Vacua, Symmetries, and Higgsing of Chern-Simons Matter Theories

This article examines the moduli space of certain classes of 3d $\mathcal{N}=3$ and $\mathcal{N}=4$ Chern-Simons-matter theories. The authors propose the use of magnetic quivers to describe the geometry of the so-called ``maximal branches" of the moduli spaces for these theories.

The results pertaining to the $\mathcal{N}=4$ theories are presented with sufficient clarity. Through the use of dualities, the authors demonstrate that the maximal branches can be identified with the Higgs and Coulomb branches of corresponding 3d $\mathcal{N}=4$ gauge theories with zero Chern-Simons levels. While explicit results are provided (e.g., Table 1 (b)), the referee does not find this aspect of the work particularly novel, as these results appear obtainable straightforwardly from well-established dualities. Nevertheless, the referee finds that the comparison between Higgsing performed via the Giveon-Kutasov duality and via the $\mathcal{T}^T$ duality, as presented for instance in Figure 4, adds scientific value to this part of the manuscript.

For the $\mathcal{N}=3$ theories, the authors limit their focus to abelian gauge theories. In the referee's opinion, the results and presentation in this section are less clear and require significant improvements.

This manuscript may be suitable for publication in SciPost, provided that the authors address the following comments satisfactorily.

The necessary improvements for the part concerning $\mathcal{N}=3$ Chern-Simons-matter theories include the following points:

1. Clarity of Scope: The authors should clearly state from the beginning of the paper that their analysis for $\mathcal{N}=3$ theories is focused specifically on abelian gauge theories.

2. Derivation Procedure: The procedure used to read off the magnetic quiver from the associated brane system is not clearly explained. For example, concerning the gauge theory presented in Eqs. (5.1)-(5.5), the authors need to provide a clear explanation of how the number of flavors in the magnetic quiver can be read off from the corresponding D3-brane segment. (As a side note, the reference to an $A_{|\kappa|-1}$ singularity below (5.1) should likely be corrected to $A_{|\kappa|+1}$ singularity).

3. Related to the previous point, the referee finds the logical progression confusing. It is unclear whether the magnetic quiver was deduced from the known result for the maximal branch ($\mathbb{C}^2/\mathbb{Z}_{|\kappa|+2}$, as stated in Eq. (4.20) of Ref. [13], arXiv:1706.00793) or if it was derived independently from the brane system. The authors should clarify this derivation path. This clarification is needed for both subsections 5.1 and 5.2.

4. Generalizability: If the magnetic quiver can indeed be read off directly from the brane system, it is not clear why the analysis was restricted to a single D3-brane (abelian case). The authors should comment on whether their method could be applied to $N$ D3-branes ($N>1$), which would correspond to non-abelian gauge theories and potentially lead to $U(N)$ gauge groups in the magnetic quivers (e.g., in Eqs. (5.1)-(5.10)). If the method is not applicable to the non-abelian case, a clear explanation for this limitation should be provided.

5. Demonstration of novelty: If the authors' approach allows for the derivation of the magnetic quiver directly from the brane system, they should demonstrate its utility by providing further interesting examples that go beyond the results already contained in Ref. [13] (arXiv:1706.00793). This would help to establish that the proposed approach is genuinely new and capable of generating new results.

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Clearly organized.

Extremely technical.

The paper provides a systematic framework for analyzing the maximal branches of the moduli space of 3d supersymmetric Chern-Simons Matter (CSM) theories with N=4 or N=3 supersymmetry. While several previous results on the subject exist, the paper analyzes the problem using recent techinques, like magnetic quivers and dualization algorithms, and provides general results for the vacua and RG structure of 3d supersymmetric CSM theories in regimes previously hard to tackle, like CS levels greater than one or N=3 supersymmetry. The analysis is demonstrated with many examples and tables. This is an extremely technical paper addressed to a relatively small community of people working on similar topics, but it is scientifically sound and it contains useful information. I think that it meets the SciPost criteria for acceptance and I recommend it for publication.

It would be useful to have in the introduction a more expanded discussion about the motivations of this paper in particular about what we can learn from the maximal branch structure of the moduli space.

The paragraph starting with "In N =4 CSM theories, the presence of Chern-Simons couplings modifies this picture." in the introduction is a bit confusing. Most of the things said below are also true for N=4 theories without CS. Please reformulate it.

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The paper discusses a novel approach to characterize the maximal branches of the moduli spaces of vacua of 3d $\mathcal{N}=4$ Chern-Simons-matter theories.
Early examples where analyzed by Jafferis-Yin and Hosomichi-Lee-Lee-Lee-Park. However, due to the inherent difficulties of a pure field-theoretic analysis, most recent checks focused on the study of Hilbert series, plus (sporadic) examples of sphere partition functions for suitably chosen Chern-Simons levels.
The combination of brane setups, magnetic quiver and Hilbert series dualization employed in the paper allows for a more systematic approach. In particular, for theories that are dual to 3d N=4 quivers without CS couplings, the authors construct the Hasse diagram of the two maximal branches thanks to a detailed analysis of the duality map, combined with the more standard techniques to analyze the moduli spaces of vacua without CS terms. The authors observe from the brane setup that certain monopole operators must have equal fluxes w.r.t gauge nodes between the two NS5-branes, which is a neat and efficient way to uncover this feature, that would be hard to systematize by constructing gauge-invariant monopole operators ``by hand''.
For theories which do not admit a description without CS couplings, this work proposes a magnetic quiver. The proposal is motivated by a number of explicit examples. The authors further extend the magnetic quiver proposal to theories with only $\mathcal{N}=3$ supersymmetry, one magnetic quiver per maximal branch, and successfully check their proposal against earlier work of Assel. However, in this less supersymmetric case the authors only discusses Abelian examples.
The content is innovative and interesting for theoretical physicists working in the field; it may also be of interest to mathematicians working on the foundations of 3d $\mathcal{N}=4$ Coulomb branches. The paper is clearly written, the necessary steps are explained in detail but in a neat and not verbose manner. Besides the main claims about moduli spaces of vacua of CS-matter theories, the paper includes various additional interesting results, such as the explicit calculations in Appendix B and the no-go theorem in Appendix C.7.
Overall, I consider that the content and presentation of the work meet the acceptance criteria. I strongly recommend the paper be published in SciPost Physics, provided some minor clarifications are incorporated. Please refer to the attached pdf file for a list of small requested changes

Main requested changes:
- In the computation of the superconformal index, it would be good to explain how the sum over magnetic fluxes is dealt with.
For example in Eq.(2.6) the authors compute the index perturbatively in $x$, thus each term should include a sum over topological sectors from the value of the gauge curvature on $S^2$ (e.g. the sum over $l$ in [Ref. 52, Eq.(2.1)]). What is the fugacity associated to such summation (e.g. the analogue of the fugacity $w$ in [Ref. 52, Eq.(2.1)])? Are the results at each order in $x$ a resummation of the contribution from (infinitely many) topological sectors? Or do only finitely many values of the sum contribute at each order?
The same applies to the other computations of the index. I would expect that, for $p$ unitary gauge nodes, each order in $x$ fixes a linear combination of the integers $l_i$ ($i=1, \dots, p$) and there are $p-1$ remaining series to resum.

- Relatedly, the authors exchange the perturbative expansion in $x$ with the infinite sum over topological sectors. Do the authors also exchange the sum over topological sectors with the residue integral? If so, they should explain why this is justified.

For other minor corrections please refer to the attached pdf.

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