Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase

We study the (3+1)d boundary theory of Z_N beyond cohomology symmetry protected topological (SPT) phase in (4+1)d, which exhibits a new kind of quantum anomaly in (3+1)d systems with Z_N symmetry. We show that when N ∉ {2,4,8,16}, there cannot be any symmetry-preserving gapped boundary states. When N ∈ {2,4,8,16}, we study a related system with C_N symmetry via the crystalline correspondence principle, and explicitly construct gapped boundary topological order for the beyond-cohomology SPT states. The boundary of the C_N SPT state can be disentangled everywhere except on the rotation axis, where there are chiral edge states described by the (E₈)₁ theory. To gap them out, we introduce N copies of (2+1)d gapped layers all terminating at the rotation axis. We study the problem of symmetry-preserving gapped interface at the rotation axis and in particular show that for N ∈ {2,4,8,16}, if each layer has the topological order of Spin(16/N)₁, the (3+1)d boundary can be completely gapped out while preserving the C_N symmetry.