The importance of being discrete

Abstract: Although continuous symmetries may be more appealing, especially the local gauge ones, I argue that discrete symmetries may still play a fundamental role in shaping our understanding of the physics beyond the Standard Model. I exemplify this on the case of left-right symmetry and recall how its spontaneous breaking leads naturally to new weak interactions, and even more interestingly, to neutrino mass and the seesaw mechanism behind its smallness. The minimal realization of this program is shown to be a self-contained theory of the origin and nature of neutrino mass, and of quark mixing angles. I also discuss some new results on the extension of the theory based on Pati-Salam quark-lepton symmetry, and its embedding in $SO(10)$ grand unified theory, where LR symmetry becomes gauged discrete transformation in the form of charge conjugation.