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 Z n . It is shown that, under the typical assumption of Haag duality, the monoidal C^* C ∗ -categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of Zⁿ 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 R¹× Sⁿ⁻¹ R 1 × S n - 1 . While the sphere Sⁿ⁻¹ S n - 1 arises geometrically as the angular coordinates of cones, the origin of the line R¹ R 1 is analytic and rooted in Haag duality. The usual braided (for $$n=2$$ n = 2 ) or symmetric (for n≥ 3 n ≥ 3 ) monoidal C^* C ∗ -categories of superselection sectors are recovered by removing a point of the sphere R¹× (Sⁿ⁻¹ pt) Rⁿ R 1 × ( S n - 1 \ pt ) ≅ R n and using the equivalence between Eₙ E n -algebras and locally constant prefactorization algebras defined on open disks in Rⁿ <mml:math xmlns: