Gauging spatial symmetries and the classification of topological crystalline phases

A \emph{topological crystalline} phase of matter is a topological phase protected by space-group symmetries. The prototypical examples are the so-called ``topological crystalline insulators''. For strongly interacting topological crystalline phases, there is as yet no systematic theory. This is in contrast to the case of \emph{internal} symmetries, where coupling to a background gauge field allows one to derive a systematic classification. In this work, we elucidate what it means to gauge a spatial symmetry, allowing us to give a systematic classification of topological crystalline phases. Our work applies to a subset of topological crystalline phases which we call ``topological crystalline liquids''; we conjecture that this subset includes nearly all topological crystalline phases, with the exception of states with exotic fracton excitations such as the ``Haah code''. As an example, we classify bosonic topological crystalline liquids for all 230 space groups.