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

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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 $N\notin \{2,4,8,16\}$, 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 $N$.