Systematic construction of interfaces and anomalous boundaries for fermionic symmetry-protected topological phases

We use the pullback trivialization technique to systematically construct gapped interfaces and anomalous boundaries for fermionic symmetry-protected topological (FSPT) states by extending their symmetry group ${G}_{f}={\mathbb{Z}}_{2}^{f}{\ifmmode\times\else\texttimes\fi{}}_{{\ensuremath{\omega}}_{2}}G$ to larger groups. These FSPT states may involve decoration layers of both Majorana chains and complex fermions. We derive general consistency formulas explicitly for ($2+1$)-dimensional (D) and ($3+1$)D systems, where nontrivial twists arise from fermionic symmetric local unitaries or ``gauge transformations'' that ensure coboundaries vanish at the cochain level. Additionally, we present explicit example for a ($3+1$)D FSPT of symmetry group ${G}_{f}={\mathbb{Z}}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ with Majorana chain decorations.