Exploring $G$-ality defects in 2-dim QFTs
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2024
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| author | Lu, Da-Chuan Sun, Zhengdi Zhang, Zipei |
| author_facet | Lu, Da-Chuan Sun, Zhengdi Zhang, Zipei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exploring $G$-ality defects in 2-dim QFTs Lu, Da-Chuan Sun, Zhengdi Zhang, Zipei High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics 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. |
| title | Exploring $G$-ality defects in 2-dim QFTs |
| topic | High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2406.12151 |