Extending fusion rules with finite subgroups: A general construction of $Z_N$ extended conformal field theories and their orbifoldings

We construct the $Z_N$ symmetry extended fusion ring of bulk and chiral theories and the corresponding modular partition functions with nonanomalous subgroup $Z_n(\subset Z_N)$. The chiral fusion ring provides fundamental data for $Z_N$- graded symmetry topological field theories and also provides algebraic data for smeared boundary conformal field theories, which describe the zero modes of the extended models. For more general multicomponent or coupled systems, we also obtain a new series of extended theories. By applying the folding trick, their partition functions correspond to charged or gapped domain walls or massless renormalization group flows preserving quotient group structures.