Anomalies of Non-Invertible Symmetries in (3+1)d

TL;DR

This work develops a higher-dimensional framework to diagnose anomalies of finite non-invertible symmetries in 3+1d by embedding the theory in a 4+1d Abelian two-form gauge theory with a 0-form permutation symmetry. It identifies two obstruction levels: a first, based on Lagrangian subgroups and polarization constraints, and a second, governed by generalized Frobenius-Schur indicators ω and ω_f that arise from decorated-domain-wall 4+1d SPTs. The authors show that certain non-invertible Kramers-Wannier–like symmetries in 3+1d are anomaly-free only after suitable symmetry extensions or non-invertible extensions, and they provide explicit 2+1d defect theories and 3+1d lattice models illustrating these phenomena. The results yield concrete criteria for when non-invertible symmetries can be realized in symmetric gapped or invertible gapped phases, and they reveal rich structures such as non-invertible time-reversal symmetries emerging from gauging charge conjugation. This framework links boundary dynamics, bulk SPT data, and lattice realizations, offering a versatile toolkit for exploring higher-dimensional anomalies and their physical consequences in gauge theories and condensed matter systems.

Abstract

Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d "inflow" symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the "Frobenius-Schur indicator" of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing $\mathbb{Z}_{4}$ ordinary symmetry host worldvolume theories with time-reversal symmetry $\mathsf{T}$ obeying the algebra $\mathsf{T}^{2}=C$ or $\mathsf{T}^{2}=(-1)^{F}C,$ with $C$ a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.

Figures (8)

• Figure 1: Two time-reversal defects in a duality domain wall, circled in pink, fuse to the FS indicator $\epsilon=\pm 1$. This means that the nontrivial FS indicator, given by $\epsilon=-1$, corresponds to $\mathsf{T}^2=-1$ on defects.
• Figure 2: Non-invertible time reversal symmetry of $O^{N,p}$, which is the $\mathcal{A}^{N,p}$ after gauging $C$. $\mathsf{T}$ means reversing the orientation, and $S$ means coupling to a $\mathbb{Z}_2$ gauge theory: $S[X]:={X\times (\mathbb{Z}_2)_{2p}\over \mathbb{Z}_2}$ for general 2+1d theories $X$ that has non-anomalous $\mathbb{Z}_2$ one-form symmetry. We have omitted the transparent fermion $\{1,f\}$ in the figure.
• Figure 3: After gauging the 1-form symmetry generated by $\prod_eX_e^{\sigma_e}$ surfaces (vertex term of $H_1$), and then shifting from the dual lattice (faces) to the original lattice (edges) with $X\leftrightarrow Z$, $H_0$ gets mapped to $H_1$. $H_0$ describes the SPT of class $p$ with $\mathbb{Z}_N$ 1-form symmetry Tsui:2019ykk while $H_1$ is dual to the 2-form gauge theory with topological action $p$ in Figure \ref{['fig:2-form']} on the dual lattice. Here, we show the lattice models for even $p$ for simplicity, the model for odd $p$ can be similarly constructed and is given in Tsui:2019ykk. While $H_0$ and $H_1$ are individually commuting Hamiltonians, $H=J_0H_0+J_1H_1$ is not commuting for nonzero $J_0$ and $J_1$.
• Figure 4: Modified Gauss law constraint that implements $ST^{n}$ gauging. Each of these terms are set to be 1, so the symmetry transformation on a matter field gets mapped to transformations on neighboring gauge fields. Here we list the model for even $n$, while the model for odd $n$ can be obtained similarly from the SPT models in Tsui:2019ykk.
• Figure 5: Lattice Hamiltonian for pure 2-form gauge theory with topological action $p$ on cubic lattice. Each face has a $\mathbb{Z}_N$ degree of freedom, acted by the generalization of Pauli operators that satisfying (\ref{['operators']}). The model can be obtained by gauging the 1-form symmetry in $\mathbb{Z}_N$ 1-form symmetry SPT phase of class $p$. The first row is the Gauss law term, while the second row is the flux term. Here we list the model for even $p$, while the model for odd $p$ can be obtained similarly from the SPT models in Tsui:2019ykk.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2