Noninvertible Global Symmetries in the Standard Model

We identify infinitely many noninvertible 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 $\mathrm{U}(1{)}_{\mathrm{A}}$ axial symmetry because of the Adler-Bell-Jackiw anomaly, for every rational angle $2\pi p/N$, 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 noninvertible fusion algebra. They act invertibly on all local operators as axial rotations, but noninvertibly on the ’t Hooft lines. We further generalize our construction to QCD, and show that the coupling ${\pi }^{0}F\wedge F$ in the effective pion Lagrangian is necessary to match these noninvertible 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.