Infrared properties of two-dimensional SU(N)/H nonlinear $\sigma$ models at nonzero $\theta$ angles

1. General approach to nonlinear sigma model with topological terms using the perturbed Wess-Zumino-Novikov-Witten (WZNW) model to identify massive or massless flows.
2. Symmetry based approach to identify relevant and marginal operators.
3. Explicit constructions of the perturbing operators using operator product expansions.

Necessity to rely on path integrals and semiclassical approximation to establish the connection between nonlinear sigma model and WZNW model.

The article considers nonlinear sigma models SU(N)/H with non-trivial topological terms with H=U(1)^N (flag manifold) H=SO(N) and in the last case, the Grassmanian, H=SU(N/2)xSU(N/2) with even N. The general strategy is to obtain the nonlinear sigma model with its topological term by considering the classical action of a SU(N) WZNW perturbed by operators of appropriate symmetry. When the perturbation is strong, the matrices in the WZNW action have to be taken in a restricted subset of SU(N) and the WZNW action reduces to the one of the nonlinear sigma model with topological terms. In a second step, the perturbation is assumed weak, and operator product expansions of WZNW primary operators
are used to identify the perturbing operators from the action. A renormalization group flow is obtained and used to identify the ground state of the perturbed WZNW model. In the case of the flag manifold and SU(N)/O(N) the flow is massless. This might be a weakness of the paper: in order to identify the WZNW and the nonlinear sigma model, one needs to consider relatively strong perturbations to contrain the WZNW matrix field. But while nonlinear sigma models flow to a strong coupling regime that must be non-trivial because of the topological term, the perturbed WZNW model flows to weak coupling. So one needs to assume some continuity in the perturbed WZNW model between strong enough coupling where the mapping to nonlinear sigma model is applicable, and the weak coupling regime where the one-loop RG shows that the model flows to weak coupling. In the case of the Grassmanian or CP^{N-1} models, relevant operators are present, and no such concern exists.
Overall, I have found the paper clearly written, and its approach, as suggested in the conclusion, could be applicable to other (1+1)-dimensional nonlinear sigma models with Lie group symmetry. I thus support publication in SciPost.
There is however in Sec. 4.1 on p. 16 a paragraph starting "For bosonic QFTs, this can be clarified by the SO-bordism group..." and ending "the effective Lagrangian on the classical moduli becomes" that I have found quite impenetrable. A few references to the mathematical litterature to precise the notations (for instance are H2 and H3 homology or cohomology groups ? is there an explicit integral form for the Stiefel-Whitney class of p. 49 ? where have SO-bordism or spin bordism been defined ? where is the Atiyah-Hirzebruth spectral sequence defined ?)

Add some references to defined SO-bordism, integral homology, Stiefel-Whitney class, Spin-bordism, Atiyah-Hirzebuth spectral sequence in the paragraph p. 16. Indicate the meaning of the group notation H_2 and H_3.

Publish (meets expectations and criteria for this Journal)

1- Important topic: studying the phase structure of nonlinear sigma models with topological terms
2- Interesting method: relating classical/quantum perturbations of WZNW conformal field theory
3- Clearly written; details of calculations are provided
4- Scope of applicability is discussed, together with possible applications in realistic systems

Overall it is not fully clear why the predictions of
1) a classical model in the strong coupling limit
should match those of
2) a quantum model (perturbed CFT) in the weak coupling limit.
However, I believe this is a standard extrapolation in the field and the critique does not diminish the interesting observations of the present paper. Besides, the authors explicitly acknowledge this in the conclusion.
In this respect it would indeed be interesting to apply the technique of the present paper to sigma models not covered in the literature, such as the ones based on SO and Sp groups (as mentioned in the conclusion) making some predictions for those.

The authors study the IR behavior of sigma models with homogeneous target spaces in the presence of topological terms. This is clearly a fascinating topic, which provides a window into non-perturbative phenomena in quantum field theories and is also closely related to spin chain physics that nowadays may be simulated numerically or even experimentally.
Some hints into the IR behavior of sigma models can be acquired via their relation to the continuum limits of spin chains (for the latter exact or numerical results are often available). A pure field-theoretic approach relies on the so-called discrete 't Hooft anomalies: predictions based on the latter seem to match the spin chain results in the known cases. Whenever such anomalies are present, it is believed that the sigma models cannot be 'trivially' gapped in the IR, meaning that they either flow to some CFT or have a degenerate ground state.
The approach of the present paper deals precisely with this situation when the IR behavior is assumed nontrivial. The idea is to start from a conformal WZNW model (based on a group G) in the UV and perturb it, while preserving the important symmetries. At the classical level the added perturbation confines (at low energies) the fields to the submanifold of minima of the potential, which is a homogeneous space G/H that emerges as the target space of an effective sigma model. On the other hand, if quantum mechanically (at one loop) the perturbation is irrelevant, one believes that RG flow will not deviate substantially from the original WZNW model, so that the IR properties of the sigma model should be governed by the CFT. The paper studies various examples of this phenomenon in concrete sigma models.

p.14 line 4: says 'N-1 massless compact bosons' but the numbering is m=0, 1, ... , N-1 indicating there are N bosons
p.14, last paragraph: the authors discuss what would happen if one flips the sign of all \lambda's. However what would happen if one flips the sign of just a few \lambda's? From (41) it seems that the effective coupling won't change much, whereas the classical minima of (23) might be altered substantially. Could you comment on this?
p. 15 and later on: the notation SU(N)/USp(N) seems somewhat bizarre. I would suggest writing explicitly SU(2M)/USp(2M) since only even values of N (=2M) are allowed here
p.15: I believe (45) is the Cartan embedding of SU(N)/SO(N) into SU(N). The Cartan involution is simply complex conjugation in this case, and the stability subgroup is therefore SO(N). Perhaps worth mentioning this.
p.16, before (51): `we can prove that there is a global section using that Spin(N) is simply connected'. I think what you have in mind here is that the bundle is trivial; this seems like a known fact and it would be worth providing a math reference.
p.21, (75): Could you give a proof that these are two disconnected components? To me it seems that the components should be distinguished by the sign of the Pfaffian of J rather than \pm J.
p.21, (73)-(75): Could you also insert some comment on what happens when N is odd?
p.23: Why is (79) the manifold of minima of the potential in (78)? As footnote 5 on p.25 indicates, the corresponding analysis for the case of CP^(N-1) seems rather involved, so why is it simpler here?
p.25: (84) and footnote 5: the potential in (84) seems rather simple, so what is the main obstacle in proving (85)-(87)? Perhaps could you add an Appendix with an explicit study of low-N cases (N=3,4)?

Ask for minor revision

Dear Editor,

I have read through the paper by P. Lecheminant, Y. Tanizaki and K. Totsuka submitted for publication to SciPost. In it the authors consider a broad class of sigma models with $\theta$ terms, where the target space is a coset of the group ${\rm SU}(N)$. The focus of the study is the physically interesting IR regime of the QFTs. In particular, the authors attempt to answer the question of whether the systems are gapless or they acquire a gap in the IR limit using a novel approach.

The method proposed in the submitted manuscript, as I see it, can be summarized as follows. At the first step the authors consider a different model than the one of interest, namely, the ${\rm SU}(N)$ WZNW model at level $k=1$ with an additional potential term proportional to $\lambda$. This term is specifically chosen so that at $\lambda=\infty$ the theory is reduced to the $\sigma$ model of interest. The next step was very confusing to me. The parameter $\lambda$ is taken to be small and it is verified whether or not the potential, treated as a perturbation of the ${\rm SU}(N)$ WZNW model, is (marginally) relevant or irrelevant. Based on the results the authors deduce whether the IR limit of the $\sigma$ model corresponding to the point $\lambda=\infty$ is gapless or not. While the computations
themselves are convincing, it is difficult for me to accept that a lowest order perturbation theory analysis near $\lambda=0$ can be used to infer the behaviour of the model at $\lambda=\infty$.

One finds many examples of theories in QFT and statistical mechanics, where the IR behaviour changes depending on the domain of the parameters. The simplest case is perhaps the Heisenberg XXZ spin-$\frac{1}{2}$ chain, which is gapless when the anisotropy parameter $\Delta$ is near zero and possesses a gap as $|\Delta|> 1$. In my reading of the paper, I never found a plausible explanation for why the IR behaviour of the ${\rm SU}(N)$ WZNW model with potential term should be the same near $\lambda=0$ and at $\lambda=\infty$.

Connected with the above is that sometimes the wording of the paper is not sufficiently precise at key points, which makes the arguments more difficult to follow. For instance, in the abstract it is written that: "$\ldots$ we realize these nonlinear $\sigma$ models through perturbations added to the ${\rm SU}(N)_1$ conformal field theory." Strictly speaking, I would say this is not true. The sigma models appear in the limit when the potential added to the ${\rm SU}(N)$ WZNW model becomes infinitely large, and hence can not be considered as a perturbation.

In my opinion, the main argument put forward in the paper to determine whether or not a given $\sigma$ model is gapped or gapless in the IR limit is incomplete. As such, although it turns out to reproduce many of the results in the literature, it has uncertain scientific value. Since the approach forms the centrepiece of the submitted manuscript I believe that, in its present form, the paper is not suitable for publication.

Reject

The main strength of the paper is the simplicity and consistency of its approach. It is also very well written.

The authors consider several important nonlinear sigma models with topological terms and suggest a method how to determine whether the ground state is critical. The sigma models in question are treated as perturbed SU_1(N) WZNW models. The sigma model descriptions emerge at strong coupling. In the weak coupling limit the perturbations almost always are reduced to marginal current-current ones and depending on the sign of the couplings may scale either to strong and weak coupling that is back to SU_1(N) critical point. The derivation is greatly simplified by the fact that SU_1(N) model can be treated by Abelian bosonization. The later fact adds to simplicity and elegance of the approach.

I do not see any weaknesses worth discussing.

I greatly enjoyed reading the paper. I think it meets all criteria.

Publish (surpasses expectations and criteria for this Journal; among top 10%)