Invertible phases of matter with spatial symmetry

Abstract

We propose a general formula for the group of invertible topological phases on a space $Y$, possibly equipped with the action of a group $G$. Our formula applies to arbitrary symmetry types. When $Y$ is Euclidean space and $G$ a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.

13 pages, 3 figures