Residual group-like symmetries in selection rules without group actions

Abstract: We analyze loop-induced group-like symmetries in theories where fields are labeled by basis elements of a fusion algebra constructed from the conjugacy classes of finite groups. Although the fusion rules for conjugacy classes are in general violated at loop level, residual group-like symmetries, including both Abelian and non-Abelian ones, remain exact through a procedure referred to as ``groupification''. By examining various conjugacy classes of finite groups realized in heterotic string theory on non-Abelian orbifolds, we identify an approximate discrete symmetry that controls the magnitude of loop-induced couplings. As a result, most parameters appearing in non-invertible selection rules are natural in the sense of 't Hooft. Furthermore, we discuss anomalies of the groupification symmetry, which can impose additional constraints on models with non-invertible fusion rules.