Extend the lattice analysis to fermionic and non-Abelian gauge theories

Develop a lattice gauge theory framework that extends the modified Villain-type formulation with a periodic scalar field (the QED axion) used to study the axial U(1) non-invertible symmetry to systems that (i) contain dynamical fermions and (ii) possess non-Abelian gauge symmetry.

Background

The paper analyzes how the symmetry operator of the axial U(1) non-invertible symmetry acts on the ’t Hooft line operator using a modified Villain-type lattice formulation. To avoid difficulties associated with introducing ’t Hooft lines in lattice theories with fermionic axial anomalies, the authors model the anomaly with a compact scalar (“QED axion”). Within this bosonic setup, they construct gauge-invariant, dressed ’t Hooft and axion string operators and derive an anomalous Ward–Takahashi identity.

They find that the symmetry operator leaves no effect when sweeping out a ’t Hooft line, a conclusion that appears inequivalent to continuum-theory expectations. However, extending this analysis beyond the bosonic axion model to include dynamical fermions and to non-Abelian gauge groups remains unresolved. Standard lattice controls of the axial anomaly for fermions rely on admissibility conditions that forbid monopole worldlines, complicating the inclusion of ’t Hooft operators; hence a general framework that incorporates fermions and non-Abelian gauge symmetry is still lacking.