Classification and surface anomaly of glide symmetry protected topological phases in three dimensions

We study glide protected topological (GSPT) phases of interacting bosons and fermions in three spatial dimensions certain on-site symmetries. They are crystalline SPT phases, which are distinguished from a trivial product state only in the presence of non-symmorphic glide symmetry. We classify these GSPT phases with various on-site symmetries such as $U(1)$ and time reversal, and show that they can all be understood by stacking and coupling two-dimensional short-range-entangled phases in a glide-invariant way. Using such a coupled layer construction we study the anomalous surface topological orders of these GSPT phases, which gap out the two-dimensional surface states without breaking any symmetries. This framework is demonstrated in many examples, including the non-symmorphic topological insulator with "hourglass fermion" surface states.