Additivity, Haag duality, and non-invertible symmetries

TL;DR

The paper relates algebraic quantum field theory via local nets to non-invertible global symmetries in 1+1D, showing that restricting to symmetry sectors tends to violate additivity when invertible elements are present and Haag duality when non-invertible elements are present. Using the Ising CFT with Verlinde lines and diagonal RCFTs, it derives precise criteria tied to the fusion category structure (invertible vs non-invertible lines) and corroborates these findings with lattice models that host non-invertible defects. The results illuminate how global symmetry content imprints on local algebras, connecting modular data, bulk TQFT perspectives, and region-based algebraic properties, with implications for modular invariance and generalized symmetries in QFT. They also outline lattice realizations (KW and Rep(D8)) where the predicted violations manifest, and propose avenues for extending the framework to higher-form and spacetime/crystalline symmetries.

Abstract

The algebraic approach to quantum field theory focuses on the properties of local algebras, whereas the study of (possibly non-invertible) global symmetries emphasizes global aspects of the theory and spacetime. We study connections between these two perspectives by examining how either of two core algebraic properties -- "additivity" or "Haag duality" -- is violated in a 1+1D CFT or lattice model restricted to the symmetric sector of a general global symmetry. For the Verlinde symmetry of a bosonic diagonal RCFT, we find that additivity is violated whenever the symmetry algebra contains an invertible element, while Haag duality is violated whenever it contains a non-invertible element. We find similar phenomena for the Kramers-Wannier and Rep(D$_8$) non-invertible symmetries on spin chains.

Figures (12)

• Figure 1: Additivity is violated by a pair of $\mathbb{Z}_2$-odd operators.
• Figure 2: Using the relation $U=\prod_{\ell=1}^L X_\ell=1$, we can rewrite the disorder operator $U(\ell_1,\ell_2)$ in a presentation so that it is supported in $\mathcal{R}'$.
• Figure 3: The disorder operator $U(\ell_1,\ell_2)$ does not commute with a pair of $Z_\ell$'s in $\mathcal{R}'$.
• Figure 4: The commutation relation between the invertible $\mathbb{Z}_2$ topological line operator ${\cal L}_\epsilon$ (dashed line), the non-invertible Kramers-Wannier line operator ${\cal L}_\sigma$ (red line), and the local primaries $\epsilon$ and $\sigma$Frohlich:2004ef.
• Figure 5: The symmetric sector with respect to the full Ising fusion category in the Ising CFT violates both additivity and Haag duality. Left: A pair of the energy operators $\epsilon$ (with $(h,\bar{h})=(\frac{1}{2},\frac{1}{2})$) violates additivity. Right: A pair of left-moving free fermions $\psi$ (with $(h,\bar{h})=(\frac{1}{2},0)$) connected by a $\mathbb{Z}_2$ line ${\cal L}_\epsilon$ (shown in dashed line) violates Haag duality.