The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker (1), but the mathematicians found their proof insufficient, so that, FQS (2) and Langlands (3), published in the same time a complete proof.

There is an error in the paper of Langlands : (3) lemma 7b page 148 (see also here 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) lemma 2 (ii) p 648), without fixing.

References :
(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) 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) 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) F. Sauvageot, Représentations unitaires des super-algèbres de Ramond et de Neveu-Schwarz. Comm. Math. Phys. 121 (1989), no. 4, 639--657.


This morning I just discovered these corrigenda 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) 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 (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.


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