Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography

The algebraic or ring structure of anyons, called the fusion rule, is one of the most fundamental research interests in contemporary studies on topological orders (TOs) and the corresponding conformal field theories (CFTs). Recently, the algebraic structure realized as generalized symmetry, including non-invertible and categorical symmetry, captured attention in the fields. Such non-abelian anyonic objects appear in a bulk CFT or chiral CFT (CCFT), but it has been known that the construction of a CCFT contains theoretical difficulties in general. In this work, we propose the fusion rules in $Z_{N}$ extended chiral and bulk conformal field theories and the corresponding TOs. We explicitly construct a nontrivial expression of subalgebra structure in the fusion rule of a bulk CFT. We name this subalgebra ``bulk semion". This corresponds to the fusion rule of the CCFT and categorical symmetry of the TO or $Z_{N}$ graded symmetry topological field theory (SymTFT). This gives a bulk-edge correspondence based on the symmetry analysis and corresponds to an anyon algebraic expression of topological holography in the recent literature. The recent topological holography is expected to apply to systems in general space-time dimensions. Moreover, we present a concise way of unifying duality (or fractional supersymmetry), generalized or categorical symmetry, and Lagrangian subalgebra. Our method is potentially useful to formulate and study general TOs, fundamentally only from the data of bulk CFTs or vice versa, and gives a clue in understanding CCFT (or ancillary CFT more generally).