Discrete gauging and Hasse diagrams

1 - Novel results in a topic of current interest
2 - High level of sophistication
3 - Cross-checks of the main claims
4 - Clarity of exposition and self-containedness
5 - Limitations and open problems clearly spelled out.

1 - No discussion or tests of whether the wreathed magnetic quivers are 3d mirrors of the $\widetilde{SU}(N)$ SQCD theories.

This is an interesting, timely and well written paper on the Higgs branch of SQCD with principally extended $\widetilde{SU}(N)$ gauge group and eight supercharges. The authors derive the Hasse diagram which encodes the stratification of the Higgs branch from a detailed study of the Higgsing patterns. They also conjecture that the corresponding magnetic quivers are recently introduced wreathed quivers and provide detailed checks of this conjecture using Hilbert series.

Challenges in finding magnetic quivers for $\widetilde{SU}(N)_{II}$ theories remain, as the author clearly explain. The article also contains interesting remarks on the relations among wreathed quivers, folded quivers, discrete gaugings and twisted compactifications, which help clarify the status of the subject.

I believe that the article is worthy of publication subject to a minor revision to address a few minor points, which I list below.

1 - Do the authors expect that the proposed magnetic quivers are also the 3d mirrors of the principally extended gauge theories? Please include a discussion.
2 - Transverse slices to leaves of the Higgs branch are deduced from the number of singlets in decompositions such as (3.4), which however only counts the dimension of the slice. For the benefit of uninitiated readers, please explain briefly how to discriminate among different slices of equal dimension.
3 - A few typos need to be corrected:
> p 2 last para: no trivial -> non-trivial;
> p 3 last para: missing of in first line; no-trivial -> non-trivial;
> p 5, first line of section 2.1: setup -> set up;
> p 6 footnote 3: $"$ -> $``$ before pseudo;
> p 20 fifth line of section 4: Coulomb branch of the is missing before wreathed quivers;
> p 31 fourth line: $SU(3)$ or $\widetilde{SU}(3)$?

1- Studies in detail the representations and their branchings for the $\widetilde{SU}_{I,II}$ groups

2- Proposes a magnetic quiver for the $\tilde{SU}_I$ groups, and provides, as a strong check of the conjecture, evidence for the matching of the corresponding Hilbert series

1- No real weakness

In this paper the authors study theories based on disconnected gauge groups obtained as principal extensions of unitary groups, which correspond to gauging charge conjugation. These groups come in two varieties dubbed $\widetilde{SU}(N)_{I,II}$, of which the $\widetilde{SU}(N)_{II}$ requires $N$ to be even.

The paper has two parts: in the first part some relevant representations and their branching rules are studied. Using this, the Hasse diagrams for Higgs branches of theories based on these groups with fundamental matter is constructed. In the second part the magnetic quiver for the $\widetilde{SU}(N)_I$ with fundamental matter is constructed. It turns out to be a wreathed version of the magnetic quiver for the ``parent" theory with $SU(N)$ group.

I think that the paper is an interesting technical development and thus I recommend it for publication.

1- Since quiver wreathing plays a very relevant role, a few more words on quiver wreathing would be acknowledged to make the paper more self-contained.

2- The identification of the wreathed quiver as magnetic dual to the $\widetilde{SU}$ theories with fundamental matter is very interesting, and the fact that the Higgs branch Hilbert series is reproduced is a strong test. However, as the authors point, this doesn't necessarily imply that the magnetic quiver captures the same symplectic singularity. Nevertheless, below the three points $\alpha)-\gamma)$ on page 24 the authors conjectured that they are indeed isomorphic. It would be good if the authors could elaborate a bit more on the motivation of this conjecture.

1. This article contains several new interesting results.
2. The presentation is clear and readable.
3. The proposals were tested and checked using a number of exact computations.

This article investigates the 4d $\mathcal{N}=2$ supersymmetric QCD with a disconnected gauge group that is a semidirect product between $SU(N)$ and the charge conjugation symmetry. There are two types of such products. The authors studied the structure of the Higgs branch of the theories of both types as a symplectic singularity using Hasse diagrams. The magnetic quivers for one of the types were proposed. It turned out that they are wreathed quivers, recently discovered by one of the authors along with other collaborators. In the referee's opinion, the article deserves publication after minor corrections and improvements.

1. The authors proposed the Hasse diagrams in Figures 3 and 5 for the case of $N_f \geq N_c$. The magnetic quivers for such a case were also depicted in Figures 7 and 8. I would like to ask the authors to comment on the case of $N_f < N_c$. What would be the corresponding Hasse diagrams and magnetic quivers? Can these be determined using the technique, such as in [arXiv:1909.00667]?
2. The authors presented the unrefined Hilbert series in (4.19). How can this be refined? Can one turn on the relevant fugacities in (4.1) and (4.4)? The authors should demonstrate how to do so at least to a few orders in the power series of the Hilbert series. Moreover, are there any fugacities associated with discrete symmetries one can turn on?
3. Some minor typos on page 20: $\widetilde{SU}_I(N)$ should be changed to $\widetilde{SU}(N)_I$ for consistency.