Noninvertible anomalies in SU(N) × U(1) gauge theories

Abstract We study 4-dimensional SU(N) × U(1) gauge theories with a single massless Dirac fermion in the 2-index symmetric/antisymmetric representations and show that they are endowed with a noninvertible 0-form $$ {\overset{\sim }{\mathbb{Z}}}_{2\left(N\pm 2\right)}^{\upchi} $$ ℤ ~ 2 N ± 2 χ chiral symmetry along with a 1-form $$ {\mathbb{Z}}_N^{(1)} $$ ℤ N 1 center symmetry. By using the Hamiltonian formalism and putting the theory on a spatial three-torus $$ \mathbbm{T} $$ T 3, we construct the non-unitary gauge invariant operator corresponding to $$ {\overset{\sim }{\mathbb{Z}}}_{2\left(N\pm 2\right)}^{\upchi} $$ ℤ ~ 2 N ± 2 χ and find that it acts nontrivially in sectors of the Hilbert space characterized by selected magnetic fluxes. When we subject $$ \mathbbm{T} $$ T 3 to $$ {\mathbb{Z}}_N^{(1)} $$ ℤ N 1 twists, for N even, in selected magnetic flux sectors, the algebra of $$ {\overset{\sim }{\mathbb{Z}}}_{2\left(N\pm 2\right)}^{\upchi} $$ ℤ ~ 2 N ± 2 χ and $$ {\mathbb{Z}}_N^{(1)} $$ ℤ N 1 fails to commute by a ℤ2 phase. We interpret this noncommutativity as a mixed anomaly between the noninvertible and the 1-form symmetries. The anomaly implies that all states in the torus Hilbert space with the selected magnetic fluxes exhibit a two-fold degeneracy for arbitrary $$ \mathbbm{T} $$ T 3 size. The degenerate states are labeled by discrete electric fluxes and are characterized by nonzero expectation values of condensates. In an appendix, we also discuss how to construct the corresponding noninvertible defect via the “half-space gauging” of a discrete one-form magnetic symmetry.