(3+1)-dimensional path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)-dimensional topological phases

The authors construct here a lattice path integral for a (3+1)D topological phase with U(1) symmetry, realizing (2+1)D topological order on its boundary. The U(1) background gauge field can possibly have a nontrivial Chern number. This allows the derivation of various formulas for anomalies of topological order involving U(1) symmetry, and leads to Walker-Wang construction of a topological insulator and superconductor. Mathematically, they correspond to new combinatorial state sums for invariants of topologically nontrivial, curved U(1) bundles over 4-manifolds, including spin${}^{c}$, pin${}^{c}$ structures.