C*-categorical prefactorization algebras for superselection sectors and topological order

I will present a geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice $Z^n$. I will show that, under certain assumptions which are implied by Haag duality, the monoidal C*-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of $Z^n$. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder $R^1 \times S^{n-1}$. While the sphere $S^{n-1}$ arises geometrically as the angular coordinates of cones, the origin of the line $R^1$ is analytic and rooted in Haag duality. The usual braided (for $n=2$) or symmetric (for $n>2$) monoidal C*-categories of superselection sectors are recovered by removing a point of the sphere and using the equivalence between $E_n$-algebras and locally constant prefactorization algebras defined on open disks in $R^n$. The non-trivial homotopy groups of spheres induce additional algebraic structures on these $E_n$-monoidal C*-categories, which in the simplest case of $Z^2$ is given by a braided monoidal self-equivalence arising geometrically as a kind of `holonomy' around the circle $S^1$.

This talk is based on joint work with Marco Benini, Victor Carmona and Pieter Naaijkens.