Gauging spatial symmetries and the classification of topological   crystalline phases
  We put the theory of interacting topological crystalline phases on a systematic footing. These are topological phases protected by space-group symmetries. Our central tool is an elucidation of what it means to "gauge" such symmetries. We introduce the notion of a "crystalline topological liquid", and argue that most (and perhaps all) phases of interest are likely to satisfy this criterion. We prove a Crystalline Equivalence Principle, which states that in Euclidean space, crystalline topological liquids with symmetry group $G$ are in one-to-one correspondence with topological phases protected by the same symmetry $G$, but acting *internally*, where if an element of $G$ is orientation-reversing, it is realized as an anti-unitary symmetry in the internal symmetry group. As an example, we explicitly compute, using group cohomology, a partial classification of bosonic symmetry-protected topological (SPT) phases protected by crystalline symmetries in (3+1)-D for 227 of the 230 space groups. For the 65 space groups not containing orientation-reversing elements (Sohncke groups), there are no cobordism invariants which may contribute phases beyond group cohomology, and so we conjecture our classification is complete. 
Ryan Thorngren and Dominic V. Else
cond-mat.str-el cond-mat.mtrl-sci math-ph math.MP