Non-invertible Global Symmetries in the Standard Model

TL;DR

This paper identifies an infinite set of non-invertible generalized global symmetries in massless QED and QCD, realized by topological operators ${\\cal D}_{p/N}$ that combine axial rotations with fractional quantum Hall data. The operators are gauge-invariant and conserved but do not form a group under fusion, instead obeying fusion algebras with 2+1d TQFT coefficients. The construction extends to QCD, where matching UV non-invertible symmetries with IR dynamics requires a $\\pi^0 F\\wedge F$ coupling, offering a symmetry-based reinterpretation of neutral pion decay. The results provide a new lens on helicity conservation and pion physics, and open questions about the full fusion structure, spontaneous symmetry breaking, and non-abelian generalizations in the Standard Model.

Abstract

We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the $U(1)_\text{A}$ axial symmetry because of the ABJ anomaly, for every rational angle $2πp/N$, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic $U(1)$ gauge field. These conserved symmetry operators do not obey a group multiplication law, but a non-invertible fusion algebra over TQFT coefficients. They act invertibly on all local operators as axial rotations, but non-invertibly on the 't Hooft lines. These non-invertible symmetries lead to selection rules, which are consistent with the scattering amplitudes in QED. We further generalize our construction to QCD, and show that the coupling $π^0 F\wedge F$ in the effective pion Lagrangian is necessary to match these non-invertible symmetries in the UV. Therefore, the conventional argument for the neutral pion decay using the ABJ anomaly is now rephrased as a matching condition of a generalized global symmetry.

Figures (2)

• Figure 1: Action of the three-dimensional non-invertible symmetry operator $\mathcal{D}_{\frac{p}{N}}$ on the 't Hooft line $H$. (a) As one sweeps the operator $\mathcal{D}_{\frac{p}{N}}$ past the 't Hooft line $H$, the latter is attached to the topological surface operator $\exp \left( i \frac{p}{N} \int F \right)$ which is stretched between $\mathcal{D}_{\frac{p}{N}}$ and $H$. The whole configuration is gauge-invariant. (b) When the 't Hooft line is supported on a contractible loop $\gamma$ such that $\gamma= \partial \Sigma$ for a two-dimensional disk $\Sigma$, the surface operator $\exp \left( i \frac{p}{N} \int F \right)$ can be deformed to be supported on $\Sigma$. The operator $\exp \left( i \frac{p}{N} \int_\Sigma F \right)$ can be thought of as a fractional Wilson line on $\gamma$.
• Figure 2: Action of the non-invertible Kramers-Wannier duality line $\mathcal{D}$ on the spin operator $\sigma$ in the 1+1d Ising CFT. When $\mathcal{D}$ sweeps past $\sigma$, the latter becomes the disorder operator $\mu$ attached to the topological $\mathbb{Z}_2^{(0)}$ line.