3+1d Boundary Topological Order of 4+1d Fermionic SPT state

We generalize the previous gapped boundary construction of 4+1d beyond-cohomology bosonic SPT phase to the fermionic SPT phase, which is now protected by a finite group symmetry including the fermion parity $Z_2^F$ symmetry. Via the crystalline correspondence, we related the classification of $Z_{2N}^F$ fermionic SPT phases with that of crystal rotational $C_N imes Z_2^F$ symmetry, where $C_N$ is the $N$-fold rotation. We construct TQFT boundaries for those SPT phases that do not forbid it and propose an exotic "$Z_K$ gauge theory" as a boundary TQFT, where the anomalous symmetry is implemented by codimension-1 invertible topological defects. We also give an explicit gapped boundary construction. In particular, for $N=2$, we relate the classification of $Z_4^{TF}$ with that of 3+1d topological superconductor classified by $Z_16$. For odd $ u in Z_{16}$, we give possible constructions of gapped phases but whose low energy theory is possibly beyond the conventional TQFT. For all other possible gapped phases, i.e. even $ u in Z_{16}$’s, we can further construct and identify the low energy TQFTs.