3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems

Understand

We study bosonic systems on a spacetime lattice defined by path integrals of commuting fields.

• We introduce branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the spacetime simplicial complex, even for a spacetime with boundaries.
• In contrast, a generic lattice bosonic (GLB) system's path integral may depend on the branch structure.
• We find the invertible topological order characterized by the Stiefel-Whitney cocycle (e.g., 4+1d w$_2$w$_3$) to be nontrivial for BIB systems, but this topological order and a trivial gapped tensor product state belong to the same phase for GLB systems.