Non-Invertible Chiral Symmetry and Exponential Hierarchies

We elucidate the fate of classical symmetries which suffer from abelian Adler-Bell-Jackiw anomalies. Instead of being completely destroyed, these symmetries survive as non-invertible topological global symmetry defects with worldvolume anyon degrees of freedom that couple to the bulk through a magnetic one-form global symmetry as in the fractional hall effect. These non-invertible chiral symmetries imply selection rules on correlation functions and arise in familiar models of massless quantum electrodynamics and models of axions (as well as their non-abelian generalizations). When the associated bulk magnetic one-form symmetry is broken by the propagation of dynamical magnetic monopoles, the selection rules of the non-invertible chiral symmetry defects are violated non-perturbatively. This leads to technically natural exponential hierarchies in axion potentials and fermion masses.