SciPost logo

SciPost Submission Page

Emergent Supersymmetry on the Edges

by Jin-Beom Bae and Sungjay Lee

This is not the latest submitted version.

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Jinbeom Bae
Submission information
Preprint Link: scipost_202106_00046v2  (pdf)
Date submitted: 2021-08-16 17:42
Submitted by: Bae, Jinbeom
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


The WZW models describe the dynamics of the edge modes of Chern-Simons theories in three dimensions. We explore the WZW models which can be mapped to supersymmetric theories via the generalized Jordan-Wigner transformation. Some of such models have supersymmetric Ramond vacua, but the others break the supersymmetry spontaneously. We also make a comment on recent proposals that the Read-Rezayi states at filling fraction $\nu=1/2,~2/3$ are able to support supersymmetry.

Author comments upon resubmission

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.

List of changes

To Referee 1:

Regarding major comments, please see the following:

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.

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.

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

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:

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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2021-9-24 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202106_00046v2, delivered 2021-09-24, doi: 10.21468/SciPost.Report.3561


The authors' improvements to the v1 were satisfactory except the following two points:


1. The referee still thinks that the title does not appropriately convey the content of the paper; the bulk of the paper discusses the full CFT, while the title focuses on the chiral half of the CFT.

2. The referee still thinks that if we pick a single spin 3/2 primary $G$ (from $2^6$ in the GTVW model) , it is forced to satisfy the N=1 super-Virasoro OPE. The rough argument is that $G(z) G(0)$ can only have singularities of the form $\sim 1/z^3$, $J/z^2$ and $T/z$;
if the coefficient of $J$ is nonzero, this means the $JGG$ 3-pt function is nonzero, meaning that $G$ is charged under the U(1) generated by $J$, which is impossible because we chose $G$ to be real.

Was it not the case?

It would also be nice if the authors explicitly describe a model which has a primary of $(h,\bar h)=(3/2,0)$ but still has $Z_{RR}\neq 0$.


The referee is prepared to agree to disagree with the authors on the point 1, but the referee would like the authors to take care of the point 2.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Jinbeom Bae  on 2021-09-28  [id 1789]

(in reply to Report 1 on 2021-09-24)
answer to question

We thank the referee for the comments.

  1. As explained in the previous letter to the referee, the main purpose of the analysis done in the main context is to show the emergence of supersymmetry in the chiral part of the full CFT. More precisely, when a non-chiral CFT can be fermionized to a supersymmetric theory, we can argue that the chiral part of the given full non-chiral CFT has supersymmetry. We thus believe that the current title can convey the main idea of the paper well and wish to use the title as it is now.

  2. We agree with the referee that an OPE of a real primary of $(h,\bar h)=(3/2,0)$ G involves only two singular terms $\frac{1}{z^3}$ and $\frac{1}{z}$. However, we are not sure that the OPE of $G(z)$ has to be identical to that of the supersymmetry current with the correct OPE coefficients.

As an illustrative example, let us consider a product of two $c=1$ CFTs, $((U(1)_4)/Z_2)^2$, as a bosonic theory $\mathcal{B}$. Performing the femionization (for details, please see a note attached to the letter), one can show that there exists a conserved current of spin-$3/2$ in the NS-sector. However, one can show that the fermion theory $\mathcal{F}$ has a non-constant Ramond-Ramond(RR) partition function and violates the supersymmetry unitarity condition $h_R \ge c/24$. In fact, the RR partition function is proportional to that of the Ising model.

Based on the above example, we believe that the mere presence of a spin-$3/2$ primary does not guarantee the existence of supersymmetry.



Anonymous on 2021-09-30  [id 1792]

(in reply to Jinbeom Bae on 2021-09-28 [id 1789])

I am the referee 1; I would like to thank the authors to prepare a separate note concerning the second point.

The model in the attached note can be described also as follows: you take two compact bosons, and fermionize one of them into two Majorana fermions. So you have operators $X$, $\psi_1$ and $\psi_2$. Your chosen spin-$3/2$ current is $\psi_1 \partial X$. Clearly this generates a standard N=1 super Virasoro acting on $X$ and $\psi_1$, but does not do anything on $\psi_2$. Therefore, if you take the RR partition function, you simply get the partition function of the theory of $\psi_2$, the Ising model.

Touché, indeed. You can always consider a direct sum of a susy sector and a non-susy sector, and then perform some discrete identifications.

I am still not convinced if there is a truly nontrivial example where a spin-3/2 current does not just generate a standard N=1 super Virasoro acting on an almost decoupled susy sector, but I get the main point by the authors.

I am now satisfied and the paper can be published on SciPost.

Login to report or comment