When $\mathbb{Z}_2$ one-form symmetry leads to non-invertible axial symmetries

TL;DR

This paper investigates how a $Z_2$ one-form symmetry in certain non-abelian gauge theories leads to non-invertible symmetries through mixed 't Hooft anomalies, revealing a universal TQFT structure and a rank-dependent pattern.

Contribution

It demonstrates that non-trivial mixed anomalies induce non-invertible symmetries with a universal $U(1)_2$ CS TQFT dressing, depending on the gauge group's rank.

Findings

Non-invertible symmetries arise from non-trivial mixed anomalies.

Universal $U(1)_2$ CS TQFT dresses the non-invertible defects.

Presence of non-invertible defects depends on gauge group rank.

Abstract

We study non-abelian gauge theories with fermions in a representation such that the surviving electric 1-form symmetry is $\mathbb{Z}_2$. This includes $SU(N)$ gauge theories with matter in the (anti)symmetric and $N$ even, and $USp(2N)$ with a Weyl fermion in the adjoint, i.e. ${\cal N}=1$ SYM. We study the mixed 't Hooft anomaly between the discrete axial symmetry and the 1-form symmetry and show that when it is non-trivial, it leads to non-invertible symmetries upon gauging the $\mathbb{Z}_2$. The TQFT dressing the non-invertible symmetry defects is universal to all the cases we study, namely it is always a $U(1)_2$ CS theory coupled to the $\mathbb{Z}_2$ 2-form gauge field. We uncover a pattern where the presence or not of non-invertible defects depends on the rank of the gauge group.