Anomalies and Symmetry Fractionalization

TL;DR

This work shows that the presence of a one-form symmetry Γ enriches the usual ’t Hooft anomaly structure of a zero-form symmetry G by introducing symmetry fractionalization classes in $H^2_ ho(G, obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak vert obreak obreak obreak obreak obreak obreak obreak obreak)$ that can shift G’s anomaly in a theory with Γ anomalies or mixed G–Γ anomalies. The authors provide a concrete framework to compute these anomalies across 0+1, 1+1, 2+1, and 3+1 dimensions, illustrate the mechanism with QCD-like theories and adjoint QCD in multiple dims, and show how twisted gauge transformations, background B-fields, and SPT inflows capture the fractionalization data. They demonstrate that distinct fractionalization classes can be realized physically by heavy states or by gauging outer automorphism twists, leading to multiple, inequivalent anomaly values for G. The paper also explores how fractionalization can drive two-group symmetry structures and how magnetic (gauged one-form) symmetries alter the symmetry algebra and anomaly interpretation, with detailed consistency checks against proposed IR dualities and domain-wall physics. Overall, this work provides a robust, dimension-spanning toolkit for analyzing how higher-form symmetries reshape anomaly constraints and the infrared behavior of strongly coupled gauge theories.

Abstract

We study ordinary, zero-form symmetry $G$ and its anomalies in a system with a one-form symmetry $Γ$. In a theory with one-form symmetry, the action of $G$ on charged line operators is not completely determined, and additional data, a fractionalization class, needs to be specified. Distinct choices of a fractionalization class can result in different values for the anomalies of $G$ if the theory has an anomaly involving $Γ$. Therefore, the computation of the 't Hooft anomaly for an ordinary symmetry $G$ generally requires first discovering the one-form symmetry $Γ$ of the physical system. We show that the multiple values of the anomaly for $G$ can be realized by twisted gauge transformations, since twisted gauge transformations shift fractionalization classes. We illustrate these ideas in QCD theories in diverse dimensions. We successfully match the anomalies of time-reversal symmetries in $2+1d$ gauge theories, across the different fractionalization classes, with previous conjectures for the infrared phases of such strongly coupled theories, and also provide new checks of these proposals. We perform consistency checks of recent proposals about two-dimensional adjoint QCD and present new results about the anomaly of the axial $\mathbb{Z}_{2N}$ symmetry in $3+1d$ ${\cal N}=1$ super-Yang-Mills. Finally, we study fractionalization classes that lead to 2-group symmetry, both in QCD-like theories, and in $2+1d$ $\mathbb{Z}_2$ gauge theory.

Figures (5)

• Figure 1: The type of line operator can change from $L$ to $g\cdot L$ when it pierces the codimension-one symmetry generator labelled by $g\in G$.
• Figure 2: The collection of line operators can realize the symmetry projectively (on the Hilbert space on $S^{d}$), while the action of $G$ symmetry that permutes the label of lines is linear.
• Figure 3: Junctions of co-dimension one symmetry defects are important in the study of 't Hooft anomalies. In theories with one-form symmetry, a co-dimension two invertible topological defect can be inserted in the junction.
• Figure 4: $1+1d$ SPT attached to line operator captures the $G$ anomaly on the line.
• Figure :