Noninvertible anomalies in $SU(N)\times U(1)$ gauge theories

We study $4$-dimensional $SU(N)\times U(1)$ 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 $\widetilde {\mathbb Z}_{2(N\pm 2)}^{\chi}$ chiral symmetry along with a $1$-form $\mathbb Z_N^{(1)}$ center symmetry. By using the Hamiltonian formalism and putting the theory on a spatial three-torus $\mathbb T^3$, we construct the non-unitary gauge invariant operator corresponding to $\widetilde {\mathbb Z}_{2(N\pm 2)}^{\chi}$ and find that it acts nontrivially in sectors of the Hilbert space characterized by selected magnetic fluxes. When we subject $\mathbb T^3$ to $\mathbb Z_N^{(1)}$ twists, for $N$ even, in selected magnetic flux sectors, the algebra of $\widetilde {\mathbb Z}_{2(N\pm 2)}^{\chi}$ and $\mathbb Z_N^{(1)}$ fails to commute by a $\mathbb Z_2$ 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 $\mathbb T^3$ size. The degenerate states are labeled by discrete electric fluxes and are characterized by nonzero expectation values of condensates.
Comment: 18 pages