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

TL;DR

The paper reframes symmetry as the algebra of local symmetric operators and introduces transparent patch operators whose algebra encodes a non-degenerate braided fusion n-category, i.e., a topological order in one higher dimension. This Symm/TO correspondence provides a unified framework for ordinary, higher, anomalous, and non-invertible (algebraic) symmetries, and leads to a topological holographic principle whereby boundary operator algebras uniquely determine the bulk. Through explicit 1D and 2D constructions, the authors derive dualities between anomalous and anomaly-free symmetries (e.g., $\mathbb{Z}_2\times\mathbb{Z}_2$ mixed anomaly $\sim$ $\mathbb{Z}_4$), and show how holography recasts symmetry as bulk topological data. The work also synthesizes a holographic point of view on symmetry, enabling a systematic classification of equivalent symmetry types via their categorical centers and bulk orders, with practical implications for identifying dualities and emergent symmetries. Overall, finite symmetry shadows of higher-dimensional topological orders provide a powerful, generalizable language for symmetry, duality, and anomaly in quantum many-body systems.

Abstract

Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds 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 non-degenerate 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, non-invertible (higher) symmetries (also known as algebraic higher symmetries), and non-invertible 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 symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: \emph{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 $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry with mixed anomaly and $\mathbb{Z}_4$ symmetry, as well as between many other symmetries, in 1-dimensional space.

Figures (21)

• Figure 1: A 2d lattice bosonic model, whose degrees of freedom live on the vertices and are labeled by the elements in a set: $g_i\in G$
• Figure 2: The matrix elements of a string-like tensor network operator, $O_{m_1,m_2,\cdots;n_1,n_2,\cdots}$, can be given by a contraction of rank-4 tensors $T_{n_2,m_2,a,b}$, . Each tensor is represented by a vertex, where the legs of the vertex correspond to the indices of the tensor. The connected legs have the same index and is summed over (which correspond to the tensor contraction). This is just one representation of tensor network operator.
• Figure 3: (a) "Fusion" of two string operators. (b) Non-trivial "braiding" between two string operators. (c) Trivial "braiding" between two string operators.
• Figure 4: Non-trivial "braiding" between two string operators, the patch symmetry operator (the solid line) and the patch charge operator (the dashed-line), measures the symmetry charge carried by boundary of patch charge operator, if the patch symmetry operator generates the symmetry.
• Figure 5: Two ways to fuse three particles $a,b,c$ into $abc$, as operator product. The phase difference of the two resulting operators is $F(a,b,c)$. The horizontal lines and the corresponding 45$^\circ$ lines correspond to hopping operators. For example $1\xrightarrow{b} 2 \sim T_b(1\to 2)$. The hopping operators with higher location are applied first. Thus we have the relation $T_c(3\to 1) T_b(2\to 1) T_a(0\to 1) = F(a,b,c) T_a(0\to 1) T_c(3\to 1) T_b(2\to 1)$.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2