Consequences of Symmetry Fractionalization without 1-Form Global Symmetries

TL;DR

The paper develops a framework to study symmetry fractionalization of $0$-form symmetries on line operators in theories without exact $1$-form symmetries, introducing the disk operator as a practical detector of projective actions. It derives exact consequences, including selection rules in twisted sectors and exact finite-volume vacuum degeneracies driven by mixed anomalies, and shows how RG flows can preserve or match these fractionalization patterns either via emergent $1$-form symmetry or decoupled quantum mechanics. The authors provide concrete realizations (e.g., Scalar QED with $N$ flavors) and discuss multiple gauge-theory and nonlinear sigma-model examples to illustrate these ideas. They also analyze RG-flow matching conditions, including 2-group extensions and anomaly inflow, highlighting how fractionalization constraints the IR phase structure and potential emergent topological order. Overall, the work extends the landscape of symmetry fractionalization beyond traditional 1-form symmetries and clarifies how endable lines encode and preserve global symmetry data along RG flows.

Abstract

We study the fractionalization of 0-form global symmetries on line operators in theories without 1-form global symmetries. The projective transformation properties of line operators are renormalization group invariant, and we derive constraints which are similar to the consequences of exact 1-form symmetries. For instance, symmetry fractionalization can lead to exact selection rules for line operators in twisted sectors, and in theories with 't Hooft anomalies involving the fractionalization class, these selection rules can further imply that certain twisted sectors have exact finite-volume vacuum degeneracies. Along the way, we define topological operators on open codimension-1 manifolds, which we call `disk operators', that provide a convenient way of encoding the projective action of 0-form symmetries on lines. In addition, we discuss the possible ways symmetry fractionalization can be matched along renormalization group flows.

Figures (16)

• Figure 1: An open line operator which transforms projectively under $G^{(0)}$ is 'charged' under the disk operator. Here, the two ways of unlinking the disk operator -- passing through the line (above) or through the end point (below) -- lead to a consistent action by a phase $\rho$.
• Figure 2: 0-form symmetries can act projectively in the presence of a line operator. The junction of symmetry operators labeled by $g,h\in G^{(0)}$ can act on a line $\mathscr{L}$ by a phase $\omega_{g,h}\in U(1)$. The group cohomology class $[\omega]\in H^2(BG^{(0)},U(1))$ measures the projective representation carried by the line, which can be interpreted as the projective transformation properties of internal (spin-defect) degrees of freedom on the line which are traced out to yield the operator $\mathscr{L}$. The arrows denote the orientations of the lines/surfaces.
• Figure 3: The $G^{(0)}$ symmetry can act projectively on the operators ${\cal O}$ living at endpoints of lines $\mathscr{L}$, as in (a). By moving the topological $G^{(0)}$ junction around the endpoint of an open line as in (b), we obtain a consistency condition relating the projective phase $\omega_{g,h}$ to the representation $R$ under which the endpoint operators transform.
• Figure 4: Two 0-form symmetry defects labeled by elements that commute in $G^{(0)}$ may only commute up to a phase when pierced by a line.
• Figure 5: Collapsing $G^{(0)}$ defects to create a disk operator.