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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something -- not ready yet for public consumption, but I need to save <a href="https://ncatlab.org/nlab/revision/parafermion/1">v1</a>, <a href="https://ncatlab.org/nlab/show/parafermion">current</a>

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2022
   • (edited May 30th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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](https://doi.org/10.1142/S0217751X00002354), [arXiv:cond-mat/9905192](https://arxiv.org/abs/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](https://ncatlab.org/nlab/show/level+%28Chern-Simons+theory%29#LevelRenormalization). Including this would instead seem to give $$ Z_{k\color{blue}+2} \leftrightarrow SU(2)_k \,. $$ Is this discussed anywhere? <a href="https://ncatlab.org/nlab/revision/diff/parafermion/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/parafermion/2">v2</a>, <a href="https://ncatlab.org/nlab/show/parafermion">current</a>

   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 $!A = \sum_{E: V} (E \to A}[$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?

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 14th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave 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](https://arxiv.org/abs/1404.2840), [doi:10.1103/PhysRevLett.113.066401](https://doi.org/10.1103/PhysRevLett.113.066401)$]$ * A. M. Tsvelik, *$\mathbf{Z}_N$ parafermion zero modes without Fractional Quantum Hall effect* $[$[arXiv:1407.4002](https://arxiv.org/abs/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](https://ncatlab.org/nlab/show/topological+order#ViaDegenerateAnyonicGroundStates)) <a href="https://ncatlab.org/nlab/revision/diff/parafermion/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/parafermion/4">v4</a>, <a href="https://ncatlab.org/nlab/show/parafermion">current</a>

   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)

   diff, v4, current

4. • CommentRowNumber4.
   • CommentAuthorperezl.alonso
   • CommentTimeAug 6th 2023
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexadded reference on twisted parafermions <a href="https://ncatlab.org/nlab/revision/diff/parafermion/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/parafermion/6">v6</a>, <a href="https://ncatlab.org/nlab/show/parafermion">current</a>

   added reference on twisted parafermions

   diff, v6, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 29th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these pointers: * [[Doron Gepner]], *New conformal field theories associated with Lie algebras and their partition functions*, Nuclear Physics B **290** (1987) 10-24 &lbrack;<a href="https://doi.org/10.1016/0550-3213(87)90176-3">doi:10.1016/0550-3213(87)90176-3</a>&rbrack; * [[Peter Bouwknegt]], Bolin Han, *Coupled free fermion conformal field theories* &lbrack;[arXiv:2403.03471](https://arxiv.org/abs/2403.03471)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/parafermion/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/parafermion/7">v7</a>, <a href="https://ncatlab.org/nlab/show/parafermion">current</a>

   added these pointers:

   • Doron Gepner, New conformal field theories associated with Lie algebras and their partition functions, Nuclear Physics B 290 (1987) 10-24 [doi:10.1016/0550-3213(87)90176-3]

   • Peter Bouwknegt, Bolin Han, Coupled free fermion conformal field theories [arXiv:2403.03471]

   diff, v7, current