Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories

TL;DR

The paper develops a comprehensive cohomological framework to classify symmetry-enriched (3+1)d U(1) gauge theories with a global symmetry G, capturing how symmetry acts on electric charges and magnetic monopoles via data (ρ, [ν], p, n). It identifies two levels of anomalies: a deconfinement obstruction governed by $\mathcal{H}^3_\rho[G,U(1)]$ and a more familiar ’t Hooft anomaly governed by $\mathcal{H}^5_s[G,U(1)]$, with physical realizations corresponding to boundaries of 4+1d bulk states (long-range entangled or SPT) respectively. The authors derive explicit 4-cocycle structures for $\mathcal{H}^4[\mathcal{G},U_T(1)]$, introduce an obstruction $[\gamma]$ and a related $[\mathcal{O}]$, and provide simplified forms in key cases (unitary finite G, anti-unitary G, and lattice translations). Applying the framework to concrete groups such as $\mathbb{Z}_2$, $\mathrm{SO}(3)$, and $\mathbb{Z}_2\times\mathbb{Z}_2^{\mathsf{T}}$, they reproduce known anomalies, classify new classes, and illuminate LSMHO-type constraints as well as fermionic interacting SPT boundaries. The work offers a unified path to diagnose when symmetry-enriched U(1) QSLs are realizable in 3D and highlights rich bulk-boundary connections in higher-dimensional topological phases.

Abstract

We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group $G$. We find that, in general, a symmetry-enrichment pattern is specified by 4 pieces of data: $ρ$, a map from $G$ to the duality symmetry group of this $\mathrm{U}(1)$ gauge theory which physically encodes how the symmetry permutes the fractional excitations, $ν\in\mathcal{H}^2_ρ[G, \mathrm{U}_\mathsf{T}(1)]$, the symmetry actions on the electric charge, $p\in\mathcal{H}^1[G, \mathbb{Z}_\mathsf{T}]$, indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor $n$ over $\mathcal{H}^3_ρ[G, \mathbb{Z}]$, the symmetry actions on the magnetic monopole. However, certain choices of $(ρ, ν, p, n)$ are not physically realizable, i.e. they are anomalous. We find that there are two levels of anomalies. The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly. States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state. If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion. States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state. We apply these results to some interesting physical examples.

Figures (2)

• Figure 1: The notion of symmetry-protected distinction between two phases. These two phases can be smoothly connected if the system lacks certain symmetry, but they are necessarily separated by a phase transition in the presence of the symmetry.
• Figure 2: Upper: the possible values of the electric and magnetic charges of an excitation, $(q_e, q_m)$, form a charge-monopole lattice. This figure shows the charge-monopole lattice at $\theta=2\pi N$, where $N$ is an integer. Lower: When $\theta\neq 0$, the positions of the fractional excitations in the charge-monopole lattice are shifted due to the Witten effect. More precisely, the excitation with magnetic charge $q_m$ will get additional electric charge $\frac{\theta q_m}{2\pi}$. In the above figure, the lengths and directions of the red arrows indicate how the positions of the corresponding excitations change.