Anomalies of Generalized Symmetries from Solitonic Defects

TL;DR

The paper develops a defect-based framework to study ’t Hooft anomalies of generalized global symmetries in 3d, parameterizing anomalies in terms of charges of solitonic defects like vortex lines and monopole operators. It unifies 0-form, 1-form, and 2-group symmetries by analyzing how defects source background fields and how gauging modifies both the spectrum of defects and the symmetry structure. The authors provide a systematic method to compute monopole and defect charges from Chern–Simons terms and fermion content, and they demonstrate how various anomalies arise as obstructions or phase factors associated with defect backgrounds. Extensive 3d gauge theory examples illustrate 1-form, 0-form, and 2-group anomalies, and the work outlines how these techniques extend to gauging procedures and to higher-dimensional generalizations. The results offer a practical, defect-centric path to understanding generalized symmetries and their anomalies in strongly coupled or non-Lagrangian settings, with potential applications in condensed matter and SCFT contexts.

Abstract

We propose the general idea that 't Hooft anomalies of generalized global symmetries can be understood in terms of the properties of solitonic defects, which generically are non-topological defects. The defining property of such defects is that they act as sources for background fields of generalized symmetries. 't Hooft anomalies arise when solitonic defects are charged under these generalized symmetries. We illustrate this idea for several kinds of anomalies in various spacetime dimensions. A systematic exploration is performed in 3d for 0-form, 1-form, and 2-group symmetries, whose 't Hooft anomalies are related to two special types of solitonic defects, namely vortex line defects and monopole operators. This analysis is supplemented with detailed computations of such anomalies in a large class of 3d gauge theories. Central to this computation is the determination of the gauge and 0-form charges of a variety of monopole operators: these involve standard gauge monopole operators, but also fractional gauge monopole operators, as well as monopole operators for 0-form symmetries. The charges of these monopole operators mainly receive contributions from Chern-Simons terms and fermions in the matter content. Along the way, we interpret the vanishing of the global gauge and ABJ anomalies, which are anomalies not captured by local anomaly polynomials, as the requirement that gauge monopole operators and mixed monopole operators for 0-form and gauge symmetries have non-fractional integer charges.

Figures (17)

• Figure 1: Depiction of a solitonic line operator $L$, which is pierced by a codimension 1 ball $D_{d-1}$. Integrating $w_2, B_2, B_w$ over this disk determines the background for 0- and 1-form as well as 2-group symmetries by the solitonic line operator.
• Figure 2: A genuine local operator $O$ transforming in a representation $R$ of the 0-form symmetry group $\mathcal{F}$ needs to be attached to a background Wilson line in the representation $R$ of $\mathcal{F}$.
• Figure 3: The figure studies correlation functions involving a non-genuine local operator $O$ arising at the end of a line defect $L$ and transforming in a representation $R$ of $F$. The operator is attached to a background Wilson line in representation $R$ whose locus is displayed as a dashed line. The representation $R$ carries a charge $\alpha_L\in\widehat{\mathcal{Z}}$ under $\mathcal{Z}$. In blue is shown a piece in the $(d-2)$-cycle Poincaré dual to the background 2-cocycle $w_2$ defined in the text, which carries an element $\beta\in\mathcal{Z}$. (1) and (3) are the same correlation functions as we have just moved $\beta$ without crossing any objects. This correlation function can be related to the correlation function without the $\beta$ insertion in two different ways. Starting from the configuration (1), we can collapse $\beta$ onto $L$ to reach the configuration (2), where $\phi$ is an a priori unknown phase obtained by moving $\beta$ across $L$. Or starting from (3), we can collapse $\beta$ onto the background Wilson line $R$ to reach the configuration (4), where we know that moving $\beta$ across the $R$ line changes correlation function by the phase $\langle\alpha_L,\beta\rangle$ using the natural map $\langle\cdot,\cdot\rangle:\widehat{\mathcal{Z}}\times\mathcal{Z}\to U(1)$. Consistency demands that the two phase factors must be equal leading to the determination of $\phi$, which is $\phi=\langle\alpha_L,\beta\rangle$. This justifies the correlation function jump (\ref{['r0fsj']}).
• Figure 4: The charges of a line defect $L$ under various elements $\gamma\in\Gamma^{(1)}$ can be described in terms of a particular element $\widehat{\gamma}_L\in\widehat{\Gamma}^{(1)}$. Here $\widehat{\gamma}_L(\gamma)\in U(1)$ describes the image of $\gamma$ under the homomorphism $\widehat{\gamma}_L:~\Gamma^{(1)}\to U(1)$.
• Figure 5: The figure studies correlation functions involving a non-genuine local operator $O$ arising at a junction between two line defects $L_1$ and $L_2$. In blue is shown a topological codimension-two defect corresponding to a 1-form symmetry element $\gamma$. (1) and (3) are the same correlation functions as we have just moved $\gamma$ without crossing any objects. This correlation function can be related to the correlation function without the $\gamma$ insertion in two different ways. Starting from the configuration (1), we can collapse $\gamma$ onto $L_1$ to reach the configuration (2). In the process, we generate an additional phase $\phi(\gamma,L_1)$ which is the charge of $L_1$ under $\gamma$. Or starting from (3), we can collapse $\gamma$ onto $L_2$ to reach the configuration (4), which generates a phase $\phi(\gamma,L_2)$ which is the charge of $L_2$ under $\gamma$. Consistency demands that the two phase factors must be equal. Thus two line defects lying in the same equivalence class have same charge under all 1-form symmetries, and hence the possible 1-form symmetries form a group isomorphic to the Pontryagin dual of the group $\widehat{\Gamma}^{(1)}$ of equivalence classes of line defects.