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.
Categories: cond-mat.str-el hep-lat hep-th