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

In this work we study gapped boundary states of ${\mathbb{Z}}_N$ bosonic symmetry-protected topological (SPT) phases in 4+1d, which are characterized by mixed ${\mathbb{Z}}_N$-gravity response, and the closely related phases protected by $C_N$ rotation symmetry. We show that if $N\mathrm{\notin }\{2,4,8,16\}$, any symmetry-preserving boundary theory is necessarily gapless for the root SPT state. We then propose a (3+1)$d$ ${\mathbb{Z}}_2$ 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 $C_N$ 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 $N$.