Superselection theory in generalized geometry

Superselection theory is a branch of algebraic quantum field theory which has been used to characterize anyonic excitations in quantum lattice models of 2+1D topological phases of matter. While rigorous mathematical treatments of these lattice models are usually restricted to a planar geometry, it is well-understood that the geometry of the underlying spatial manifold has non-trivial physical implications. For example, the topology of the spatial manifold dictates the number of ground states. At an even more basic level, the manifold provides us with a notion of locality. For example, we may associate each region in the plane with the algebra of quantum operators supported on it. Each inclusion of these regions carries with it an inclusion of algebras, thereby encoding locality in the quantum model by representing a geometric poset structure algebraically. In this talk, we provide a general formalism for superselection theory by axiomatizing the geometric properties which impose locality on nets of von Neumann algebras of observables. This allows for the study of superselection sectors in systems with unconventional geometries and elucidates the key properties of locality from which topological excitations emerge.