Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 7

The first of two special issues in honor of Cumrun Vafa’s 60th birthday

Invertible phases of matter with spatial symmetry

Pages: 1773 – 1788



Daniel S. Freed (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Michael J. Hopkins (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


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.

This material is based upon work supported by the National Science Foundation under Grant Numbers DMS-1158983, DMS-1160461, DMS-1510417, and DMS-1611957.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Published 8 September 2021