Topological phases and twisted equivariant vector bundles

This is the first of two talks on the paper "Twisted Equivariant Matter" by Freed and Moore. The topological space of equivariant gapped Hamiltonians will be defined, which will put topological band theory on a rigorous ground. Topological phases are its connected components. The primary goal of the talk is then to relate topological phases to twisted equivariant vector bundles on the Brillouin torus. Hence part of the lecture will be devoted to defining such objects and their equivalence classes.