Emergent Supersymmetry on the Edges

This is an interesting paper, studying a basic question in CFT: when do WZW models have a hidden supersymmetry? Or, more precisely, when do the fermionized versions of these models have supersymmetry?

This is an interesting question and the paper is clearly written and certainly worthy of publication in SciPost. I have only minor comments and corrections.

First, the title does not seem to fit the main result of the paper. The results are very much focussed on 2d CFTs. Their role as the edge modes of a CS theory, let alone of quantum Hall states, seems to be secondary at best. Moreover, the fermionization story involve a Z_2 quotient of the CFT, coupled with Arf invariant and I didn't understand the importance of this from the bulk or from the quantum Hall perspective.

Other comments. "Laurant modes" is misspelled, as is "supersymemtric". (This latter error is particularly egregious when discussing anyons because [e,m]\neq 0.)

Finally, there is an important Arf identity written between equations (2.6 ) and (2.7) . I wondered how it was proven but the paper cited ([21] in the bibliography) simply sends the reader to an old paper of Atiyah. It may be better to cite this directly.

Recently the relation between bosonic 2d CFTs with Z2 symmetry and fermionic 2d CFTs has been greatly clarified in the literature.
In this paper the authors considered the fermonized WZW models and studied when they have supersymmetry. It is an interesting paper, and will easily pass the criteria to be published in this journal, once the issues listed below are addressed.

There are a few major comments:

In general, the title and the abstract emphasizing supersymmetry on the edges and the content which studies the fermionized non-chiral WZW models do not match very well. The referee thinks that the latter is by itself an interesting mathematical physics question worth being published here, and that it is closely related to the supersymmetry on the edges, but the authors should emphasize that they are not quite the same. The edge theory is inherently chiral; if we put the Chern-Simons theory $G_k$ on a disk with an insertion of a single anyon $\lambda$ at the center, the edge excitation has the Hilbert space whose character is $\chi_\lambda$. Although no extensive discussion is available in the literature, if we have a suitable Z2 1-form symmetry in the bulk which reduces to a suitable Z2 0-form symmetry on the edge, then we can gauge it to make it fermionic both on the edge and in the bulk at the same time. But the edge theory will not be the fermionized non-chiral WZW models as discussed in this paper, but will rather be a chiral model whose NS vacuum character would be the original vacuum character plus the character whose h=3/2.
This distinction (between the non-chiral WZW theories and the chiral edge theories) needs to be clearly made in the paper.

In Sec. 2.2, why do the authors list the conditions 1.2.3 as necessary conditions? If the NS sector partition function contains an state of dimension $(h_L,h_R)=(3/2,0)$, it becomes a spin 3/2 operator G via the state-operator correspondence, and G, T together are forced to have the OPE (2.8) simply from consistency. So the referee thinks that this condition is also sufficient.

In Sec.2, it would be nice to review the possible anomalies of Z2 symmetry in bosonic 2d theories, and how to distinguish them. It is not indispensable, but as Sec.2 is already such a nice review of various materials, the referee thinks that it would be nice to have this too.
One nice source to cite is https://arxiv.org/abs/1802.04445 , see in particular their Sec 4.4. There it is explained that the Z2 symmetry in 2d bosonic theories is either non-anomalous or anomalous, and that it can be distinguished in the case of Verlinde lines just by looking at its spin, so that it is non-anomalous when h=0 or 1/2 mod 1 and is anomalous when h=1/4 or 3/4 mod 1.

In Sec.3, it would be nice if the authors explain why $SU(N)_k$ with Nk=12 works for A-type and D-type. Is there a deep reason behind this observation?

The referee also has various minor comments:

p.1: The authors write "novel ideas such as hierarchy states [11,12]", but they are from the early eighties. I strongly doubt if the authors were even born back then, so they might want to drop the adjective "novel"

p.2: The authors refer to (R,NS) etc. without specifying which is the temporal and which is the spatial spin structure. Please do specify.

p.2: The authors refer to the nontrivial phase of the Kitaev chain simply as the Kitaev chain. This is often done in hep-th, but is frowned upon on the cond-mat side of the community. Please do add the qualification "the nontrivial (or topological) phase of" to the Kitaev chain, at least at the first mention.

p.3: consistent to -> consistent with

p.4: In (2.3) the authors refer to PP, AP etc, while they use R, $\tilde R$ etc in the other parts of the paper. Please unify the notations.

p.6: supersymemtry -> supersymmetry

p.8: in correspond with -> in correspondence with

p.9: by implementing the SageMath package -> "by writing a program in the SageMath", or if there is a particular SageMath package which does the computation of the affine character already, please mention it by name.

p.10: The logic in the last paragraph is strange. First, the sentence "The group elements of center B(G) are commutes" is grammatically incorrect. It should have been "The group elements of center B(G) all commute", but then B(G) is not defined at this point, if it is the center there is nothing to show about why its elements commute with any element of G, etc. The referee thinks that the authors should first define B(G) via (2.23) and show that it is a subgroup of the center.

p.11: It would be nicer to have a discussion of which Z2 subgroup of O(G) is (non-)anomalous.

p.13: $n^2/2$ -> $n^2/(2k)$

p.14: this originates that -> this implies that

p.14: there is one Z2 which is not mathbb

p.16: focus on discussion -> focus on the discussion

p.27: there is a comma at the start of a line

p.27: It would be nice to have a Lagrangian way to understand that the fermionized $SO(N)_3$ is always supersymmetric.

p.35: The authors say that the fermionic RCFTs are not always a $Z_2$ orbifold of WZW model. That is reasonable. But at least in the c=39/2 case mentioned here, what happens if we perform the reverse Jordan-Wigner transformation to get the bosonic theory? By construction it would contain an $(E_6)_4$ algebra, so it would presumably be a non-diagonal $(E_6)_4$ WZW invariant, is it not?

p.36: The authors say that "... provides further evidence for an emergent SUSY". It would be nice to cite [36], [37] again here.