The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker ([1](http://prl.aps.org/abstract/PRL/v52/i18/p1575_1)), but the mathematicians found their proof insufficient, so that, FQS ([2](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-107/issue-4/Details-of-the-nonunitarity-proof-for-highest-weight-representations-of/cmp/1104116227.full)) and Langlands ([3](http://books.google.fr/books/about/Infinite_Dimensional_Lie_Algebras_and_th.html?id=PI7gmgEACAAJ&redir_esc=y)), published in the same time a complete proof.


There is an error in the paper of Langlands : ([3](http://books.google.fr/books/about/Infinite_Dimensional_Lie_Algebras_and_th.html?id=PI7gmgEACAAJ&redir_esc=y)) lemma 7b page 148 (see also [here](https://sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/VirasoroAlg-ps.pdf) page 7) :


* $p=2$, $q=1$, $m=2$, $h\_{p,q}(m)= \frac{5}{8}$, $M=4$
* $p=4$, $q=1$, $m=3$, $h\_{p,q}(m)= \frac{7}{2}$, $M=13$
* ...


yield case $(B)$, but $(p,q) \ne (1,1)$ and $m \ngtr q+p-1$.


**Remark** : In fact, we need to distinguish between $q \ne 1$ and $q=1$, not between
$(p,q) \ne (1,1)$ and $(p,q)=(1,1)$).


This lemma is used in the rest of the paper.



> 
> **Question**: Is there a way to fix the rest of the paper ?
> 
> 
> 


**Remark** : This way was used by Sauvageot (([4](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-121/issue-4/Repr%C3%A9sentations-unitaires-des-super-alg%C3%A8bres-de-Ramond-et-de-Neveu/cmp/1104178251.full)) lemma 2 (ii) p 648), without fixing.


**References** :  

([1](http://prl.aps.org/abstract/PRL/v52/i18/p1575_1)) D. Friedan, Z. Qiu, S. Shenker, *Conformal invariance, unitarity, and critical exponents in two dimensions.* Phys. Rev. Lett. 52 (1984), no. 18, 1575--1578.


([2](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-107/issue-4/Details-of-the-nonunitarity-proof-for-highest-weight-representations-of/cmp/1104116227.full)) D. Friedan, Z. Qiu, S. Shenker, *Details of the nonunitarity proof for highest weight representations of the Virasoro algebra.* Comm. Math. Phys. 107 (1986), no. 4, 535--542.



> 
> ([3](http://books.google.fr/books/about/Infinite_Dimensional_Lie_Algebras_and_th.html?id=PI7gmgEACAAJ&redir_esc=y)) R. P. Langlands, *On unitary representations of the Virasoro
> algebra.* Infinite-dimensional Lie algebras and their applications
> (Montreal, PQ, 1986), 141--159, World Sci. Publ., Teaneck, NJ, 1988.
> 
> 
> 


([4](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-121/issue-4/Repr%C3%A9sentations-unitaires-des-super-alg%C3%A8bres-de-Ramond-et-de-Neveu/cmp/1104178251.full)) F. Sauvageot, *Représentations unitaires des super-algèbres de Ramond et de Neveu-Schwarz.* Comm. Math. Phys. 121 (1989), no. 4, 639--657.



Is this error in this paper of Langlands fixable?
This morning I just discovered [these corrigenda](https://web.archive.org/web/20210506191547/http://math.univ-lyon1.fr/%7Eiohara/Corr/Corr-Chap11.pdf) of K. Iohara and Y. Koga (in which my name is cited in acknowledgement). In fact, three years ago, I have contacted K. Iohara (author, with Y. Koga, of the book *[Representation Theory of the Virasoro Algebra](http://link.springer.com/book/10.1007/978-0-85729-160-8/page/1)*) about this error (but I didn't know they fixed it).  

I do not yet read these corrigenda into details, but I guess it's ok (I hope).


**Another unfixed error for the Ramond algebra** :  

As I said in remark, F. Sauvageot gave a proof 'à la Langlands' of the FQS criterion for the superVirasoro algebras ($N=1$): the Neveu-Schwarz algebra and the Ramond algebra.  

Seven years ago, during my PhD, I have discovered this error in this paper of Langlands, reproduced in this paper of Sauvageot, so I decided to write a proof "à la FQS" of the FQS criterion for the Neveu-Schwarz and Ramond algebras.  

In fact I discovered that this criterion runs for the Neveu-Schwarz case,
but **not for the Ramond case** :  

We can prove lemma 4.19 p 22 of [this paper](https://arxiv.org/abs/1010.0077) (Neveu-Schwarz case)
thanks to the curves $h=h^{m}\_{pp}$, but the Ramond case doesn't have these curves !  

So it's ok for the Neveu-Schwarz case (and I guess Sauvageot's paper is fixable in this case by using the corrigenda above), but for the Ramond case,
the FQS criterion gives the discrete series plus some representations of
charge $c\_{m}$ with $m$ non-interger !


**Sketch of fixing** :  

For excluding these last representations we can use an argument of
fusion:  

$$ (R) \boxtimes (R) \to (NS) $$


It's known that the fusion of two representations of the Ramond algebra (R), in the discrete series at central charge $c\_{m}$, give a (discrete series) representation of the Neveu-Schwarz algebra (NS) at the same central charge $c\_{m}$ (the Ramond algebra is given by a twisted vertex module over the vertex operator algebra of the Neveu-Schwarz algebra). But we know
that the Neveu-Schwarz case doesn't contain such representations at central
charge $c\_{m}$ with $m$ non-interger, the result follows.


