Bootstrapping Mixed MN Correlators in 3D

1. This paper advances the frontier of the numerical bootstrap by studying a set of crossing equations for MN models that have not previously been explored.
2. The required new representation theory is explained well despite being quite complex.
3. Details on the representation theory for a wider class of theories than studied in this paper are also worked out. This can be very useful for future numerical bootstrap studies.
4. Characteristics of solutions to (truncated) crossing equations located close to previously discovered and new kinks in bounds are studied numerically. Under reasonable assumptions isolated islands are found. This provides (some) evidence for the existence of a corresponding CFT located close to these kinks.
5. These numerical results are of physical interest and relevant to answering various experimental and methodological questions (although they do not provide conclusive answers).
6. Useful appendices (and an ancillary .nb file) are provided so that the numerical setup can more easily be reproduced.

1. The motivation of the work could be expanded on. The paper reads as a follow up to arXiv:1904.00017 [hep-th]. That work includes more on the physical motivation for studying some of the models also studied here. It also motivates well why understanding the properties of the kinks studied in this work is important for solving some open physical questions (validity of the epsilon expansion / existence of non-perturbative fixed points / new CFTs / matching experimental results). The physical motivation is explained much more briefly in this work (probably to avoid redundancy). Perhaps it would be appropriate to offer a summary of these motivations or an explicit statement that the these can be found in the previous paper.

The included motivation,
"Our motivation is to find the minimal set of conditions
that allow us to isolate allowed regions in parameter space at the positions of kinks of the single correlator bounds obtained in [24].", does not sound like the strongest motivation. (See also point 1. of the requested changes.)

The authors find "such kinks to be strong indicators for the presence of actual CFTs". Therefore I would expect the motivation to be to study the characteristics of these approximate solutions in order to:
1.) Determine whether its properties are compatible with the expectations of a true CFT.
2.) To learn about physical quantities in this CFT so that it can be identified and so that these quantities can be matched with data from experiments and or alternative computational methods.

This paper presents new original analytic work as well as new numerical bounds on the operator dimensions in a family of physically interesting conformal field theories. It also facilitates future conformal bootstrap research by working out the representation theory to study a wider class of theories. I recommend publication (possibly after the minor revisions / clarifications that I recommend in "Requested changes" ).

1. The reason that saturation of single correlator bounds is demanded in the multi-correlator setup for some figures could be clarified.

a.) What advantage does this have over alternative methods of estimating $\Delta_S$ at the kink - such as the extremal functional method (EFM).

b.) What is the role of $\Lambda$ in these considerations?

Note that without a change in $\Lambda$ the spectrum at saturation is uniquely fixed. The EFM would instantly provide the sought for "isolated region" also in the $(\Delta_\phi,\Delta_S)$ plane. The "minimal set of conditions required in order to find an isolated region" would be only the saturation of this bound.

The benefit of the author's methods seems to be that it allows one to use information obtained in the single correlator bootstrap at large values of $\Lambda$ as input in multi-correlator bootstrap setups where similar values of $\Lambda$ are unpractical to reach. It would be useful to comment on this.

It is also interesting to look at for example Fig. 8 realizing that if $\Lambda$ were increased from 35 to 45 the island would have to shrink to a line. If this line is too expensive to compute we can instead ask what line you get when you demand $\Delta_X$ to saturate the $\Lambda=35$ bound instead of the $\Lambda=45$ one. This should not be too hard to compute (easier than the computed islands) and it would be interesting to see whether any features stand out in this line.

Related: The authors demand saturation of the $\Delta_X$ bound. However, the true solution could reasonably be expected to be located slightly below this bound. It would be nice to know how sensitive the results are to the assumed $\Delta_{X_{\text{sat}}}$. Would lowering this value slightly significantly alter the bounds.

2. The motivation of the work could possibly be expanded on. (See Weaknesses.)

3. The $Z\times Z$ bootstrap seems to be organized as if it were a separate section, yet it is found inside the section "3. Numerical Results". It even contains a subsection 3.5 also named "Numerical Results". I would propose that either subsections 3.3 and 3.4 are included in section 2. Or that subsection 3.3-3.5 become a new section on the $Z\times Z$ bootstrap.

4. Do I understand correctly that the $Z \times Z$ OPE exchanges itself? If so, was it imposed that the external operator $Z$ corresponds to the lowest operator of its kind? This assumption can be made without loss of generality and could change the position of the kinks in various plots.

P.S.: Access to [29] is restricted by Kousvos, S. I am not sure whether this is intended.

1-Novel work presented in a format that is instructive and useful to a reader within the field
2-Pedagogical presentation (such as going through examples of the $\phi_i^a\times\phi_j^b$ OPE) to make the work accessible
3-Lengthy and extensive appendices to document the intermediate technical results
4-General intermediate results that can be used in broader outside contexts (i.e., the $G^n\rtimes S_n$ OPE expansion results)

1-Results themselves (the bounds and islands) are presented in a slightly confusing format that can become hard to follow
2-Some of the text could be better organized

The manuscript presents novel results in bootstrapping the 3D $O(m)^n\rtimes S_n$ CFTs with mixed correlators. The results extend the conformal bootstrap's success in the $O(N)$ CFTs, which can be seen as the $n=1$ case of this manuscript's results. The authors find several rigorous bounds and less rigorous islands which will be of interest outside of the bootstrap community and leaves open room for further projects in this direction. Along the way, they derive more general OPE decompositions for operators transforming under representations of $G^n\rtimes S_n$, which will be useful for other bootstrap studies in future.

The authors do a good job of going through what amounts to fairly dense algebraic work in a way that is both useful and accessible to readers of varying technical expertise. Both an expert and an inexperienced student would be able to learn from section 2 especially. Additionally, the authors present multiple intermediate results in the appendix so that the various insights of this work can be used to not only reproduce but extend the work of this manuscript. In short, this manuscript makes a strong contribution to the broader conformal bootstrap project.

The main results of section 3 are impressive and robust and demonstrate a thorough understanding of what was studied. However, I found myself slightly confused due to the number of similar but distinct plots presented serially for the different models, as that hid the relations between the figures. I might suggest putting the $\Delta_\phi$-$\Delta_S$ and $\Delta_\phi$-$\Delta_X$ plots of the same model as subfigures within the same figure. In other words, have only one figure per model, with subfigures for each individual plot. Additionally, I had some trouble relating e.g. Figs 5 and 6 due to the changing axis scales: it was not immediately apparent to me that these were referring to the same parameter space. With this said, it was very interesting to have both of these plots back to back so as to show which "lobe" in fig 5 corresponded to which rough range of gap assumptions. I would suggest combining figs 5 and 6 into a single plot, if possible: I recognize that these are at different $\Lambda$, but perhaps an intersection plot would be a nice way of presenting this data.

Having done calculations of this sort, I can say that the ZZZZ single correlator work is heroic for the sheer volume of algebra that was successfully managed and presented. It's clear that the authors have spent a great deal of time and effort thinking through how best to present this dense subject matter. With this said, I found the exact placement of the results within the paper (in sections 3.3-5) to be slightly confusing; within the numerical results section, the authors spend a few pages on algebraic results that might have better fit in section 2 with the other OPE work. Relatedly, the numerical results subsection (3.5) within the numerical results section made it difficult for me when jumping around in the paper to find specific results. Besides moving the algebraic results to section 2, the authors might also consider organizing the paper with one or two sections for the $\phi$-$X$ systems along with their numerical results followed by another one or two sections for the $Z$ systems and their numerical results. I suspect it should be relatively easy to implement and would better showcase the excellent work done by the authors.

There were some minor grammatical or stylistic errors in the manuscript, but it is perfectly legible so it's not a problem.

In short, this manuscript is a useful, accessible, and important contribution to the field which will open new directions of research. With the above minor revisions, it achieves all of the criteria for acceptance.

1-Condense or otherwise reorganize the presentation of the plots so as to make clear which plots are related and which plots are distinct.
2-Modify sections 3.3-5 so as to prevent algebraic work from being presented in the midst of numerical work.