We study the fractionalization of space group symmetries in two-dimensional topologically ordered phases. Specifically, we focus on
Z2-fractionalized phases in two dimensions whose deconfined topological excitations transform trivially under translational symmetries but projectively under glide reflections, whose quantum numbers are hence fractionalized. We accomplish this by generalizing the dichotomy between even and odd gauge theories to incorporate additional symmetries inherent to nonsymmorphic crystals. We show that the resulting fractionalization of point group quantum numbers can be detected in numerical studies of ground state wave functions. We illustrate these ideas using a microscopic model of a system of bosons at integer unit cell filling on a lattice with space group
p4g that can be mapped to a half-magnetization plateau for an
S=1/2 spin system on the Shastry-Sutherland lattice.
It is demonstrated that fermionic/bosonic symmetry-protected topological (SPT) phases across different dimensions and symmetry classes can be organized using geometric constructions that increase dimensions and symmetry-reduction maps that change symmetry groups. Specifically, it is shown that the interacting classifications of SPT phases with and without glide symmetry fit into a short exact sequence, so that the classification with glide is constrained to be a direct sum of cyclic groups of order 2 or 4. Applied to fermionic SPT phases in the Wigner-Dyson class AII, this implies that the complete interacting classification in the presence of glide is ${\mathbb{Z}}_{4}\ensuremath{\bigoplus}{\mathbb{Z}}_{2}\ensuremath{\bigoplus}{\mathbb{Z}}_{2}$ in three dimensions. In particular, the hourglass-fermion phase recently realized in the band insulator KHgSb must be robust to interactions. Generalizations to spatiotemporal glide symmetries are discussed.