Generalized Matter Parities from Finite Modular Symmetries

We classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries ${\mathrm{\Gamma }}_N$. Since all the couplings in supersymmetric theories of finite modular symmetries ${\mathrm{\Gamma }}_N$ 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 $1\mathrm{/}M$, we find ${\mathbb{Z}}_{2M}$ symmetries, including the generalized baryon and lepton parities, $R$-parity, ${\mathbb{Z}}_3$ baryon triality and ${\mathbb{Z}}_6$ proton hexality. Such ${\mathbb{Z}}_{2M}$ symmetries are enlarged to ${\mathbb{Z}}_{2M}⋊{\mathbb{Z}}_2^{\text{CP}}$ symmetries together with the CP transformation.