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

Abstract

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]

Modified instanton sum in QCD and higher-groups

“…This is indeed the slight generalization of the anomaly of CP N −1 model computed in Ref. [24] for the charge-p model and by restricting the P SU (N ) symmetry to Z N × Z N [34]. This explains why 2d charge-p CP N −1 model studied in Refs.…”

Undoing decomposition

“…In passing, it is also worth mentioning that gauging the flavor symmetry in these models can also be of interest, see [41, section 2.4] and [42][43][44].…”

Generalized ’t Hooft anomalies on non-spin manifolds

“…In this Section, we consider two examples of vector-like theories and check whether putting them on non-spin manifolds and turning on the most general background fluxes imposes 10 For the sake of completeness, we recall that the conditions (3.7) are replaced on T 4 by:…”

“…The study of these generalized anomalies is a currently active area of research with contributions coming from the high-energy, condensed matter, and mathematical communities. We do not claim to be in command of all points of view and only give a list of references written from a (largely) high-energy physics perspective and pertaining to theories somewhat similar to the ones discussed in this paper [7][8][9][10][11][12][13][14][15][16][17][18][19].…”