Quasi-local algebras and invertible phases of matter

In the algebraic approach to quantum field theory, observables localized in bounded regions form a net of von Neumann algebras generating what is known as a quasi-local algebra. I will discuss the classification of quasi-local algebras under a notion of equivalence that generalizes Brauer equivalence and captures only large-scale properties. I will argue that this problem coincides with the classification of invertible phases of lattice systems in one dimension higher, providing a way to relate quantum field-theoretic and microscopic descriptions. As a byproduct, I will describe a candidate for the generalized cohomology theory associated with invertible phases, as conjectured by Kitaev.