On 2-Group Global Symmetries and Their Anomalies

TL;DR

The paper develops a comprehensive framework for 2-group global symmetries, unifying ordinary 0-form and 1-form symmetries through the data (G, A, ρ, [β]). It formalizes symmetry defects and 2-group background fields, derives anomaly inflow descriptions across dimensions, and clarifies the distinction between operator-valued structures and bona fide anomalies. A central contribution is the explicit procedure to extract 2-group data from theories and to couple them to generalized backgrounds, with detailed treatments of RG flows, emergent symmetries, and various 3d/4d examples, including TQFTs and CS-matter theories. The work also connects 2-group obstructions to familiar phenomena like symmetry fractionalization, Green-Schwarz-type couplings, and time-reversal or spacetime symmetries, providing a versatile toolkit for analyzing gapped and gapless QFTs with higher-form symmetries. Overall, it furnishes a robust, cohomology-based language for classifying 2-group anomalies and for implementing generalized symmetry couplings in diverse quantum field theories.

Abstract

In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related 't Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the "obstruction to symmetry fractionalization" discussed in some condensed matter literature is really an instance of 2-group global symmetry.

Figures (6)

• Figure 1: A junction (in red) where three 0-form symmetry defects of type ${\bf g}$, ${\bf h}$, ${\bf g}{\bf h}$ meet in codimension 2. This configuration is generic in spacetime dimension 2 and above. These junctions encode the group law of the 0-form symmetry.
• Figure 2: When a symmetry operator of type $a \in \mathcal{A}$ crosses a codimension-1 symmetry operator of type ${\bf g}\in G$, it emerges transformed by an automorphism $\rho_{\bf g}$ of $\mathcal{A}$.
• Figure 3: A transformation of symmetry defects can be used to detect 2-group global symmetry. The lines (labelled ${\bf g},{\bf h},{\bf k}, {\bf g}{\bf h}{\bf k}$) are codimension-1 symmetry operators of $G$. Transforming from left to right (also called an $F$-move), a 1-form symmetry defect $\beta({\bf g},{\bf h},{\bf k})\in \mathcal{A}$ is created (blue dot). In $d$ dimensions, all objects span the remaining $d-2$ dimensions. A probe line passing through the 0-form symmetry defects, detects $[\beta]$ and hence sees the non-associativity of the 0-form symmetry defects.
• Figure 4: A junction where 0-form symmetry defects of type ${\bf g}$, ${\bf h}$, ${\bf k}$, ${\bf g}{\bf h}{\bf k}\in G$ meet in codimension 3. This configuration is generic in spacetime dimension 3 and above. The junctions of three codimension-1 defects are in red, and their intersection is the black point. At the codimension-3 intersection, a 1-form symmetry defect $\beta({\bf g},{\bf h},{\bf k})$ emanates, signaling the 2-group symmetry. In $d$ dimensions, all objects span the remaining $d-3$ dimensions.
• Figure 5: Left: Minimal triangulation necessary to encode the configuration of codimension-1 symmetry defects in Figure \ref{['fig: 2-group 3D']}. The vertices $\{i,j,k,l,m,n\}$ are ordered; in $d$ dimensions the remaining $d-3$ dimensions are implicit. The configuration is made of an upper "pyramid" $\{ijlmn\}$ and a lower upside-down pyramid $\{ijkmn\}$. Right: 2D section of the configuration of symmetry defects, seen from above. The nodes contained in each domain are indicated. The dashed line corresponds to the lower portion of Figure \ref{['fig: 2-group 3D']} which contains the node $k$; the dotted line corresponds to the upper portion of Figure \ref{['fig: 2-group 3D']} which contains the node $l$. Therefore, the triangulation can encode a bordism between the two configurations in Figure \ref{['fig: obstructed F-move']}.