On the action of non-invertible symmetries on local operators in 3+1d

Abstract

Most of the known non-invertible symmetries of quantum field theories in three and four spacetime dimensions act invertibly on local operators. An exception is coset symmetries, which can be constructed from gauging a non-normal subgroup of an invertible symmetry. In this paper, we study the action of a general finite non-invertible symmetry on local operators in four dimensions. We show that non-invertible symmetries without topological line operators necessarily act invertibly on local operators. Using this result, we argue that the action of a general non-invertible symmetry in 3+1d on local operators can be decomposed into the invertible action of some operators composed with the action of a gauging interface. We use this result to study when such a symmetry is anomaly-free. We find a necessary condition for a finite non-invertible symmetry in 3+1d to be anomaly-free, and show that anomaly-free non-invertible symmetries without topological line operators are non-intrinsically non-invertible.

Figures (8)

• Figure 1: The action of a topological codimension 1 operator on a local operator $O$ is defined by sliding the topological operator past $O$. If two codimension 1 operators $M_1$ and $M_2$ are connected by a topological interface $I$, then $I$ can be freely deformed so as to implement either the action of $M_1$ or that of $M_2$ on $O$. Consequently, the actions of $M_1$ and $M_2$ on $O$ must coincide.
• Figure 2: If the topological operators $M_1$ and $M_1$ are related by a topological interface $I$, then $M_2$ can be constructed by putting a network of $S:=I^{\dagger}\times I$ lines on it. Since $S$ is a codimension-2 operator, it does not act on local operators.
• Figure 3: The gapped boundary $\mathcal{B}_{\mathfrak{C}}$ can be understood as an interface between the TQFT $\mathcal{Z}(\mathfrak{C})$ and a $G$-SPT obtained from gauging the $\Omega \mathcal{Z}(\mathfrak{C})\cong \text{Rep}(G)$ symmetry. A surface operator $S$ on $\mathcal{B}_{\mathfrak{C}}$ is attached to a membrane operator $M_g$ in the $G$-SPT implementing the dual 0-form symmetry. We have suppressed one dimension of all operators in the diagram.
• Figure 4: Consider a surface operator $S\in \mathfrak{C}$ on $S^{1} \times \mathds{R}$. At the limit where the radius of $S^1$ goes to zero, we get a line operator $L_S$. $L_S$ must be topological as the operator $S$ is topological.
• Figure 5: Folding a topological surface operator $S$ gives an interface between the surface operator $S\times \overline{S}$ and the identity surface operator $\mathds{1}$. This shows that the outcome of the fusion rule $S\times \overline{S}$ must contain at least one condensation surface operator.