3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
Abstract
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 w2w3) 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. The invertible topological orders in GLB systems 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. 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 BIB systems to realize the w2w3 topological order, and its 3+1d gapped or gapless boundaries. A 3+1d Z2 gauge theory with (1) fermionic Z2 gauge charge particle trivializes w2 and (2) fermionic Z2 gauge flux line trivializes w3. In particular, if the flux loop's worldsheet is unorientable, then an orientation-reversal 1d worldline corresponds to a fermion worldline carrying no Z2 gauge charge. Spin and Spinc structures trivialize the w2w3 global pure gravitational anomaly to zero (which helps to construct 3+1d Z2 and all-fermion U(1) gauge theories), but the Spinh and Spin×Z2Spin(n ≥ 3) structures modify the w2w3 into a global mixed gauge-gravitational anomaly, which helps to constrain Grand Unifications (e.g., n=10,18) or construct new models.
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