Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory

We consider compact $U^\kappa(1)$ gauge theory in 3+1D with the $2\pi$-quantized topological term ${\sum_{I, J =1}^\kappa\frac{K_{IJ}}{4\pi}\int_{M^4}F^I\wedge F^J}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry, where $k_i$ are the diagonal elements of the Smith normal form of $K$ and $\mathbb{Z}_{0}^{(1)}$ is regarded as $U(1)^{(1)}$. In the $U^\kappa(1)$ confined phase, the boundary's IR properties are described by Chern-Simons field theory and has a ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this field theory, thus acting as its UV completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry. We find that a gapped phase of the lattice model, corresponding to the confined phase of the $U^\kappa(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry, whose SPT invariant is ${e^{i\pi\sum_{I, J}K_{IJ}\int B_I\smile B_J+B_I\underset{1}{\smile} d B_J}e^{i\pi\sum_{I< J}K_{IJ}\int d B_I\underset{2}{\smile}d B_J}}$. Here, the background 2-cochains $B_I$ satisfy ${d B_I=\sum_I B_{I}K_{IJ} = 0}$ mod $1$ and describe the symmetry twist of the ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the non-trivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.