  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.
