Operator Algebra of Local Symmetric Operators and Braided Fusion n-Category*

Symmetries are ordinarily described by groups or higher groups. A symmetry selects a set of local symmetric operators which form an algebra. The algebra of all local symmetric operators determines the possible quantum phases and phase transitions, as well as all other properties allowed by the symmetry. Isomorphic algebras give rise to the same physical properties, and are regarded as equivalent symmetries. We refer to such isomorphic classes of algebra as categorical symmetries, which, by definition, describe all symmetries. We argue that an algebra of all local symmetric operators in n-dimensional space determines a non-degenerate braided fusion n-category (NBF-n-C), i.e. topological orders in one higher dimension, and isomorphic algebras give rise to the same NBF-n-C. We suggest that this NBF-n-C is a better description of symmetry than groups, since (anomalous) symmetries described by different groups can be equivalent. This description naturally includes symmetries beyond group and higher group, such as algebraic (higher) symmetries. We discuss explicit examples in the case of n=1 to make our case.