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I am after the relation between $Z_k$ parafermions and $SU(2)$-Chern-Simons/WZW theory.
The article
gives the identification
$Z_k \leftrightarrow SU(2)_k \,,$i.e. that $Z_k$ parafermions are essentially described by $SU(2)$ conformal blocks at level $k$.
But it seems to me that (in their section 2) the authors are neglecting the Chern-Simons level renormalization. Including this would instead seem to give
$Z_{k\color{blue}+2} \leftrightarrow SU(2)_k \,.$Is this discussed anywhere?
have added these tow references, provividing an integrable model for $\mathbb{Z}_N$ parafermion anyons:
A. M. Tsvelik, An integrable model with parafermion zero energy modes, Phys Rev. Lett. 113 066401 (2014) $[$arXiv:1404.2840, doi:10.1103/PhysRevLett.113.066401$]$
A. M. Tsvelik, $\mathbf{Z}_N$ parafermion zero modes without Fractional Quantum Hall effect $[$arXiv:1407.4002$]$
the second of these refers to topologically ordered ground states as the “modern day philosopher’s stone”.
This strikes me as an interesting association, so I have added mentioning of this phrase at topological order (here)
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