Symmetry as a shadow of topological order and a derivation of topological holographic principle

Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corre-sponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators- transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in n-dimensional space gives rise to a nondegenerate braided fusion n-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, noninvertible (higher) symmetries (also known as algebraic higher symmetries), and noninvertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to as symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: boundary uniquely determines the bulk, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between Z2 x Z2 symmetry with mixed anomaly and Z4 symmetry, as well as between many other symmetries, in 1-dimensional space.

Automated summary

Symmetry is usually defined by transformations described by a (higher) group. However, symmetries can also be described by an algebra of local symmetric operators, which directly affect the properties of the system. This paper introduces the concept of transparent patch operators, a special class of extended operators within the algebra of local symmetric operators, which reveal the selection sectors and corresponding symmetries. The algebra of these transparent patch operators in n-dimensional space gives rise to a nondegenerate braided fusion n-category, providing a unified and systematic description of generalized symmetries.