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.2 By a trivial ground state we mean that the system is gapped and ground state is non-degenerate, while the nontrivial ground state is either gapless, breaks some global symmetry or has topological degeneracy arXiv:1805.11423v2 [cond-mat.str-el]

Anomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known exception is an 't Hooft anomaly involving one-form symmetries as in pure SU(N ) Yang-Mills theory at θ = π. Recent development about large-N volume independence, however, gives us a circumstantial evidence that 't Hooft anomalies can also remain under circle compactifications in some theories without one-form symmetries. We develop a systematic procedure for deriving an 't Hooft anomaly of the circle-compactified theory starting from the anomaly of the original uncompactified theory without one-form symmetries, where the twisted boundary condition for the compactified direction plays a pivotal role. As an application, we consider Z N -twisted CP N −1 sigma model and massless Z N -QCD, and compute their anomalies explicitly.

We consider the SU (N ) Yang-Mills theory, whose topological sectors are restricted to the instanton number with integer multiples of p. We can formulate such a quantum field theory maintaining locality and unitarity, and the model contains both 2πperiodic scalar and 3-form gauge fields. This can be interpreted as coupling a topological theory to Yang-Mills theory, so the local dynamics becomes identical with that of pure Yang-Mills theory. The theory has not only Z N 1-form symmetry but also Z p 3-form symmetry, and we study the global nature of this theory from the recent 't Hooft anomaly matching. The computation of 't Hooft anomaly incorporates an intriguing higher-group structure. We also carefully examine that how such kinematical constraint is realized in the dynamics by using the large-N and also the reliable semiclassics on R 3 × S 1 , and we find that the topological susceptibility plays a role of the order parameter for the Z p 3-form symmetry. Introducing a fermion in the fundamental or adjoint representation, we find that the chiral symmetry becomes larger than the usual case by Z p , and it leads to the extra p vacua by discrete chiral symmetry breaking. No dynamical domain wall can interpolate those extra vacua since such objects must be charged under the 3-form symmetry in order to match the 't Hooft anomaly.