Twisted equivariant matter

Understand

We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation.

• This framework generalizes both the classical 3-fold way of real/complex/quaternionic representations as well as a corresponding 10-fold way which has appeared in condensed matter and nuclear physics.
• We establish a foundation for discussing continuous families of quantum systems.
• Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians.