Symmetry, Symmetry Topological Field Theory and von Neumann Algebra

TL;DR

This paper investigates the properties of von Neumann algebras in quantum field theories with global symmetries, linking non-local operators and topological field theory structures, and demonstrates these concepts with 2D examples.

Contribution

It generalizes previous results on additivity and Haag duality violations to arbitrary dimensions, connecting algebraic properties with symmetry topological field theories.

Findings

Additivity and Haag duality can be violated under certain algebraic conditions.

The connection between non-local operators and symmetry topological field theories is established.

Concrete 2D examples illustrate the theoretical framework.

Abstract

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.