Noninvertible 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 noninvertible topological global symmetry defects with world volume anyon degrees of freedom that couple to the bulk through a magnetic 1-form global symmetry as in the fractional Hall effect. These noninvertible 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 1-form symmetry is broken by the propagation of dynamical magnetic monopoles, the selection rules of the noninvertible chiral symmetry defects are violated nonperturbatively. This leads to technically natural exponential hierarchies in axion potentials and fermion masses.