Symmetry, symmetry topological field theory and von Neumann algebra

We study the additivity and Haag duality of the von Neumann algebra of a quantum field theory with 0-form (and the dual (d − 2)-form) (non)-invertible global symmetry. We analyze the symmetric (uncharged) sector von Neumann algebra of with the inclusion of bi-local and bi-twist operators in it. We establish the connection between the existence of these non-local operators in and certain properties of the Lagrangian algebra of the extended operators in the corresponding symmetry topological field theory (SymTFT). We prove that additivity or Haag duality of the symmetric sector von Neumann algebra is violated when satisfies specific criteria, thus generalizing the result of Shao, Sorce and Srivastava to arbitrary dimensions. We further demonstrate the SymTFT construction via concrete examples in two dimensions.