Classifying loops of symmetry-protected states

The classification of states of quantum lattice systems is a well -defined mathematical endeavour which started with the discovery of the quantum Hall effect. In this talk, I will discuss the topology of a simple cl ass, the so-called invertible states, which I will
define. It is by definition a connected set, and we shall explore its further topological properties. Specifically, I will be interested in what can be identified with its fundamental group; Physically, this is about classifying cycles of physical processes, or pumps. I will
present a classification of such loops of invertible state that have a local symmetry, which can be proved to be complete. This is joint work with Wojciech De Roeck, Martin Fraas and Tijl Jappens.