理解非分裂2-群对称性：(3+1)维SymTFT、反常和配边
Understanding Non-Split 2-Group Symmetry: (3+1)D SymTFT, Anomaly and Bordism
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中文总结 AI 辅助
该研究系统分析含ℤ₂ 0-形式与1-形式对称性的非分裂2-群𝒢，计算d≤5时的配边群以分类其反常，在d=3时结合融合2-范畴与TQFT作用研究其Symmetry TFT并分类边界条件，实施范畴化朗道范式。
中文摘要 AI 辅助
我们对具有非平凡Postnikov类的有限非分裂2-群对称性开展系统研究，聚焦于最简单的示例：含ℤ₂ 0-形式对称性与ℤ₂ 1-形式对称性，二者通过H³(Bℤ₂;ℤ₂)≅ℤ₂中的非平凡Postnikov类相互交织，记为𝒢。我们针对d维时空的物理理论，计算定向配边群Ω_{d+1}^{SO}(B𝒢)与自旋配边群Ω_{d+1}^{Spin}(B𝒢)（d≤5），以此对该2-群对称性的反常进行分类。对于d=3的情形，我们采用融合2-范畴语言，结合无2-群反常或由Hom(Ω̃₄^{SO}(B𝒢),U(1))≅H⁴(B𝒢;U(1))≅ℤ₂分类的2-群反常的(3+1)维TQFT作用，研究2-群对称性𝒢的Symmetry TFT/TO。我们对Symmetry TFT的最小拓扑与物理边界条件进行分类，并针对这类非分裂有限2-群实施范畴化朗道范式。
英文摘要
We present a systematic study of finite, non-split 2-group symmetries with a non-trivial Postnikov class, focusing on the simplest example with a $\mathbb{Z}_2$ 0-form symmetry and a $\mathbb{Z}_2$ 1-form symmetry, intertwined together by the non-trivial Postnikov class in $H^3(B\mathbb{Z}_2;\mathbb{Z}_2)\cong\mathbb{Z}_2$, denoted by $\mathcal{G}$. We classify the anomalies of this 2-group symmetry for physical theories in $d$-dimensional spacetime, by computing the oriented bordism groups $Ω_{d+1}^{\rm SO}(B\mathcal{G})$ and the spin bordism groups $Ω_{d+1}^{\rm Spin}(B\mathcal{G})$ for $d\leq 5$. For the case of $d=3$, we investigate the Symmetry TFT/TO of the 2-group symmetry $\mathcal{G}$ using the language of fusion 2-categories, as well as (3+1)D TQFT actions, for the cases without or with the 2-group anomaly classified by $\operatorname{Hom}\left(\widetildeΩ_4^{\rm SO}(B\mathcal{G}),U(1)\right) \cong H^4(B\mathcal{G};U(1))\cong\mathbb{Z}_2$. We classify the minimal topological and physical boundary conditions of the Symmetry TFTs, and carry out the categorical Landau paradigm for such a non-split finite 2-group.
