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 $e^{i\pi {\sum }_{I,J}{K}_{IJ}\int B_I⌣B_J+B_{I}d{B}_J}{e}^{i\pi {\sum }_{I<J}{K}_{IJ}\int d{B}_{I}d{B}_J}$. 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.