Matter Symmetries in Supersymmetric Standard Models from Noninvertible Selection Rules

Abstract We discuss the phenomenological implications of noninvertible selection rules in the framework of the supersymmetric standard model. We find that a remnant $\mathbb {Z}_2$ symmetry of fusion algebras that holds at all-loop order plays the role of R-parity, forbidding baryon- and lepton-violating operators. In addition, a combination of Standard Model gauge symmetry and the noninvertible selection rules lead to baryon triality and proton hexality that can protect the proton from decay. In general, our selection rules cannot realize the same results as conventional $\mathbb {Z}_N$ symmetries in the supersymmetric standard model. We also clarify the assignments of matter fields under the fusion algebras that are consistent with $SU(4) \times SU(2) \times SU(2)$, $SU(5)$, and $SO(10)$ grand unified theories.