Emergent Supersymmetry on the Edges

We thank the referees for carefully reading our manuscript and giving us valuable comments and suggestions. In the revised manuscript, we made the corrections for further clarification following the referee's comments. We hope this revision meets the referee's requests.

To Referee 1:

Regarding major comments, please see the following:

1.)
Let us denote a given bosonic theory as ${\cal B}$, its chiral part as $\chi$, and a fermion theory in correspondence to ${\cal B}$ via the generalized Jordan-Wigner transformation as ${\cal F}$. A couple of paragraphs clarifying the relation between the emergent supersymmetry of the chiral theory $\chi$ and the supersymmetry of a fermion theory ${\cal F}$ were added right below the first paragraph on page 2. We also demonstrated a well-known example, the tricritical Ising model, to understand the above relation.

2.)
We believe that a primary of $(h,\bar h)=(3/2,0)$, although it is conserved, does not imply supersymmetry. (Note that Haag-Łopuszański-Sohnius theorem is not valid in two-dimension.) More precisely, it is not guaranteed that such a primary automatically satisfies the supersymmetry current OPE. For instance, let us consider a ${\cal B}=(SU(2)_1)^6$ which can be fermionized to a supersymmetric model {\cal F}, the Gaberdiel-Taormina-Volpato-Wendland model (K3 sigma model). The GTVW model has 2^6 primaries of $(h,\bar h)=(3/2,0)$. However, only four of them can satisfy N=4 supersymmetry currents, as shown in arXiv:1309.4127 and arXiv:2003.13700. We also have a few examples where primaries of $(h,\bar h)=(3/2,0)$ in the NS sector are present, but the partition functions in the Ramond-Ramond sector become nontrivial, i.e., $Z_{RR}\neq const.$. Thus, we propose that conditions 1,2, and 3 are necessary conditions.

3.)
We have put the reference the referee suggested after presenting the equation (2.45).

4.)
Unfortunately, we do not have an answer to the referee's question. Indeed it would be nice if there is a mathematical/physical reason behind the observation $Nk=12$.

Regarding two of the referee's minor comments, please see our responses below:

1.)
We agree with the referee that it would be nice to understand why the supersymmetry emerges for the fermionized SO(N)_3 models for any N in a Lagrangian way.
Due to the lack of our understanding, we however leave it as a conjecture for now in this manuscript.

2. )
One can show that the fermion theory with $c=39/2$ is mapped to a bosonic CFT whose partition function is the non-diagonal modular invariant of the (E_6)_4 WZW model. For instance, the vacuum character of the bosonic CFT can be expressed as a linear combination of two characters for the vacuum and the primary of $h=2$ of (E_6)_4 WZW model.

However, the bosonic CFT of our interest cannot be obtained by Z_2 orbifolding the (E_6)_4 WZW model. This is partly understood from the fact that the Verlinde line of $h=2$ cannot be associated with a Z_2 symmetry. More precisely, the action of the Verlinde line of $h=2$ on each primary is not always given by $\pm 1$.

Following the rest of the referee's minor comments, several typos were fixed and minor corrections were made.

To Referee 2:

We agree that our results very much focused on the 2d CFTs. We added a couple of paragraphs clarifying the relation between the emergent supersymmetry of the chiral theory and the supersymmetry of a fermion theory on page 2. Also, we fixed the typos and added the reference following the referee's suggestion.