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Explore a geometric framework for encoding algebraic structures on superselection sectors in algebraic quantum field theory through this conference talk from Harvard CMSA's Workshop on Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras. Discover how monoidal C*-categories of localized superselection sectors carry the structure of locally constant prefactorization algebras over cone-shaped subsets of the n-dimensional lattice Z^n under assumptions implied by Haag duality. Learn about the extraction of underlying locally constant prefactorization algebras defined on open disks in the cylinder R^1 x S^{n-1} using higher algebra techniques, where the sphere S^{n-1} represents angular coordinates of cones and the line R^1 has analytic origins rooted in Haag duality. Understand how braided (for n=2) or symmetric (for n>2) monoidal C*-categories of superselection sectors are recovered by removing a point from the sphere and utilizing the equivalence between E_n-algebras and locally constant prefactorization algebras on open disks in R^n. Examine how non-trivial homotopy groups of spheres induce additional algebraic structures on E_n-monoidal C*-categories, with the Z^2 case featuring a braided monoidal self-equivalence arising as geometric 'holonomy' around the circle S^1. The presentation draws from collaborative research with Marco Benini, Victor Carmona, and Pieter Naaijkens, offering insights into the intersection of quantum field theory, topological order, and operator algebras.
