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

Dear Editor,

We have improved our manuscript thanks to the referees' suggestions. We hope that our manuscript can now be published in SciPost Phys. Thank you in advance for your considerations.

Sincerely,

P. Lecheminant, Y. Tanizaki, and K. Totsuka

Reply to the Referee 1:

We thank the referee for his/her very positive judgment about our work and his/her recommendation for publication.

Reply to the Referee 2:

We thank the referee for taking his/her time to read our paper. We, however, respectfully disagree with his/her opinion, and the main critics presented in his/her report that we now answer.

Let us begin with a formal issue. The referee says: "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: " ... we realize these nonlinear $\sigma$ models through perturbations added to the 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 SU($N$) WZNW model becomes infinitely large, and hence can not be considered as a perturbation."

Reply: We used "perturbation" or "perturbed CFT" in the commonly used terminology in CFT that we deviate the system from a fixed point by adding a certain operator whose coupling constant is not necessarily assumed infinitesimal. The concept of "perturbed CFT" or "deformation of CFT" has been introduced in the late eighties when A. B. Zamolodchikov proposed a program to understand the global phase structure of 2d field theories by adding "perturbations" to fixed-point CFTs. For instance, if we take an appropriate perturbation, the resulting theory is integrable and well-defined even for finite values of the coupling constant (we can use the truncated conformal space approach in the absence of integrability). In the revised version of the paper, we have adopted another commonly used terminology "deformation(s)" or "deformed WZNW CFT" to emphasize that the coupling constant is not necessarily small, addressing the referee's criticism.

The referee says: "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 SU($N$) WZNW model, ... 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 $\lambda = \infty$. ... I never found a plausible explanation for why the IR behaviour of the SU($N$) WZNW model with potential term should be the same near $\lambda = 0$ and at $\lambda = \infty$."

Reply: First of all, let us emphasize that we never claimed that our strategy is fully justified with mathematical rigor. It is just a prescription aimed to explaining the infrared behaviors of different nonlinear $\sigma$ models at nonzero topological angles in a consistent manner. For example, we explicitly stated in the conclusion that "while we do not have a rigrorous proof that this prescription always yields the correct answer, ...".

We tried our best to explain why our prescription is legitimate to understand the massless RG flows for the cases of the complete flag manifold and SU($N$)/SO($N$). Suppose we add a potential term which leads (in the classical sense) to the desired $\sigma$ model at strong coupling. The key point is the presence of sufficiently large UV symmetry in the deformed WZNW model (and the targeted $\sigma$ model), in particular the discrete chiral symmetry [denoted by $Z_{N})_{L}$ in our paper], which prevents the occurrence of any strongly-relevant perturbations around the SU($N$)$_1$ WZNW CFT fixed point. No intermediate fixed point can be reached from the UV SU($N$)$_1$ fixed point via a non-perturbative renormalization group flow since the non-Abelian symmetry of the model will constrain it to the SU($N$)$_k$ CFT, which is ruled out by the $c$-theorem when $k>1$. Also, an RG flow from the UV $\sigma$ models to the SU($N$)$_1$ WZNW fixed point is kinematically allowed by the anomaly-matching argument.

To establish that we do have such a massless RG flow, we use the method pioneered by Affleck and Haldane. The only symmetry-allowed "perturbation" around the SU($N$)$_1$ fixed point that may affect the IR physics is the SU($N$)-symmetric current-current interaction which leads to an integrable field theories for both signs of its coupling constant. Thanks to the Bethe-ansatz solution or the exact factorized $S$-matrices of the SU($N$) chiral Gross-Neveu model, we know that the precise value of the coupling constant does not matter and that only the sign is crucial to conclude on the nature of the IR physics of the model. The situation is very different from the example with an U(1) symmetry mentioned by the referee which does not apply here due to the presence of anomaly-free non-Abelian continuous symmetry, such as vector-like PSU($N$) symmetry, which prohibits the exactly marginal perturbation, as the Luttinger parameter in the referee's example which can give rise to a relevant perturbation. We have just used the potential term to fix the sign of the current-current interaction by a suitable regularization of the action and a free-field representation. We note that when the sign is negative, as we found, the exact massless factorized scattering approach of the SU($N$) chiral Gross-Neveu model have been exploited in Refs. [32,57,58] to predict that non-linear $\sigma$ models on SU$(2)$/U(1) and SU($N$)/SO($N$) target spaces at $\theta =\pi$ are integrable field theories. In the revised version of the paper, we have indeed added some comments and references related to this point.

Last but not least, we believe that incompleteness does not necessarily mean the uselessness of the research in physics and even in mathematics. Given that understanding the low-energy limit of 2d nonlinear $\sigma$ models with topological terms stands as one of the most challenging and interesting problems in front of us, our prescription turns out to be successful in explaining multiple examples in a unified (though non-rigorous) way, which we believe is quite valuable as stressed by the other three referees.

Reply to the Referee 3:

We thank the referee for carefully reviewing our manuscript and for recommending it for publication. We have taken into account all his/her requested changes which increase the quality of our manuscript. We thank the anonymous referee for pointing out his/her very interesting comments. About the weaknesses, one central point, made by Affleck and Haldane, in the case of a massless flow, is the existence of a discrete symmetry which forbids any possible relevant operator in the non-perturbative renormalization group flow of the action from weak to strong coupling where the non-linear sigma model is stabilized. In that case, the leading part of the WZNW deformed action can only be an SU($N$) current-current interaction which is an integrable field theory for all sign of its coupling constant. The non-perturbative spectrum is known and does not assume that the coupling constant of the current-current term is infinitesimal. The crucial point is its sign. We propose a way to fix this sign by considering a suitable regularization of the action and the use of a free-field representation. As mentioned by the referee, we are investigating similar massless flow in other series.

We now reply to the requested changes asked by the referee:

p.~14 line 4: says `$N-1$ massless compact bosons' but the numbering is $m=0, 1, ... , N-1$ indicating there are $N$ bosons.

Reply: We have corrected the typo in the revised version of the manuscript.

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?

Reply: We agree that changing the relative sign of the deformations is an interesting problem, but we think it is difficult to discuss that problem in our current framework. The main reason is that our prescription of the point-splitting regularization (21) strongly depends on the UV cutoff, and thus the relative magnitude of the deformation also depends on the UV details. As long as each deformation contributes with the same sign to the current-current interaction, this does not cause a problem, and we can safely relate the sigma-model description and the CFT with the current-current deformation. To clarify this point, we have added a new footnote (footnote 5) after Eq. (43).

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.

Reply: We have followed the recommendation. In the revised version we have explicitly used the notation SU($2k$)/USp($2k$) at the end of Section 4.4.

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.

Reply: We thank the referee for pointing it out this comment related to the Cartan embedding. It gives a clear explanation. We have thus added the comment as a footnote (footnote 6) for our expression of the coset SU($N$)/SO($N$).

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.

Reply: We agree that it should be well known, but it is too well-known to cite a specific reference. Thus, instead of adding a reference, we have changed the text as follows in Section 4.1 before Eq.~(52). "There exists a global section as the Spin($N$)-bundle on 2-manifolds is a trivial bundle thanks to $\pi_1($Spin$(N)) = 0$ :"

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 $\pm J$.

Reply: We thank the referee for pointing out our mistake. As explained by the referee, the disconnected component is characterized by the sign of Pfaffian instead of the overall sign. After Eq.~(75), we have modified our description accordingly by adding the explanation for the existence of two disconnected components.

p.~21, (73)-(75): Could you also insert some comment on what happens when $N$ is odd?

Reply: In the odd-$N$ case, there is no solution. We have added a footnote (footnote 9) related to this point.

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 $\text{CP}^{N-1}$ seems rather involved, so why is it simpler here?

Reply: We have added the explanation in the footnote (footnote 10), which explains why it gives the minimum and also proves any minima can be written in the same form. For $G$ to live in $U(2k)/[U(k)\times U(k)]$, its condition can be described by a simple formula, $G^2=-I_{2k}$ and ${\rm tr}(G)=0$, but the condition becomes more complicated for more general Grassmannians.

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 ($N=3,4$)?

Reply: We tried to give a general proof, but it becomes a problem of minimizing a sum of trigonometric functions with multi variables, and it was not successful. However, we confirmed the result by using different numerical methods, which makes us confident the correctness of the result. We have noticed that adding an extra term in the potential makes the analytical discussion easier, and we have added the comment in the footnote (footnote 11) in the revised version of the paper. We believe this provides a clearer explanation than explicitly performing the minimization for small $N$. However, we think this extra term is redundant physically and just a mathematical trick, so we didn't include it in the discussion of the main text.

Reply to the Referee 4:

We thank the referee for carefully reviewing our manuscript and for recommending it for publication. Below we answer the comment given by the referee.

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 $H_3$.

Reply: We understand that the formal derivation of the topological term of the SU($N$)/SO($N$) sigma model is quite technical. This is why we have also included a more pedagogical explanation of its derivation. We have revised our presentation to clarify the presentation and have added some references and footnotes 7 and 8 in accordance with the requested changes.

- We revised the wording in the text, replacing perturbation, perturbed CFT with deformation, deformed CFT, in response to one of the second referee’s comments.
- We added several references to answer some referees comments or suggestions.
- We added several footnotes (5,6,7,8, 9, 10, 11) to answer referees requested changes.
- In the introduction, Sec. 2.2, Sec. 2.3, we added some comments to argue that our conclusions are not restricted to infinitesimal coupling constant regime.