Symmetry, Symmetry Topological Field Theory and von Neumann Algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918115636412416 |
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| author | Jia, Qiang Tian, Jiahua |
| author_facet | Jia, Qiang Tian, Jiahua |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_17103 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetry, Symmetry Topological Field Theory and von Neumann Algebra Jia, Qiang Tian, Jiahua High Energy Physics - Theory 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. |
| title | Symmetry, Symmetry Topological Field Theory and von Neumann Algebra |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.17103 |