Naturally light invisible axion and local Z13⊗Z3 symmetries

We show that by imposing local Z₁₃Z₃ symmetries in an SU(2)U(1) electroweak model we can implement an invisible axion in such a way that (i) the Peccei-Quinn symmetry is an automatic symmetry of the classical Lagrangian, and (ii) the axion is protected from semiclassical gravitational effects. In order to be able to implement such a large discrete symmetry, and at the same time allow a general mixing in each charge sector, we introduce right-handed neutrinos and enlarge the scalar sector of the model. The domain wall problem is briefly considered.