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

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. $\mathcal{N}=1$ 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.

32 pages, 6 figures; v2: added references and minor comments