(3+1)d boundaries with gravitational anomaly of (4+1)d invertible topological order for branch-independent bosonic systems

We study bosonic systems on a space-time lattice (with imaginary time) defined by path integrals of commuting fields. We introduce a concept of branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the space-time simplicial complex, even for a space-time 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 [such as (4+1)d w(2)w(3)] to be nontrivial for branch-independent bosonic systems, but this topological order and a trivial gapped tensor product state belong to the same phase (via a smooth deformation without any phase transition) for generic lattice bosonic systems. This implies that the invertible topological orders in generic lattice bosonic systems on a space-time lattice are not classified by the oriented cobordism. The branch dependence on a lattice may be related to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories. In general, the branch structure on a discretized lattice may be related to a frame structure on a smooth manifold that trivializes any Stiefel-Whitney classes. We construct branch-independent bosonic systems to realize the w(2)w(3) topological order, and its (3+1)d gapped or gapless boundaries. One of the gapped boundaries is a (3+1)d Z(2) gauge theory with (1) fermionic Z(2) gauge charge particle which trivializes w(2) and (2) fermionic Z(2) gauge flux line trivializes w(3). In particular, if the flux loop's world sheet is unorientable, then the orientation-reversal one-dimensional world line must correspond to a fermion world line that does not carry the Z(2) gauge charge. We also explain why Spin and Spinc structures trivialize the w(2)w(3) nonperturbative global pure gravitational anomaly to zero [which helps to construct the anomalous (3+1)d gapped Z(2) and gapless all-fermion U(1) gauge theories], but the Spin(h) and Spinx Z(2) Spin(n >= 3) structures modify the w(2)w(3) into a nonperturbative global mixed gauge-gravitational anomaly, which helps to constrain grand unifications (e.g., n = 10, 18) or construct new models.

Automated summary

This article studies bosonic systems on a space-time lattice and introduces the concept of branch-independent bosonic (BIB) systems. It is found that invertible topological orders in branch-independent systems differ from those in generic lattice bosonic systems. The branch dependence on a lattice is related to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories.