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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 30th 2022

starting something – not ready yet for public consumption, but I need to save

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 30th 2022
• (edited May 30th 2022)

I am after the relation between $Z_k$ parafermions and $SU(2)$-Chern-Simons/WZW theory.

The article

• Daniel C. Cabra, Eduardo Fradkin, G. L. Rossini, F. A. Schaposnik, Section 4 of: Non-Abelian fractional quantum Hall states and chiral coset conformal field theories, International Journal of Modern Physics A 15 30 (2000) 4857-4870 $[$doi:10.1142/S0217751X00002354, arXiv:cond-mat/9905192$]$

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?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 14th 2022

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)