$$C^*$$-Categorical Prefactorization Algebras for Superselection Sectors and Topological Order

This paper presents a conceptual and efficient geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice Zn. It is shown that, under the typical assumption of 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 Zn. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder R1×Sn-1. While the sphere Sn-1 arises geometrically as the angular coordinates of cones, the origin of the line R1 is analytic and rooted in Haag duality. The usual braided (for n=2) or symmetric (for n≥3) monoidal C∗-categories of superselection sectors are recovered by removing a point of the sphere R1×(Sn-1\pt)≅Rn and using the equivalence between En-algebras and locally constant prefactorization algebras defined on open disks in Rn. The non-trivial homotopy groups of spheres induce additional algebraic structures on these En-monoidal C∗-categories, which in the case of Z2 is given by a braided monoidal self-equivalence arising geometrically as a kind of ‘holonomy’ around the circle S1. The locally constant prefactorization algebra structures discovered in this work generalize, under some mild geometric conditions, to other discrete spaces and thereby provide a clear link between the geometry of the localization regions and the algebraic structures on the category of superselection sectors.