二维量子场论中的 $G$-ality 缺陷探索
Exploring $G$-ality defects in 2-dim QFTs
AI总结：
本文利用SymTFT和群论融合范畴，系统研究了二维QFT中由扭曲规范产生的非可逆G-ality缺陷的分类与物理含义，并通过三性、p-性和S3-性三个例子展示了其异常结构和自旋选择规则。
摘要：
Tambara-Yamagami (TY) 融合范畴对称性 $\text{TY}(\mathbb{A},χ,ε)$ 描述了在规范有限阿贝尔群 $\mathbb{A}$ 下二维量子场论（QFT）的增强非可逆自对偶对称性。我们通过考虑扭曲规范（即在规范前后堆叠 $\mathbb{A}$-SPT）来推广增强非可逆对称性。这种非可逆对称性可以从三维 SymTFT 的可逆任意子置换对称性中获得。考虑由 $\mathbb{A}$ 的（非）扭曲规范形成的有限群 $G$，在 $G$ 中的拓扑操作下不变的二维 QFT 允许非可逆的 \textit{$G$-ality 缺陷}。我们利用 SymTFT 和群论融合范畴研究 $G$-ality 缺陷的分类和物理意义，并给出三个具体例子。1) 三性（Triality），其中 $\mathbb{A} = \mathbb{Z}_N \times \mathbb{Z}_N$，且 $N$ 与 $3$ 互素。该分类先前由 Jordan 和 Larson 确定，其数据类似于 TY 融合范畴，我们确定了这些融合范畴的异常（anomaly）。2) $p$-性（$p$-ality），其中 $\mathbb{A} = \mathbb{Z}_p \times \mathbb{Z}_p$，$p$ 为奇素数。我们考虑两个这样的范畴 $\mathcal{P}_{\pm,m}$，它们通过不同的对称性分数化（symmetry fractionalization）选择来区分，这是一种在 TY 分类中未出现的新数据，并表明它们具有不同的异常结构和自旋选择规则。3) $S_3$-性（$S_3$-ality），其中 $\mathbb{A} = \mathbb{Z}_N \times \mathbb{Z}_N$。我们通过 SymTFT 对 $N < 20$ 的情况进行了显式分类，并为某些 $N$ 提供了群论构造。我们发现 $N=5$ 是允许 $S_3$-ality 的最小 $N$，而 $N=11$ 是允许群论 $S_3$-ality 的最小 $N$。
英文摘要：
The Tambara-Yamagami (TY) fusion category symmetry $\text{TY}(\mathbb{A},χ,ε)$ describes the enhanced non-invertible self-duality symmetry of a $2$-dim QFT under gauging a finite Abelian group $\mathbb{A}$. We generalize the enhanced non-invertible symmetries by considering twisted gauging which allows stacking $\mathbb{A}$-SPTs before and after the gauging. Such non-invertible symmetries can be obtained from invertible anyon permutation symmetries of the $3$-dim SymTFT. Consider a finite group $G$ formed by (un)twisted gaugings of $\mathbb{A}$, a $2$-dim QFT invariant under topological manipulations in $G$ admits non-invertible \textit{$G$-ality defects}. We study the classification and the physical implication of the $G$-ality defects using the SymTFT and the group-theoretical fusion categories, with three concrete examples. 1) Triality with $\mathbb{A} = \mathbb{Z}_N \times \mathbb{Z}_N$ where $N$ is coprime with $3$. The classification was previously determined by Jordan and Larson where the data is similar to the $\text{TY}$ fusion categories, and we determine the anomaly of these fusion categories. 2) $p$-ality with $\mathbb{A} = \mathbb{Z}_p \times \mathbb{Z}_p$ where $p$ is an odd prime. We consider two such categories $\mathcal{P}_{\pm,m}$ which are distinguished by different choices of the symmetry fractionalization, a new data that does not appear in the TY classification, and show that they have distinct anomaly structures and spin selection rules. 3) $S_3$-ality with $\mathbb{A} = \mathbb{Z}_N \times \mathbb{Z}_N$. We study their classification explicitly for $N < 20$ via SymTFT, and provide a group-theoretical construction for certain $N$. We find $N=5$ is the minimal $N$ to admit an $S_3$-ality and $N=11$ is the minimal $N$ to admit a group-theoretical $S_3$-ality.
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