$C^{\ast }$-categorical prefactorization algebras for superselection sectors and topological order

Abstract

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 ${\mathbb{Z}}^n$. It is shown that, under the typical assumption of Haag duality, the monoidal $C^{\ast }$-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of ${\mathbb{Z}}^n$. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder ${\mathbb{R}}^1\times {\mathbb{S}}^{n-1}$. While the sphere ${\mathbb{S}}^{n-1}$ arises geometrically as the angular coordinates of cones, the origin of the line ${\mathbb{R}}^1$ is analytic and rooted in Haag duality. The usual braided (for $n=2$) or symmetric (for $n\ge 3$) monoidal $C^{\ast }$-categories of superselection sectors are recovered by removing a point of the sphere ${\mathbb{R}}^1\times ({\mathbb{S}}^{n-1}\setminus \mathrm{pt})\cong {\mathbb{R}}^n$ and using the equivalence between ${\mathbb{E}}_n$-algebras and locally constant prefactorization algebras defined on open disks in ${\mathbb{R}}^n$. The non-trivial homotopy groups of spheres induce additional algebraic structures on these ${\mathbb{E}}_n$-monoidal $C^{\ast }$-categories, which in the case of ${\mathbb{Z}}^2$ is given by a braided monoidal self-equivalence arising geometrically as a kind of `holonomy' around the circle ${\mathbb{S}}^1$. 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.

v3: 41 pages. Final version to appear in Communications in Mathematical Physics

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• Localized endomorphisms in Kitaev's toric code on the plane
• Quantum charges and spacetime topology: The emergence of new superselection sectors
• A derivation of braided $C^{\ast }$-tensor categories from gapped ground states satisfying the approximate Haag duality