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Superconformal algebras in two dimensions with arbitrary N
Volume 174, number 3
SUPERCONFORMAL
PHYSICS LETTERS B
10 July 1986
A L G E B R A S IN T W O D I M E N S I O N S W I T H A R B I T R A R Y N
M. B E R S H A D S K Y
Landau Institute for Theoretical Physics, 117334 Moscow, USSR Received 26 February 1986
Superconformal algebras in two dimensions are constructed for arbitrary N. The arising algebras are not Lie algebras but they are associated algebras and they possess a central extension. Local supersymmetry is generated by multiplet of spin-currents in fundamental representation SO(N) or U(N).
Introduction. Super extensions o f the conformal algebra were constructed by Ademollo and others [1 ]. F o r the O(N) symmetry these extensions demand 2N- [1-N(N1)/2] subcanonical charges(with as the only exception the SU(2) subalgebra of 0(4)). For N ~> 5 these extensions also do not possess a central extension. In this article we present super extensions of the conformal algebra with arbitrary N that are not Lie algebras in the ordinary sense in contrast to the construction o f ref. [1 ]. These algebras also allow for central extensions. These extensions are associated algebras (in other words they possess the Jacobi identity), but the commutator of two generators of the algebra cannot be expressed as a linear combination o f initial generators. In our case one has to add bilinear combinations of generators to the Lie bracket [X i X j] = fi) k X k -- gik) l x k x l,
generated by the energy tensor T(z) (see ref. [4]), local color SO(N) (or U(N)) symmetry generated by the currentsJa(z) (a = 1 , 2 .... (N 2 - N)/2 for SO(N) and a = 0, 1,2 .... (N 2 - 1) for U(N)) and local supersymmetry generated by a multiplet of spin-currents Si(z) in the fundamental representation SO(N) (or U(N)). The dimensions o f the fields T(z),Ja(z) and Si(z) are 2, 1 , 3 / 2 , respectively. We will assume that these fields satisfy an operator algebra that contains in the operator product expansion o f two spin-currents S i the normal ordered square of currents ja :
Si(z)S/(w) = ~0"(d/2) (z - w) - 3 + ~(Aa)iO~(w)
(z - w) -2
+ IT(w)6 i / _ (a/2) ( A a ) i f j a ( w )
(1)
where the X i are the initial set o f generators. To be logical one has to add operators x i f = gikJ l X k X l to the initial set of generators and consider them to be independent generators. Thus there arise multilinear combinations of initial generators X i. Similar algebras arose in refs. [2,3]. The idea that it could be possible to construct super extensions of the conformal algebra in this way is due to Zamolodchikov (the elder) [3].
SO(N)-supersymmetry. The theories that are going to be constructed possess ordinary conformal symmetry
+ .),Mij : j a j b : (w)] (z - w) -1
(2)
where the (Aa)ii are generators of SO(N) (or U(N)), Jaw denotes the derivative of the field ja, MiJb is a special projector that one has to construct from
(Aa A b ) i/and ~ i/lab : MiaJb = [(A(aAb)) i/+ f6ab6i]].
(3)
Normal ordering is defined via an integral representation:
:AB : (z) = ~
[A(z)B(x) + A(x)B(z)] .
(4) 285
Volume 174, number 3
PHYSICS LETTERS B
The spin-current and current-current operator product expansions are assumed to be the following:
J a ( z ) j b ( w ) = --fabcjc(w)(z - w) -1 + k~ab(z - w) - 2 ,
(s) Ja(z)Si(w) = (Aa)iiSJ(w) (z
-
w) -1 .
(6)
For the U(N) case the fields S i are complex. So one must introduce independent complex conjugate fields SZ:
Ja(z)Si(w) = - (Aa)fiSi(w) (z - w) -1 .
(6')
This expression determines the transformation law for the currents ja (for the U(N) case) under a complex conjugation (ja * _ja for the hermitian matrix (Aa) i/ and ja * ja for the antihermitian matrix (Aa)i]). The operator product expansion rules with energy tensor T(z) are the following: T ( z ) X ( w ) = A X ( w ) ( z - w) - 2 + X w ( W ) ( Z -
w) - 1 ,
T(z) -- ~ L n z - n - 2 , Si(z)= ~ S i
Ja(z) = ~-jSanz-n-i,
z-n-312,
. j a j b :(z)= ~ x a b z - n - 2
tl
"
#l
Y ,ia z-n-5/2
: s i J a . (z) =
(lo)
The demand that these fields constitute an associated algebra is equivalent to the fact that the operators defined by expressions (10) satisfy the Jacobi identity. The arising equations determine unknown parameters a, 7, ~', c, d of the algebra. Now we represent the result for the SO(N) case. The generators (Aa) i! are normalized by the condition
(Aa)il(Aa) kl = 8iksi l _ 8i161k.
a -
7=
(11)
:(w) + ksab(z - w) - 4 , (7)
ja(z ) : j b j c :(w) = 2 f ad(b :jc)jd : (w)(z - w) -1
2k + 4 - N 4k - 4 N + 12 '
(12a)
1 2k-2N+6'
(12b)
d = - 2 k 2k + 4 - N 2k- 2N+6' N 2 - 6k-
c=~k
T(z) : j a j b " (w) = 2 : ja jb : (w) (z - w) - 2
(12c)
10
2k- 2N+6
(12d) '
~"in (3) is equal to one, k is arbitrary. The projector M//b has the form
Mijb = (A(aAb)) i~ + 6ab8 if.
(13)
For the case N = 2 this projector is equal to zero and we obtain the U(1) algebra of a spinning string (see ref. [ l l ) . The commutator relations are the following:
+ 2kSa(bsc)rjr(w) (z - w) - 2 +fad(hue)dr Jr(w) (z - w) - 2 ,
:SiJ a : in Laurent series:
Checking the Jacobi identity one obtains
where X(w) is one of the fields Ja(w) or Si(w); A is the dimension of the field. The arising algebra except the initial fields T(z), Ja(z), Si(z) contains an infinite set of composite operators • 1 : <]b : j a j b :, (A(ab)c)ij : sfjC :, ... (multilinear combinations of the fields S i and ja). The operator product expansions of the initial fields T(z), Ja(z), Si(z) with • j a j b : (z) can be easily found with the help of the integral representation (3):
+ (z - w ) - l ( d / d w ) : j a j b
10 July 1986
(8)
[L n Lm] = ( n si(z) :JaJ b :(w) = -2(A(a6b)c)i/ :s/J e : ( w ) ( z - w)-1
rn)Ln+ m +-~c(n 3 - n)6,,,_ m,
ja [L n jam ] _ _ m n+m' _
+ (A(aAb))iSSl(w) (z - w) - 2 .
(9)
Let us define the operators Ln, -nffa,-n'qi,"-ngab, y£a, expanding the fields T(z), Ja(z), Si(z), : j a j b :,
,1 X...(ayb)... denotes the symmetrization symbol. 286
[L
=(12
- m)S;+m,
[ja S i l = (Aa)ilS~+m '
(14)
PHYSICS LETTERSB
Volume 174, number 3
[Sin Sire] = ~i]Ln+ m + ol(n - m) (Aa)i]Jan+ m
10 July 1986
J =(l/a)tr(OnA),
~f° = ( I / 3 ) t r ( Q n I ) ,
(18,18')
where the constants a and 3 are determined by the conditions:
+ .).Mia]bxab + 1 d(rt 2 "" n+m a -- ~)(SqSn, m' ab ] = (n - m)Xn+ ab m + g k(n3 - n)sabSn ,- m' [Ln X m
t<
tL °
=s;
?/+/77
'
(19,19')
[Sin xa2] =-2(A(asb)c)ifY/nC m
So one can write the expressions for the operators
+ (. + ~)(A(~b))ifS~+~, QinJ = al(AV)if J~ + (A°)iJ J°n l ,
[da xbCl = 2fad(cxb)d + 2kn 5a(bsc)dj d m • a n+m n+m + nFad(bfc)drj r a
a
/'/+/77
(14 cont'd) "
It is also possible to construct the associated algebra without the energy tensor T(z). This algebra contains only SO(N) currents and spin-currents and it is a little simpler than we have considered:
where we introduce A ° = hi and jO = hT0 (h = (13/ N~)1/2). It turns out that c~ is real and h is imaginary. So the currents Ja(z)under complex conjugation are transformed in the following way:
s~-s,
s ° * s °.
The current-current operator product expansion has to be written in a slightly different form (compare
[jta Jbrn] -- -sfabc'lc-n+m + "knab ~n,- m '
with (5)): [Jan sire] = (Aa)ifSJn+ m , [Sin S ~ ] =~'(n -
ja (z) j b (w) = - f a b c j c ( w ) (z - w ) - 1 + gab (z - w ) - 2
m)Ja+m(Aa) ij+ ~Mif ab i ab X n+m
+ -~'d(n 2 - 1)6iJ6n_m ,
(20) (15)
where," = 2 / ( N - 2 ) ; d = 4 ( N - 3 ) ~ ; k = N - 3;~" is arbitrary and determines the scaling of spin-currents Si(z). The commutator relations with the composite field :jajb :(z) are the same (see (14)).
U(N)-supersymmetry. The U(N) case is more subtle. We chose generators of the SU(N) subalgebra (AU)if (p = 1, ..., N 2 - 1) in hermitian form. The U(1) generator A ° is proportional to the unit matrix (A ° =hi), other generators are normalized by the conditions (Av)i/(AV) kl = 6il5 ]k - (1/N)fif6 kl.
(16)
The U(N) algebra possesses a discrete symmetry complex conjugation that transforms the fields into each other:
where g00 = kO,gOv = O,gUU = k6UV and fabc is zero if one Of the indices is zero. The expression for the projector M/~ will be also different (see (3))M~v (A v~if M ij ~ 6 if M i/-~ (A(UAv))if - ~6if6 uv, The J ~ O0 ' 14v explicit calculations yield M~)]0 = h - 2 ( 1 -
l/N)(½ - 1/N)8 ij,
Mio] = - h - 1 (1 _ l / N ) (A v)if, MOu = (A(UAV))if _ 16i/suv"
(21)
The essential commutator relations are the following: [L n L m ] = (n - m ) L n + m + ~ c ( n 3 -
n)fn,_m
,
[bn S~mI = -"~Jan+~, [L n S mi ] -- ( ~1n - m ) S i +nm
'
T(z) * T(z), S~(z) * ~i(z),
[L n gim I = (12 n
~i(z) * Si(z).
tg~ Sbml = - - ~r"bcgcl ' l + m + "g~%
Let us define the additional operators Qin/
[S i S/m] = . . . - (n - m)Qff+ m.
- m)Stn+rn ,
(17)
[ j a sire ] = ( A a ) g s f n + m
,
,-
m (22)
Then the currents could be expressed in terms of Qin]: 287
Volume 174, number 3
PHYSICS LETTERS B
G sg] = - ( A ° )l Sn+m ' [s il S/m] = Ln+ m 5 '7 + a(n - m) (Aa~i/ J na + Dl ", l + 7MiaJbxab n-m + l d ( n 2 -
~)6i]6n ,- m ' (22 cont'd)
10 July 1986
as happens for the case of the Virasoro algebra [4], the super-Virasoro extension with N = 1 ( N e v e u Schwarz algebra and Ramond algebra) [5], and for the case of a pure current algebra [6]. Some new string models may arise also. We think that it is important to investigate all these questions.
where
gab = ksab,
h = i[(N - 2)k] 1/2 [(N + 2k)N] -1/2,
1 7=N+k-
1'
N + 2k 1 a=N+k1 2'
N + 2k d=2kN+k1' C =-
2 3k(N+2k)+(N- 1 ) [ k ( N - 1)+ 1] N+k1 3
k is arbitrary.
Conclusions. Though we found new supersymmetry algebras we have n o t found any representation. To find them, we hope, is equivalent to connect these algebras with physics. The reducible representations, if they exist, may give a new set of statistical models,
288
The authors would like to thank A.B. Zamolodchikov (the elder) for useful advice.
References [1 ] M. Ademollo et al., Phys. Lett. B 62 (1976) 105; Nucl. Phys. B i l l (1976) 77; B144 (1976) 297, [2] E.K. Sklianin, Funkt. Anal. 17 (1983) 43. [3] A.B. Zamolodchikov, Teor, Mat. Fiz. (1984), to be pubfished. [4] A.A. Belavin, A.B. Zamolodchikov and A.M. Polyakov, Nucl. Phys. B241 (1984) 333. [5] M. Bershadsky, V. Knizhnik and M. Teitelman, Phys. Lett. B 151 (1985) 31; D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. B 151 (1985) 35. [6]V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83.
SUPERCONFORMAL
PHYSICS LETTERS B
10 July 1986
A L G E B R A S IN T W O D I M E N S I O N S W I T H A R B I T R A R Y N
M. B E R S H A D S K Y
Landau Institute for Theoretical Physics, 117334 Moscow, USSR Received 26 February 1986
Superconformal algebras in two dimensions are constructed for arbitrary N. The arising algebras are not Lie algebras but they are associated algebras and they possess a central extension. Local supersymmetry is generated by multiplet of spin-currents in fundamental representation SO(N) or U(N).
Introduction. Super extensions o f the conformal algebra were constructed by Ademollo and others [1 ]. F o r the O(N) symmetry these extensions demand 2N- [1-N(N1)/2] subcanonical charges(with as the only exception the SU(2) subalgebra of 0(4)). For N ~> 5 these extensions also do not possess a central extension. In this article we present super extensions of the conformal algebra with arbitrary N that are not Lie algebras in the ordinary sense in contrast to the construction o f ref. [1 ]. These algebras also allow for central extensions. These extensions are associated algebras (in other words they possess the Jacobi identity), but the commutator of two generators of the algebra cannot be expressed as a linear combination o f initial generators. In our case one has to add bilinear combinations of generators to the Lie bracket [X i X j] = fi) k X k -- gik) l x k x l,
generated by the energy tensor T(z) (see ref. [4]), local color SO(N) (or U(N)) symmetry generated by the currentsJa(z) (a = 1 , 2 .... (N 2 - N)/2 for SO(N) and a = 0, 1,2 .... (N 2 - 1) for U(N)) and local supersymmetry generated by a multiplet of spin-currents Si(z) in the fundamental representation SO(N) (or U(N)). The dimensions o f the fields T(z),Ja(z) and Si(z) are 2, 1 , 3 / 2 , respectively. We will assume that these fields satisfy an operator algebra that contains in the operator product expansion o f two spin-currents S i the normal ordered square of currents ja :
Si(z)S/(w) = ~0"(d/2) (z - w) - 3 + ~(Aa)iO~(w)
(z - w) -2
+ IT(w)6 i / _ (a/2) ( A a ) i f j a ( w )
(1)
where the X i are the initial set o f generators. To be logical one has to add operators x i f = gikJ l X k X l to the initial set of generators and consider them to be independent generators. Thus there arise multilinear combinations of initial generators X i. Similar algebras arose in refs. [2,3]. The idea that it could be possible to construct super extensions of the conformal algebra in this way is due to Zamolodchikov (the elder) [3].
SO(N)-supersymmetry. The theories that are going to be constructed possess ordinary conformal symmetry
+ .),Mij : j a j b : (w)] (z - w) -1
(2)
where the (Aa)ii are generators of SO(N) (or U(N)), Jaw denotes the derivative of the field ja, MiJb is a special projector that one has to construct from
(Aa A b ) i/and ~ i/lab : MiaJb = [(A(aAb)) i/+ f6ab6i]].
(3)
Normal ordering is defined via an integral representation:
:AB : (z) = ~
[A(z)B(x) + A(x)B(z)] .
(4) 285
Volume 174, number 3
PHYSICS LETTERS B
The spin-current and current-current operator product expansions are assumed to be the following:
J a ( z ) j b ( w ) = --fabcjc(w)(z - w) -1 + k~ab(z - w) - 2 ,
(s) Ja(z)Si(w) = (Aa)iiSJ(w) (z
-
w) -1 .
(6)
For the U(N) case the fields S i are complex. So one must introduce independent complex conjugate fields SZ:
Ja(z)Si(w) = - (Aa)fiSi(w) (z - w) -1 .
(6')
This expression determines the transformation law for the currents ja (for the U(N) case) under a complex conjugation (ja * _ja for the hermitian matrix (Aa) i/ and ja * ja for the antihermitian matrix (Aa)i]). The operator product expansion rules with energy tensor T(z) are the following: T ( z ) X ( w ) = A X ( w ) ( z - w) - 2 + X w ( W ) ( Z -
w) - 1 ,
T(z) -- ~ L n z - n - 2 , Si(z)= ~ S i
Ja(z) = ~-jSanz-n-i,
z-n-312,
. j a j b :(z)= ~ x a b z - n - 2
tl
"
#l
Y ,ia z-n-5/2
: s i J a . (z) =
(lo)
The demand that these fields constitute an associated algebra is equivalent to the fact that the operators defined by expressions (10) satisfy the Jacobi identity. The arising equations determine unknown parameters a, 7, ~', c, d of the algebra. Now we represent the result for the SO(N) case. The generators (Aa) i! are normalized by the condition
(Aa)il(Aa) kl = 8iksi l _ 8i161k.
a -
7=
(11)
:(w) + ksab(z - w) - 4 , (7)
ja(z ) : j b j c :(w) = 2 f ad(b :jc)jd : (w)(z - w) -1
2k + 4 - N 4k - 4 N + 12 '
(12a)
1 2k-2N+6'
(12b)
d = - 2 k 2k + 4 - N 2k- 2N+6' N 2 - 6k-
c=~k
T(z) : j a j b " (w) = 2 : ja jb : (w) (z - w) - 2
(12c)
10
2k- 2N+6
(12d) '
~"in (3) is equal to one, k is arbitrary. The projector M//b has the form
Mijb = (A(aAb)) i~ + 6ab8 if.
(13)
For the case N = 2 this projector is equal to zero and we obtain the U(1) algebra of a spinning string (see ref. [ l l ) . The commutator relations are the following:
+ 2kSa(bsc)rjr(w) (z - w) - 2 +fad(hue)dr Jr(w) (z - w) - 2 ,
:SiJ a : in Laurent series:
Checking the Jacobi identity one obtains
where X(w) is one of the fields Ja(w) or Si(w); A is the dimension of the field. The arising algebra except the initial fields T(z), Ja(z), Si(z) contains an infinite set of composite operators • 1 : <]b : j a j b :, (A(ab)c)ij : sfjC :, ... (multilinear combinations of the fields S i and ja). The operator product expansions of the initial fields T(z), Ja(z), Si(z) with • j a j b : (z) can be easily found with the help of the integral representation (3):
+ (z - w ) - l ( d / d w ) : j a j b
10 July 1986
(8)
[L n Lm] = ( n si(z) :JaJ b :(w) = -2(A(a6b)c)i/ :s/J e : ( w ) ( z - w)-1
rn)Ln+ m +-~c(n 3 - n)6,,,_ m,
ja [L n jam ] _ _ m n+m' _
+ (A(aAb))iSSl(w) (z - w) - 2 .
(9)
Let us define the operators Ln, -nffa,-n'qi,"-ngab, y£a, expanding the fields T(z), Ja(z), Si(z), : j a j b :,
,1 X...(ayb)... denotes the symmetrization symbol. 286
[L
=(12
- m)S;+m,
[ja S i l = (Aa)ilS~+m '
(14)
PHYSICS LETTERSB
Volume 174, number 3
[Sin Sire] = ~i]Ln+ m + ol(n - m) (Aa)i]Jan+ m
10 July 1986
J =(l/a)tr(OnA),
~f° = ( I / 3 ) t r ( Q n I ) ,
(18,18')
where the constants a and 3 are determined by the conditions:
+ .).Mia]bxab + 1 d(rt 2 "" n+m a -- ~)(SqSn, m' ab ] = (n - m)Xn+ ab m + g k(n3 - n)sabSn ,- m' [Ln X m
t<
tL °
=s;
?/+/77
'
(19,19')
[Sin xa2] =-2(A(asb)c)ifY/nC m
So one can write the expressions for the operators
+ (. + ~)(A(~b))ifS~+~, QinJ = al(AV)if J~ + (A°)iJ J°n l ,
[da xbCl = 2fad(cxb)d + 2kn 5a(bsc)dj d m • a n+m n+m + nFad(bfc)drj r a
a
/'/+/77
(14 cont'd) "
It is also possible to construct the associated algebra without the energy tensor T(z). This algebra contains only SO(N) currents and spin-currents and it is a little simpler than we have considered:
where we introduce A ° = hi and jO = hT0 (h = (13/ N~)1/2). It turns out that c~ is real and h is imaginary. So the currents Ja(z)under complex conjugation are transformed in the following way:
s~-s,
s ° * s °.
The current-current operator product expansion has to be written in a slightly different form (compare
[jta Jbrn] -- -sfabc'lc-n+m + "knab ~n,- m '
with (5)): [Jan sire] = (Aa)ifSJn+ m , [Sin S ~ ] =~'(n -
ja (z) j b (w) = - f a b c j c ( w ) (z - w ) - 1 + gab (z - w ) - 2
m)Ja+m(Aa) ij+ ~Mif ab i ab X n+m
+ -~'d(n 2 - 1)6iJ6n_m ,
(20) (15)
where," = 2 / ( N - 2 ) ; d = 4 ( N - 3 ) ~ ; k = N - 3;~" is arbitrary and determines the scaling of spin-currents Si(z). The commutator relations with the composite field :jajb :(z) are the same (see (14)).
U(N)-supersymmetry. The U(N) case is more subtle. We chose generators of the SU(N) subalgebra (AU)if (p = 1, ..., N 2 - 1) in hermitian form. The U(1) generator A ° is proportional to the unit matrix (A ° =hi), other generators are normalized by the conditions (Av)i/(AV) kl = 6il5 ]k - (1/N)fif6 kl.
(16)
The U(N) algebra possesses a discrete symmetry complex conjugation that transforms the fields into each other:
where g00 = kO,gOv = O,gUU = k6UV and fabc is zero if one Of the indices is zero. The expression for the projector M/~ will be also different (see (3))M~v (A v~if M ij ~ 6 if M i/-~ (A(UAv))if - ~6if6 uv, The J ~ O0 ' 14v explicit calculations yield M~)]0 = h - 2 ( 1 -
l/N)(½ - 1/N)8 ij,
Mio] = - h - 1 (1 _ l / N ) (A v)if, MOu = (A(UAV))if _ 16i/suv"
(21)
The essential commutator relations are the following: [L n L m ] = (n - m ) L n + m + ~ c ( n 3 -
n)fn,_m
,
[bn S~mI = -"~Jan+~, [L n S mi ] -- ( ~1n - m ) S i +nm
'
T(z) * T(z), S~(z) * ~i(z),
[L n gim I = (12 n
~i(z) * Si(z).
tg~ Sbml = - - ~r"bcgcl ' l + m + "g~%
Let us define the additional operators Qin/
[S i S/m] = . . . - (n - m)Qff+ m.
- m)Stn+rn ,
(17)
[ j a sire ] = ( A a ) g s f n + m
,
,-
m (22)
Then the currents could be expressed in terms of Qin]: 287
Volume 174, number 3
PHYSICS LETTERS B
G sg] = - ( A ° )l Sn+m ' [s il S/m] = Ln+ m 5 '7 + a(n - m) (Aa~i/ J na + Dl ", l + 7MiaJbxab n-m + l d ( n 2 -
~)6i]6n ,- m ' (22 cont'd)
10 July 1986
as happens for the case of the Virasoro algebra [4], the super-Virasoro extension with N = 1 ( N e v e u Schwarz algebra and Ramond algebra) [5], and for the case of a pure current algebra [6]. Some new string models may arise also. We think that it is important to investigate all these questions.
where
gab = ksab,
h = i[(N - 2)k] 1/2 [(N + 2k)N] -1/2,
1 7=N+k-
1'
N + 2k 1 a=N+k1 2'
N + 2k d=2kN+k1' C =-
2 3k(N+2k)+(N- 1 ) [ k ( N - 1)+ 1] N+k1 3
k is arbitrary.
Conclusions. Though we found new supersymmetry algebras we have n o t found any representation. To find them, we hope, is equivalent to connect these algebras with physics. The reducible representations, if they exist, may give a new set of statistical models,
288
The authors would like to thank A.B. Zamolodchikov (the elder) for useful advice.
References [1 ] M. Ademollo et al., Phys. Lett. B 62 (1976) 105; Nucl. Phys. B i l l (1976) 77; B144 (1976) 297, [2] E.K. Sklianin, Funkt. Anal. 17 (1983) 43. [3] A.B. Zamolodchikov, Teor, Mat. Fiz. (1984), to be pubfished. [4] A.A. Belavin, A.B. Zamolodchikov and A.M. Polyakov, Nucl. Phys. B241 (1984) 333. [5] M. Bershadsky, V. Knizhnik and M. Teitelman, Phys. Lett. B 151 (1985) 31; D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. B 151 (1985) 35. [6]V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83.