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 $\mathcal{T}_\mathcal{F}$ 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.