2-Group Symmetries and their Classification in 6d

TL;DR

This work identifies and classifies 2-group global symmetries in 6d SCFTs, showing that discrete $1$-form symmetries can nontrivially mix with continuous flavor symmetries via a Postnikov class $\,\Theta$ in $H^3(B\mathcal{F},\Gamma^{(1)})$. The authors develop a dimension-general framework for constructing and computing 2-groups, including a charge-matrix (Smith Normal Form) method to extract $\,\\Gamma^{(1)}$, $\\mathcal{E}$, and $\\mathcal{Z}$, and they apply this to 6d SCFTs/LSTs built from F-theory quivers. They provide a full classification: seven 6d SCFT classes carry 2-group symmetries of the type studied, while no LSTs do; the analysis hinges on the global form of the flavor group, the 1-form symmetry data, and the nontrivial Bockstein contribution to the Postnikov class. The work also identifies two higher-form anomalies (a dual $3$-form anomaly and a mixed $0$-form/$1$-form anomaly) and supplies a Mathematica tool to compute 2-group data for given quivers; it further discusses a universal dimension-independent approach with possible extensions to other dimensions via instantons and monopoles.

Abstract

We uncover 2-group symmetries in 6d superconformal field theories. These symmetries arise when the discrete 1-form symmetry and continuous flavor symmetry group of a theory mix with each other. We classify all 6d superconformal field theories with such 2-group symmetries. The approach taken in 6d is applicable more generally, with minor modifications to include dimension specific operators (such as instantons in 5d and monopoles in 3d), and we provide a discussion of the dimension-independent aspects of the analysis. We include an ancillary mathematica code for computing 2-group symmetries, once the dimension specific input is provided. We also discuss a mixed 't Hooft anomaly between discrete 0-form and 1-form symmetries in 6d.

Figures (4)

• Figure 1: Action of a topological operator associated to 0-form symmetry on topological operators associated to 1-form symmetries.
• Figure 2: If there exists a non-genuine local operator $O_{21}\neq0$ that can be used to transform a line defect $L_1$ to line defect $L_2$, then we regard $L_1$ and $L_2$ to be in the same equivalence class. Such equivalence classes form a group under OPE of line defects, which can be recognized as the Pontryagin dual group $\widehat{\Gamma}^{(1)}$ of the 1-form symmetry group $\Gamma^{(1)}$.
• Figure 3: A genuine local operator transforming in representation $R_2\otimes R_1^*$ of the flavor symmetry algebra $\mathfrak{f}$ can be regarded as transforming a flavor Wilson line in representation $R_1$ to a flavor Wilson line in representation $R_2$. The above configuration of the local operator joined to flavor Wilson lines is consistent as it is invariant under gauge transformations of a background flavor connection. If such a local operator exists, then we regard the $R_1$ and $R_2$ flavor Wilson lines to be in the same equivalence class. Such equivalence classes form a group $\widehat{\mathcal{Z}}$ with product operation being tensor product of representations.
• Figure 4: Now, consider a non-genuine local operator $O_{21}$ transitioning line defect $L_1$ to line defect $L_2$. Say $O_{21}$ transforms as $R_2\otimes R_1^*$ under the flavor algebra. Then we regard elements $(L_1,R_1)$ and $(L_2,R_2)$ (in the product set of line defects and flavor Wilson lines) to lie in the same equivalence class. Such equivalence classes form a group $\widehat{\mathcal{E}}$ with product operation being OPE and tensor product of representations.