对称性作为拓扑序的阴影及拓扑全息原理的推导
Symmetry as a shadow of topological order and a derivation of topological holographic principle
AI总结：
本文提出用高一维拓扑序统一描述普通、反常及不可逆高阶对称性，并由边界局域算符代数推导拓扑全息原理。
摘要：
对称性通常通过由（高阶）群描述的变换来定义。但对称性实际上对应于局域对称算符的代数，该代数直接约束系统的性质。在本文中，我们指出局域对称算符的代数包含一类特殊的扩展算符——透明补丁算符，它们揭示选择扇区并因此揭示相应的对称性。这些透明补丁算符在 n 维空间中的代数产生一个非退化编织融合 n-范畴，而这恰好描述了高一维中的拓扑序（对于有限对称性）。这样一种全息理论不仅描述（高阶）对称性，还描述反常（高阶）对称性、不可逆（高阶）对称性（也称为代数高阶对称性）以及不可逆引力反常。因此，高一维中的拓扑序取代群，为上述广义对称性提供统一且系统的描述。这被称为对称性/拓扑序（Symm/TO）对应。我们的方法还导出拓扑全息原理：边界唯一确定体，或更准确地说，局域边界算符的代数唯一确定体拓扑序。作为 Symm/TO 对应的应用，我们展示了在 1 维空间中，具有混合反常的 Z₂×Z₂ 对称性与 Z₄ 对称性之间，以及许多其他对称性之间的等价性。
英文摘要：
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.
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