We describe a systematic procedure for determining the identity of a 2D bosonic symmetry-protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $\omega$ of the cohomology group $H^3(G,U_T\left(1\right))$. This element $\omega \in H^3(G,U_T\left(1\right))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the $F$ symbol associated with domain walls in a symmetry-broken edge theory; this domain-wall $F$ symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories, including both lattice models and continuum field theories.

中文翻译：

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玻色子对称保护拓扑边缘理论中的异常：与符号的连接和计算方法

我们描述了一个系统的程序，用于根据边缘激发的特性确定二维玻色对称保护拓扑 (SPT) 相的身份。我们的方法适用于具有幺正或反酉对称性以及连续或离散对称群的一般玻色 SPT 相位，唯一的限制是对称性必须在现场。具体来说，我们的程序将玻色 SPT 边缘理论作为输入，并产生一个元素$\omega$ 上同调群 $H^3(G,{你}_{吨}\left(1\right))$. 这个元素$\omega \in H^3(G,{你}_{吨}\left(1\right))$可以解释为体 2D SPT 相位的标签或 SPT 边缘理论携带的异常的标签。我们方法背后的基本思想是计算$F$对称破缺边缘理论中与畴壁相关的符号；这个域名墙$F$符号正是我们希望计算的异常。我们用几个 SPT 边缘理论展示了我们的方法，包括晶格模型和连续场理论。