The Non-Invertible Axial Symmetry in QED Comes Full Circle

TL;DR

This paper explores the construction of non-invertible topological defects for axial symmetry in massless QED, extending the understanding of symmetry structures beyond traditional invertible cases using topological quantum field theories.

Contribution

It introduces a novel approach to describe non-invertible axial symmetry defects in QED, including irrational angles, and provides a comprehensive symmetry TFT framework.

Findings

Constructed non-invertible defects for all rational and irrational angles.

Developed a symmetry TFT that captures the action of these defects.

Discussed techniques for studying condensation defects for a global symmetry.

Abstract

We revisit the possibility of constructing non-invertible topological defects for the axial symmetry of massless QED, despite its ABJ anomaly. Dressing the defects with a topological quantum field theory with mixed $U(1)$ and $\mathbb{R}$-valued gauge fields, we are able to describe axial rotations of any rational or irrational angle. We confront our results with the existing proposals, in particular those that concern rational angles. We also provide the Symmetry TFT that reproduces the action of all such symmetry defects of QED. Finally, we discuss how similar techniques allow the study of condensation defects for a $\mathbb{R}$ global symmetry.