Understanding Non-Split 2-Group Symmetry: (3+1)D SymTFT, Anomaly and Bordism

Abstract

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 ${\Omega }_{d+1}^{\mathrm{SO}}(B\mathcal{G})$ and the spin bordism groups ${\Omega }_{d+1}^{\mathrm{Spin}}(B\mathcal{G})$ for $d\le 5$. 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 $\mathrm{Hom}\left({\tilde{\Omega }}_4^{\mathrm{SO}}(B\mathcal{G}),U(1)\right)\cong H^4(B\mathcal{G};U(1))\cong {\mathbb{Z}}_2$. 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.

41 pages + appendices; v3: minor changes, citation added