Categorical Continuous Symmetry
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918142240882688 |
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| author | Jia, Qiang Luo, Ran Tian, Jiahua Wang, Yi-Nan Zhang, Yi |
| author_facet | Jia, Qiang Luo, Ran Tian, Jiahua Wang, Yi-Nan Zhang, Yi |
| contents | We define the symmetry category in 1+1d for continuous 0-form $G$-symmetry to be $\textbf{Sky}^τ(G)$, the category of skyscraper sheaves of finite dimensional vector spaces with finite support on the group manifold of $G$, where $τ\in H^4(BG,\mathbb{Z})$ is the anomaly. We propose that the corresponding 2+1d SymTFT is described by the Drinfeld center of $\textbf{Sky}^τ(G)$. We show explicitly the way that $τ$ twists the convolution tensor product of the objects of $\textbf{Sky}^τ(G)$. As a concrete example, we present the $S$ and $T$-matrices for the simple anyons of the resulting $Z(\textbf{Sky}^τ(G))$ category for $G = U(1)$, both for the cases without or with anomaly and discuss the topological boundary conditions as Lagrangian algebra of $Z(\textbf{Sky}^τ(U(1)))$. We also present the definition of $\textbf{Sky}^τ(G)$ and $Z(\textbf{Sky}^τ(G))$ for the non-abelian case of $G=SU(2)$, as well as the speculated modular data. We point out that in order to have a physically relevant center and Lagrangian algebras it is necessary to generalize $\textbf{Sky}^τ(G)$ to a larger category, which we argue to be closely related to the category of quasi-coherent sheaves on $G_\mathbb{C}$ with convolution tensor product twisted by $τ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_13170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Categorical Continuous Symmetry Jia, Qiang Luo, Ran Tian, Jiahua Wang, Yi-Nan Zhang, Yi High Energy Physics - Theory We define the symmetry category in 1+1d for continuous 0-form $G$-symmetry to be $\textbf{Sky}^τ(G)$, the category of skyscraper sheaves of finite dimensional vector spaces with finite support on the group manifold of $G$, where $τ\in H^4(BG,\mathbb{Z})$ is the anomaly. We propose that the corresponding 2+1d SymTFT is described by the Drinfeld center of $\textbf{Sky}^τ(G)$. We show explicitly the way that $τ$ twists the convolution tensor product of the objects of $\textbf{Sky}^τ(G)$. As a concrete example, we present the $S$ and $T$-matrices for the simple anyons of the resulting $Z(\textbf{Sky}^τ(G))$ category for $G = U(1)$, both for the cases without or with anomaly and discuss the topological boundary conditions as Lagrangian algebra of $Z(\textbf{Sky}^τ(U(1)))$. We also present the definition of $\textbf{Sky}^τ(G)$ and $Z(\textbf{Sky}^τ(G))$ for the non-abelian case of $G=SU(2)$, as well as the speculated modular data. We point out that in order to have a physically relevant center and Lagrangian algebras it is necessary to generalize $\textbf{Sky}^τ(G)$ to a larger category, which we argue to be closely related to the category of quasi-coherent sheaves on $G_\mathbb{C}$ with convolution tensor product twisted by $τ$. |
| title | Categorical Continuous Symmetry |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2509.13170 |