Symmetry, symmetry topological field theory and von Neumann algebra

A bstract We study the additivity and Haag duality of the von Neumann algebra of a quantum field theory $${\mathcal{T}}_{\mathcal{F}}$$ with 0-form (and the dual ( d − 2)-form) (non)-invertible global symmetry $$\mathcal{F}$$ . We analyze the symmetric (uncharged) sector von Neumann algebra of $${\mathcal{T}}_{\mathcal{F}}$$ 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 $${\mathcal{T}}_{\mathcal{F}}$$ and certain properties of the Lagrangian algebra $$\mathcal{L}$$ 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 $$\mathcal{L}$$ 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.