# Emergent Supersymmetry on the Edges

### Submission summary

 As Contributors: Jinbeom Bae Preprint link: scipost_202106_00046v3 Date accepted: 2021-10-15 Date submitted: 2021-09-30 15:30 Submitted by: Bae, Jinbeom Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

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.

Published as SciPost Phys. 11, 091 (2021)

Dear Editor,

We are grateful for the referee's thoughtful comments on our manuscript. Regarding the referee's report, please find our response below.

1. As mentioned in the previous letter, we can show that the chiral part of a given full non-chiral CFT has supersymmetry when the non-chiral CFT is fermionized to the supersymmetric theory. We thus 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, 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. As the referee pointed out, this model can be also obtained by performing certain discrete identification of the direct sum of the supersymmetric and non-supersymmetric 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