  We discuss the $SU(3)/[U(1)\times U(1)]$ nonlinear sigma model in 1+1D and,
more broadly, its linearized counterparts. Such theories can be expressed as
$U(1)\times U(1)$ gauge theories and therefore allow for two topological
$\theta$-angles. These models provide a field theoretic description of the
$SU(3)$ chains. We show that, for particular values of $\theta$-angles, a
global symmetry group of such systems has a 't Hooft anomaly, which manifests
itself as an inability to gauge the global symmetry group. By applying anomaly
matching, the ground-state properties can be severely constrained. The anomaly
matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin
chain from which the field theory descends, and it forbids a trivially gapped
ground state for particular $\theta$-angles. We generalize the statement of the
LSM theorem and show that 't Hooft anomalies persist even under perturbations
which break the spin-symmetry down to the discrete subgroup $\mathbb
Z_3\times\mathbb Z_3\subset SU(3)/\mathbb Z_3$. In addition the model can
further be constrained by applying global inconsistency matching, which
indicates the presence of a phase transition between different regions of
$\theta$-angles. We use these constraints to give possible scenarios of the
phase diagram. We also argue that at the special points of the phase diagram
the anomalies are matched by the $SU(3)$ Wess-Zumino-Witten model. We
generalize the discussion to the $SU(N)/U(1)^{N-1}$ nonlinear sigma models as
well as the 't Hooft anomaly of the $SU(N)$ Wess-Zumino-Witten model, and show
that they match. Finally the $(2+1)$-dimensional extension is considered
briefly, and we show that it has various 't Hooft anomalies leading to
nontrivial consequences.
