Generalized matter parities from finite modular symmetries

We classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries Γ_N. Since all the couplings in supersymmetric theories of finite modular symmetries Γ_N are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight 1/M, we find $Z_{2M}$ symmetries, including the generalized baryon and lepton parities, R-parity, $Z_3$ baryon triality and $Z_6$ proton hexality. Such $Z_{2M}$ symmetries are enlarged to $Z_{2M}⋊Z_2^{\text{CP}}$ symmetries together with the CP transformation.