Non-Invertible Chiral Symmetry and Exponential Hierarchies

TL;DR

The paper extends the notion of global symmetries in quantum field theory by introducing non-invertible chiral symmetry defects that arise when abelian ABJ anomalies are present. These defects, realized as domain walls fused with 3d TQFTs and coupled to bulk magnetic one-form symmetry, impose exact selection rules on correlators and transform the conventional chiral symmetry into a non-invertible topological structure. The authors demonstrate the construction in massless QED, axion electrodynamics, non-abelian theories, and axion-Yang-Mills, and analyze how monopole-induced breaking of the magnetic one-form symmetry non-perturbatively relaxes these rules, producing technically natural exponential hierarchies in axion potentials and fermion masses. They further connect monopole physics to the emergence and controlled breaking of these non-invertible symmetries, offering a framework for naturally tiny mass scales and potential applications in model building and naturalness. Overall, the work provides a non-perturbative, symmetry-based mechanism to generate exponential hierarchies via monopole-induced violations of non-invertible chiral defects.

Abstract

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.

Figures (2)

• Figure 1: The action of the non-invertible symmetry $\mathcal{D}_k$ on the correlation function on $S^4$ with local operator insertions with charge $q_1,q_2$, and $q_3$. The rotation angle $\theta$ of the symmetry is $\frac{2\pi}{k}$. At the last step the defect is contracted at the other side of the sphere.
• Figure 2: A non-invertible symmetry transforms a line operator $T$ on a line $\gamma$ into a line operator attached to a surface operator.