SciPost Submission Page
Emergent Supersymmetry on the Edges
by JinBeom Bae and Sungjay Lee
 Published as SciPost Phys. 11, 091 (2021)
Submission summary
As Contributors:  Jinbeom Bae 
Preprint link:  scipost_202106_00046v3 
Date accepted:  20211015 
Date submitted:  20210930 15:30 
Submitted by:  Bae, Jinbeom 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The WZW models describe the dynamics of the edge modes of ChernSimons theories in three dimensions. We explore the WZW models which can be mapped to supersymmetric theories via the generalized JordanWigner 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 ReadRezayi states at filling fraction $\nu=1/2,~2/3$ are able to support supersymmetry.
Published as SciPost Phys. 11, 091 (2021)
Author comments upon resubmission
Dear Editor,
We are grateful for the referee's thoughtful comments on our manuscript. Regarding the referee's report, please find our response below.

As mentioned in the previous letter, we can show that the chiral part of a given full nonchiral CFT has supersymmetry when the nonchiral CFT is fermionized to the supersymmetric theory. We thus wish to use the title as it is now.

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, one can show that there exists a conserved current of spin$3/2$ in the NSsector. However, one can show that the fermion theory $\mathcal{F}$ has a nonconstant RamondRamond(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. As the referee pointed out, this model can be also obtained by performing certain discrete identification of the direct sum of the supersymmetric and nonsupersymmetric theory.
Based on the above example, we believe that the mere presence of a spin$3/2$ primary does not guarantee the existence of supersymmetry.
Best Regards, Jinbeom Bae, Sungjay Lee