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 + 1 dimensions with a general 2 pi-quantized topological term Sigma(kappa)(I,J=1) K-IJ/4 pi integral(M4) F-I boolean AND F-J, where K is an integer symmetric matrix with even diagonal elements and F-I = dA(I). At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent Z(k1)((1)) x Z(k2)((1)) x ... 1-symmetry, where k(i) are the diagonal elements of the Smith normal form of K and Z(0)((1)) is regarded as a U(1) 1-symmetry. In the U-kappa(1) confined phase, the boundary can have a phase whose infrared (IR) properties are described by Chern-Simons field theory. Such a phase has a Z(k1)((1)) x Z(k2)((1)) x ... 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this continuum field theory, thus acting as its ultraviolet completion. The lattice model in the aforementioned limit has an exact Z(k1)((1)) x Z(k2)((1)) x ... 1-symmetry. We find that the short-range entangled 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 Z(k1)((1)) x Z(k2)((1)) x ... 1-symmetry, whose SPT invariant is e(i pi Sigma I,JKIJ integral M4BI(sic)BJ+BI(sic)1dBJ) e(i pi Sigma I

Automated summary

This study investigates a compact U-kappa(1) gauge theory in 3 + 1 dimensions with a general 2π-quantized topological term and a symmetric matrix K. It shows that at energies below the gauge charges' gaps but above the monopoles' gaps, the theory exhibits a compact Z(k1)((1)) x Z(k2)((1)) x ... 1 symmetry.