Non-invertible Global Symmetries in the Standard Model

Understand

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 $2\pi p/N$, 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 non-invertible fusion algebra over TQFT coefficients.