(5¯) in their origins. In this article, we consider a possibility that they come from different irreducible components in g/hg/h. Requiring that all the Yukawa coupling constants of quarks and leptons be generated from the super-Yang–Mills interactions of G  , we found in the context of Georgi–Glashow H=SU(5)H=SU(5) unification that the minimal choice of G   is E7E7 and E8E8 is the only alternative. This idea is systematically implemented in heterotic String, M theory and F theory, confirming the absence of dimension 4 proton decay operators. Not only H=SU(5)H=SU(5) but also G constrain operators of effective field theories, providing non-trivial information.