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

In this work we study gapped boundary states of ZN bosonic symmetry-protected topological (SPT) phases in (4+1)d, which are characterized by mixed ZN -gravity response, and the closely related phases protected by CN rotation symmetry.We show that if N / ∈ {2, 4, 8, 16}, any symmetrypreserving boundary theory is necessarily gapless for the root SPT state.We then propose a (3+1)d Z2 gauge theory coupled to fermionic matter as a candidate boundary theory for N = 2, 4, 8, 16, where the anomalous symmetry is implemented by invertible topological defects obtained from gauging (2+1)d chiral topological superconductors.For the CN case, we present an explicit construction for the boundary states for N = 2, 4, 8, 16, and argue that the construction fails for other values of N .