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Anomaly and global inconsistency matching: θ angles, SU(3)/U(1)2 nonlinear sigma model, SU(3) chains, and generalizations

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We discuss the SU(3)/[U(1)×U(1)] nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as U(1)×U(1) gauge theories and therefore allow for two topological θ angles. These models provide a field theoretic description of the SU(3) chains. We show that, for particular values of θ 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 θ 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 Z3×Z3⊂SU(3)/Z3. 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 θ 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.