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.

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.

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 $m \ngtr q+p-1$ (in fact, we need to distinguish between $q \ne 1$ and $q=1$, not between $(p,q) \ne (1,1)$ and $q=(1,1)$).

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.

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.