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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, aglobal symmetry group of such systems has a 't Hooft anomaly, which manifestsitself as an inability to gauge the global symmetry group. By applying anomalymatching, the ground-state properties can be severely constrained. The anomalymatching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spinchain from which the field theory descends, and it forbids a trivially gappedground state for particular $\theta$-angles. We generalize the statement of theLSM theorem and show that 't Hooft anomalies persist even under perturbationswhich break the spin-symmetry down to the discrete subgroup $\mathbbZ_3\times\mathbb Z_3\subset SU(3)/\mathbb Z_3$. In addition the model canfurther be constrained by applying global inconsistency matching, whichindicates the presence of a phase transition between different regions of$\theta$-angles. We use these constraints to give possible scenarios of thephase diagram. We also argue that at the special points of the phase diagramthe anomalies are matched by the $SU(3)$ Wess-Zumino-Witten model. Wegeneralize the discussion to the $SU(N)/U(1)^{N-1}$ nonlinear sigma models aswell as the 't Hooft anomaly of the $SU(N)$ Wess-Zumino-Witten model, and showthat they match. Finally the $(2+1)$-dimensional extension is consideredbriefly, and we show that it has various 't Hooft anomalies leading tonontrivial consequences.

Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1)^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations