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Bosonization and multiloop calculations for the Wess-Zumino-Witten model
Volume 229, number 3
PHYSICS LETTERS B
12 October 1989
BOSONIZATION AND MULTILOOP CALCULATIONS FOR THE W E S S - Z U M I N O - W I T T E N MODEL A. M O R O Z O V Institutefor Theoretical and Experimental Physics, B. Cheremushkinskaya ul., 25, 117 259 Moscow, USSR
Received 23 March 1989; revised manuscript received 5 July 1989
The free field representation of the KaY-Moody algebra is discussed.
1. In order to find multiloop correlators in t w o - d i m e n s i o n a l conformal theory it is necessary to express t h e m in terms o f c o r r e l a t o r s o f some free fields on R i e m a n n surfaces. Let me refer to this procedure as bosonization ~l. So far b o s o n i z a t i o n was investigated mostly in the case o f m i n i m a l models, which m a y be studied in terms o f free scalar fields, which take their values in a circle [2,3]. In o r d e r to deal with generic conformal theories it m a y be useful to consider the W e s s - Z u m i n o - W i t t e n m o d e l ( W Z W ) [4,5 ], since all non-trivial rational conformal theories seem to arise from it through G K O projection [ 6 ]. 2. Let me start from the simplest W Z W model, related to the K a Y - M o o d y algebra SU ( 2 ) k. Then currents m a y be represented in terms o f three free fields Z, W. ~b,X a n d W b e i n g bosonic 0- and 1-differentials, and ~ the scalar field, which takes values in a circle [ 7 ]: J+ = W ,
H=2zW+x/~q~
,
J_ =z3W+~/2qz~O+(Z-qZ)az.
(1)
The o p e r a t o r expansions are
W(z)z(o)=+l/z+...,
~ ( z ) ~ ( 0 ) = + l o g z + ....
J+ ( z ) H ( O ) = +_2J+/z+ ...,
J+ ( z ) J _ (0) = - k / z Z + H / z +
....
H ( z ) H ( O ) = + 2 k / z 2 + ....
The central charge o f the K a 6 - M o o d y algebra is equal to k = - 2 + q2, a n d the e n e r g y - m o m e n t u m tensor is 1
T w z w = 2(k+2------~: J + J - + J - J + - ½H 2 : = -
W S Z - ½ (8g}) 2 - - - 8 2 0 .
(2)
The central charge is c = 3 k / ( k + 2 ), a n d it is a sum o f Cwx= + 2 and co = 1 - 6 / q 2 . The radius o f the circle where takes values is p r o p o r t i o n a l to q. F o r integer q2 the m o d e l is a rational conformal theory and m a y be described by the action
The correlators in this theory m a y be constructed from the products o f the correlators o f the free fields 0 a n d W, Z. The correlators o f the fields 0 on an a r b i t r a r y genus surface are described, for example, in refs. [ 3,8 ]. They ~ Note that sometimes this term is used in a more general sense: when a theory is rewritten in terms of bosonic fields, which are not free (i.e. the energy-momentum tensor is not quadratic), see e.g. ref. [ 1]. However, this does not help one to calculate correlators. 239
Volume 229, number 3
PHYSICS LETTERSB
12 October 1989
may be expressed in terms of Prime bidifferentials and theta-functions Ok+2 of level k + 2 [i.e. the period matrix is multiplied by 2 (k+ 2) ]. We omit these well known formulae here. The correlators of the bosonic free fields fle), 7 ( l -s) with spins j and 1 - j respectively and with energy-momentum tensor Tpy = -jfl 07+ (J'- 1 )7 Off are slightly less trivial than the correlators for grassmannian b, c systems [ 8,9 ]. Their form is known from ref. [ 10 ]. First of all, the bosonic fields fl, y may be represented locally as [ 11 ] flo)= 0~(0) e-~, 7 (I -s)= ~(~) e+,, u being a scalar field, which takes values in a circle, and ~, t/being anticommuting fields with spins 0 and 1:
To~=~lO~, T~=-½(Ou)2-(Zj-1)O2u. Then [ 10 ] (det 0o ) ~/2(~(Xo) -.. ~(x~)rl(y~ ) ... rl(y~) exp [al U(Zl ) ] ... exp [amU(Zm) ] ) U~
I-fE(xi, yj)I~k~tE(z~,zt)~k~'Ila(zk) (2j-~)~k
II'/=oO(-x~+Ex-Ey+Eaz-(2j-1)A,)"
(4)
Here E(x, y) is the Prime bidifferential, ~(z) is a section of a trivial bundle, 0 is the ordinary (i.e. of level 1 ) theta-function on a Riemann surface; see ref. [ 10 ] for details. In the case of interest here j = 1 and fl (') = W, 7'(°)=Z.
These formulae provide one with original material, from which conformal blocks in the SU (2) k WZW model may be constructed. Conformal blocks themselves are certain linear combinations of the correlators just described (note that integrals of unit-dimension operators along non-contractible circles should be included in the set of these linear combinations; in the case of genus 0 these integrals are expressed through generalized hypergeometric functions [ 2,3,5 ] ). 3. This construction may be easily generalized for the case of the WZW model with arbitrary group G or its GKO projection. The starting point for the derivation of bosonization formulae like (1) is the representation of G in the algebra of vector fields on the space G/H. Then the Z~ ( a = l, ..., dimc G / H ) are related to coordinates on G / H , and W~ ~ O/OZ~. In order to get ( 1 ) one should allow W~ and Z~ to depend on z, g and make a careful central extension. The WZW model with central charge c= kD/(k+ C v ) = D [ 1 - Cv/(k+ Cv)] itself arises for homogeneous spaces G / H of real dimension D - r (D and r being the dimension and rank of the algebra G). Then the independent free fields Z~, W~ (which still are bosonic 0- and 1-differentials) correspond to positive roots of the Lie algebra G, aeA+, while the scalar fields ~i (i= 1, ..., r), which take values in circles, correspond to Cartan generators. The current, which corresponds to the Cartan vector/t, has the simple form
H,,=
~
(~,a)z~W~+q~O0.
(5)
c~eA+
Therefore, the central charge of the current algebra is
k = - C v +q 2,
(6)
here (#, ~ ' ) C v - Z~zx+ (#, a ) ( o ~ , v), and the energy-momentum tensor of the fields Z~, W~ and ~i can be written as
1
~A ( J~J-~+J-,~J~- t~vl 7 H,~H~) : = -
Twzw= 2 ( k + C v ) : Z +
t/12
~+Z W,~0Y~-l(0q~) 2 - z p 0 q~.
(7)
The central charge c = D - D C v / ( k + Cv) is composed of Z~A+ cw~x~= + 2 ( D - r ) / 2 , ~o±pco = + 1 ( r - 1 ) for all ~, orthogonal to the vector p - ½ Z~+o~, and of the central charge 1 - 12pZ/q 2= 1 - D C v / ( k + Cv) of the field 0p, collinear to p. 240
Volume 229, number 3
PHYSICS LETTERSB
12 October 1989
The remaining currents, which are generators of the KaY-Moody algebra, J+~ for a e A + look rather complicated. For example, in the case of G = SU (3)~ (see fig. 1 ) they have the following commutation relations: for ~ + p e A (N~,p= _+ 1) ,
J a ( z ) J ~ ( O ) = N a , , s J a + J z + ... J ~ ( z ) H p ( O ) = ( o ~ , f l ) J , ~ / z + ...,
J~(z)J_,~(O) = - k / z 2 + H , ~ / z +
...,
H , ~ ( z ) H B ( O ) = + ( a , f l ) k / z 2 + ....
and look like JI=WI-az3W2,
Jz=W2,
J3=W3+bz1W2,
J - i =Z21 WI + b z , z2 W2 - b z i z 3 W3 +Z2 W3 +abzZz3 W2 + ( 2 + b - q 2) OZ, +qzl oq 0q~,
J-2 =XlX2 Wl "JrX2 W2 Jl-X2X3W3 +ax2x3 Wx -bzIX~ W3 + a b x ~ x 2 W2 + a(2-q2)x30X~ -b(2-q2)x1
0X3 + ( 3 - q 2) 0X2 +qx~x3(aoq -bo~3 )0~+ qx20~20~,
J - 3 = - a x ~ x 3 Wl + aX2X3 W2 + X~ W3 - X 2 W1 - a b x i x ~ W2 + ( 2 + a - q 2) 0X3 + qx30¢30~,
H~ =2Xl Wl +)~2 W2 -X3 W3 +qo~0~,
H= =X~ Wl +2X2 W2 +X3 W3 +qo~=0~,
H3 = -X~ Wl +X2 W2 + 2X3 W3 +qo~3 0~, a+b=l.
(8)
The energy-momentum tensor Twzw = - Wl 0Xl - W2 0)~2 - W3 0X3 - 1 ( 0 0 ± ) 2 _ 1 ( 0 0 p ) 2 _ ~
q
02Op.
(9)
The central charges are k = - 3 + q2, C = 8k/(k-t- 3 ) = 8 - 24/q 2. The three vectors oq, 0~2, o¢3 are three positive roots and have the same length ( 04 ¢x) ~/2 = , / ~ . The parameters a and b are related by a+b=l.
(10)
The concrete choice of a is inessential [eqs. (8) may be a bit simplified, if one takes a = 0 , b = 1 ~2. Eqs. (8) correspond to the space G / H = SU ( 3 ) / U ( 1 ) × U ( 1 ). Other choices of G / H would be related to GKO projections of the SU (3) model; the number of fields X~, W~ obviously diminishes in this case. All the fields Z~, W~ and ~ are free and independent; those correlators which are necessary to form conformal ~2 One can eliminate a ¢ 0 by the change of variables WI --, WI + cz3 W2, W3 --, W3 + cz~ W2, Z2---'7.2=Z2 - cglza, a-* a - c, b ~ b + c, which leaves (9) invariant.
J3; W3
X2;J_2
W2;J2
,'4
X3; J-3
Fig. 1.
241
Volume 229, number 3
PHYSICS LETTERS B
12 October 1989
blocks are described i n the b e g i n n i n g o f this paper. I n particular, the o n e loop characters are l i n e a r c o m b i n a t i o n s o f ~/D(0k+ Cv)r0D-r" However, quite as it h a p p e n s i n the case o f m i n i m a l models, the selection o f p r o p e r l i n e a r c o m b i n a t i o n s is n o t quite trivial. Details, i n c l u d i n g the case o f a r b i t r a r y simple algebras, correlators a n d character formulae, will be p u b l i s h e d elsewhere. Recently, eqs. ( 8 ) for the p a r t i c u l a r v a l u e o f b = 1 a p p e a r e d also in ref. [ 12 ]. F o r s u b s e q u e n t g e n e r a l i z a t i o n s to other s i m p l e algebras, see ref. [ 13 ]. I a m i n d e b t e d to A. G e r a s i m o v , V. D o t z e n k o , V. Fateev, V. Fock, A. Marshakov, A. M i r o n o v , G. Moore, A. Rosly, S. Shatashvili a n d A. Schwarz for i l l u m i n a t i n g discussions.
References [ 1 ] K. Li and N. Warner, CERN preprint CERN-TH 5047/88; P. Griffin and D. Nemeschansky, SLAC preprint SLAC-PUB-4666/88; E. Kiritsis, CALT preprint ( 1988 ). [2] V. Dotzenko and V. Fateev, Nucl. Phys. B 240 (1984) 312; T. Jayaraman and K. Narain, CERN preprint CERN-TIt 5166/88; G. Felder, Zurich preprint ( 1988 ). [3] A. Gerasimov et al., ITEP preprint ITEP-139/88. [4] E. Witten, Commun. Math. Phys. 92 (1984) 455. [ 5 ] V. Knizhnik and A. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [6] M. Halpern, Phys. Rev. D 4 (1971) 2398; P. Goddard, A. Kent and D. Olive, Phys. Lett. B 152 (1985) 88; G. Moore and N. Seiberg, IASSNS preprint (1989). [7] M. Wakimoto, Commun. Math. Phys. 104 (1986) 605; A. Zamolodchikov, unpublished; B. Feigin and E. Frenkel, Usp. Mat. Nauk 43 ( 1988 ) 227; V. Dotzenko and V. Fateev, ITP preprint ( 1988); A. Alekseev and S. Shatashvili, LOMI preprint LOMI E-16-88. [ 8 ] L. Alvarez-Gaum6, J. Bost, G. Moore, P. Nelson and C. Vafa, Commun. Math. Phys. 112 (1987) 503. [9] V. Knizhnik, Phys. Lett. B 180 (1986) 247. [ 10] E. Verlinde and H. Verlinde, Phys. Lett. B 192 (1987) 95; J. Atick and A. Sen, SLAC preprint SLAC-PUB-4292/88. A. Semikhatov, FIAN preprint (1989); A. Morozov, Nucl. Phys. B 303 (1988) 343. [ 11 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. 13271 (1986) 93. [ 12] M. Bershadsky and H. Ooguri, IASSNS preprint IASSNS-HEP-89/09. [ 13 ] B. Feigin and E. Frenkel, to appear; A. Gerasimov et al., ITEP preprints ITEP-89/64,70,72,74.
242
PHYSICS LETTERS B
12 October 1989
BOSONIZATION AND MULTILOOP CALCULATIONS FOR THE W E S S - Z U M I N O - W I T T E N MODEL A. M O R O Z O V Institutefor Theoretical and Experimental Physics, B. Cheremushkinskaya ul., 25, 117 259 Moscow, USSR
Received 23 March 1989; revised manuscript received 5 July 1989
The free field representation of the KaY-Moody algebra is discussed.
1. In order to find multiloop correlators in t w o - d i m e n s i o n a l conformal theory it is necessary to express t h e m in terms o f c o r r e l a t o r s o f some free fields on R i e m a n n surfaces. Let me refer to this procedure as bosonization ~l. So far b o s o n i z a t i o n was investigated mostly in the case o f m i n i m a l models, which m a y be studied in terms o f free scalar fields, which take their values in a circle [2,3]. In o r d e r to deal with generic conformal theories it m a y be useful to consider the W e s s - Z u m i n o - W i t t e n m o d e l ( W Z W ) [4,5 ], since all non-trivial rational conformal theories seem to arise from it through G K O projection [ 6 ]. 2. Let me start from the simplest W Z W model, related to the K a Y - M o o d y algebra SU ( 2 ) k. Then currents m a y be represented in terms o f three free fields Z, W. ~b,X a n d W b e i n g bosonic 0- and 1-differentials, and ~ the scalar field, which takes values in a circle [ 7 ]: J+ = W ,
H=2zW+x/~q~
,
J_ =z3W+~/2qz~O+(Z-qZ)az.
(1)
The o p e r a t o r expansions are
W(z)z(o)=+l/z+...,
~ ( z ) ~ ( 0 ) = + l o g z + ....
J+ ( z ) H ( O ) = +_2J+/z+ ...,
J+ ( z ) J _ (0) = - k / z Z + H / z +
....
H ( z ) H ( O ) = + 2 k / z 2 + ....
The central charge o f the K a 6 - M o o d y algebra is equal to k = - 2 + q2, a n d the e n e r g y - m o m e n t u m tensor is 1
T w z w = 2(k+2------~: J + J - + J - J + - ½H 2 : = -
W S Z - ½ (8g}) 2 - - - 8 2 0 .
(2)
The central charge is c = 3 k / ( k + 2 ), a n d it is a sum o f Cwx= + 2 and co = 1 - 6 / q 2 . The radius o f the circle where takes values is p r o p o r t i o n a l to q. F o r integer q2 the m o d e l is a rational conformal theory and m a y be described by the action
The correlators in this theory m a y be constructed from the products o f the correlators o f the free fields 0 a n d W, Z. The correlators o f the fields 0 on an a r b i t r a r y genus surface are described, for example, in refs. [ 3,8 ]. They ~ Note that sometimes this term is used in a more general sense: when a theory is rewritten in terms of bosonic fields, which are not free (i.e. the energy-momentum tensor is not quadratic), see e.g. ref. [ 1]. However, this does not help one to calculate correlators. 239
Volume 229, number 3
PHYSICS LETTERSB
12 October 1989
may be expressed in terms of Prime bidifferentials and theta-functions Ok+2 of level k + 2 [i.e. the period matrix is multiplied by 2 (k+ 2) ]. We omit these well known formulae here. The correlators of the bosonic free fields fle), 7 ( l -s) with spins j and 1 - j respectively and with energy-momentum tensor Tpy = -jfl 07+ (J'- 1 )7 Off are slightly less trivial than the correlators for grassmannian b, c systems [ 8,9 ]. Their form is known from ref. [ 10 ]. First of all, the bosonic fields fl, y may be represented locally as [ 11 ] flo)= 0~(0) e-~, 7 (I -s)= ~(~) e+,, u being a scalar field, which takes values in a circle, and ~, t/being anticommuting fields with spins 0 and 1:
To~=~lO~, T~=-½(Ou)2-(Zj-1)O2u. Then [ 10 ] (det 0o ) ~/2(~(Xo) -.. ~(x~)rl(y~ ) ... rl(y~) exp [al U(Zl ) ] ... exp [amU(Zm) ] ) U~
I-fE(xi, yj)I~k~tE(z~,zt)~k~'Ila(zk) (2j-~)~k
II'/=oO(-x~+Ex-Ey+Eaz-(2j-1)A,)"
(4)
Here E(x, y) is the Prime bidifferential, ~(z) is a section of a trivial bundle, 0 is the ordinary (i.e. of level 1 ) theta-function on a Riemann surface; see ref. [ 10 ] for details. In the case of interest here j = 1 and fl (') = W, 7'(°)=Z.
These formulae provide one with original material, from which conformal blocks in the SU (2) k WZW model may be constructed. Conformal blocks themselves are certain linear combinations of the correlators just described (note that integrals of unit-dimension operators along non-contractible circles should be included in the set of these linear combinations; in the case of genus 0 these integrals are expressed through generalized hypergeometric functions [ 2,3,5 ] ). 3. This construction may be easily generalized for the case of the WZW model with arbitrary group G or its GKO projection. The starting point for the derivation of bosonization formulae like (1) is the representation of G in the algebra of vector fields on the space G/H. Then the Z~ ( a = l, ..., dimc G / H ) are related to coordinates on G / H , and W~ ~ O/OZ~. In order to get ( 1 ) one should allow W~ and Z~ to depend on z, g and make a careful central extension. The WZW model with central charge c= kD/(k+ C v ) = D [ 1 - Cv/(k+ Cv)] itself arises for homogeneous spaces G / H of real dimension D - r (D and r being the dimension and rank of the algebra G). Then the independent free fields Z~, W~ (which still are bosonic 0- and 1-differentials) correspond to positive roots of the Lie algebra G, aeA+, while the scalar fields ~i (i= 1, ..., r), which take values in circles, correspond to Cartan generators. The current, which corresponds to the Cartan vector/t, has the simple form
H,,=
~
(~,a)z~W~+q~O0.
(5)
c~eA+
Therefore, the central charge of the current algebra is
k = - C v +q 2,
(6)
here (#, ~ ' ) C v - Z~zx+ (#, a ) ( o ~ , v), and the energy-momentum tensor of the fields Z~, W~ and ~i can be written as
1
~A ( J~J-~+J-,~J~- t~vl 7 H,~H~) : = -
Twzw= 2 ( k + C v ) : Z +
t/12
~+Z W,~0Y~-l(0q~) 2 - z p 0 q~.
(7)
The central charge c = D - D C v / ( k + Cv) is composed of Z~A+ cw~x~= + 2 ( D - r ) / 2 , ~o±pco = + 1 ( r - 1 ) for all ~, orthogonal to the vector p - ½ Z~+o~, and of the central charge 1 - 12pZ/q 2= 1 - D C v / ( k + Cv) of the field 0p, collinear to p. 240
Volume 229, number 3
PHYSICS LETTERSB
12 October 1989
The remaining currents, which are generators of the KaY-Moody algebra, J+~ for a e A + look rather complicated. For example, in the case of G = SU (3)~ (see fig. 1 ) they have the following commutation relations: for ~ + p e A (N~,p= _+ 1) ,
J a ( z ) J ~ ( O ) = N a , , s J a + J z + ... J ~ ( z ) H p ( O ) = ( o ~ , f l ) J , ~ / z + ...,
J~(z)J_,~(O) = - k / z 2 + H , ~ / z +
...,
H , ~ ( z ) H B ( O ) = + ( a , f l ) k / z 2 + ....
and look like JI=WI-az3W2,
Jz=W2,
J3=W3+bz1W2,
J - i =Z21 WI + b z , z2 W2 - b z i z 3 W3 +Z2 W3 +abzZz3 W2 + ( 2 + b - q 2) OZ, +qzl oq 0q~,
J-2 =XlX2 Wl "JrX2 W2 Jl-X2X3W3 +ax2x3 Wx -bzIX~ W3 + a b x ~ x 2 W2 + a(2-q2)x30X~ -b(2-q2)x1
0X3 + ( 3 - q 2) 0X2 +qx~x3(aoq -bo~3 )0~+ qx20~20~,
J - 3 = - a x ~ x 3 Wl + aX2X3 W2 + X~ W3 - X 2 W1 - a b x i x ~ W2 + ( 2 + a - q 2) 0X3 + qx30¢30~,
H~ =2Xl Wl +)~2 W2 -X3 W3 +qo~0~,
H= =X~ Wl +2X2 W2 +X3 W3 +qo~=0~,
H3 = -X~ Wl +X2 W2 + 2X3 W3 +qo~3 0~, a+b=l.
(8)
The energy-momentum tensor Twzw = - Wl 0Xl - W2 0)~2 - W3 0X3 - 1 ( 0 0 ± ) 2 _ 1 ( 0 0 p ) 2 _ ~
q
02Op.
(9)
The central charges are k = - 3 + q2, C = 8k/(k-t- 3 ) = 8 - 24/q 2. The three vectors oq, 0~2, o¢3 are three positive roots and have the same length ( 04 ¢x) ~/2 = , / ~ . The parameters a and b are related by a+b=l.
(10)
The concrete choice of a is inessential [eqs. (8) may be a bit simplified, if one takes a = 0 , b = 1 ~2. Eqs. (8) correspond to the space G / H = SU ( 3 ) / U ( 1 ) × U ( 1 ). Other choices of G / H would be related to GKO projections of the SU (3) model; the number of fields X~, W~ obviously diminishes in this case. All the fields Z~, W~ and ~ are free and independent; those correlators which are necessary to form conformal ~2 One can eliminate a ¢ 0 by the change of variables WI --, WI + cz3 W2, W3 --, W3 + cz~ W2, Z2---'7.2=Z2 - cglza, a-* a - c, b ~ b + c, which leaves (9) invariant.
J3; W3
X2;J_2
W2;J2
,'4
X3; J-3
Fig. 1.
241
Volume 229, number 3
PHYSICS LETTERS B
12 October 1989
blocks are described i n the b e g i n n i n g o f this paper. I n particular, the o n e loop characters are l i n e a r c o m b i n a t i o n s o f ~/D(0k+ Cv)r0D-r" However, quite as it h a p p e n s i n the case o f m i n i m a l models, the selection o f p r o p e r l i n e a r c o m b i n a t i o n s is n o t quite trivial. Details, i n c l u d i n g the case o f a r b i t r a r y simple algebras, correlators a n d character formulae, will be p u b l i s h e d elsewhere. Recently, eqs. ( 8 ) for the p a r t i c u l a r v a l u e o f b = 1 a p p e a r e d also in ref. [ 12 ]. F o r s u b s e q u e n t g e n e r a l i z a t i o n s to other s i m p l e algebras, see ref. [ 13 ]. I a m i n d e b t e d to A. G e r a s i m o v , V. D o t z e n k o , V. Fateev, V. Fock, A. Marshakov, A. M i r o n o v , G. Moore, A. Rosly, S. Shatashvili a n d A. Schwarz for i l l u m i n a t i n g discussions.
References [ 1 ] K. Li and N. Warner, CERN preprint CERN-TH 5047/88; P. Griffin and D. Nemeschansky, SLAC preprint SLAC-PUB-4666/88; E. Kiritsis, CALT preprint ( 1988 ). [2] V. Dotzenko and V. Fateev, Nucl. Phys. B 240 (1984) 312; T. Jayaraman and K. Narain, CERN preprint CERN-TIt 5166/88; G. Felder, Zurich preprint ( 1988 ). [3] A. Gerasimov et al., ITEP preprint ITEP-139/88. [4] E. Witten, Commun. Math. Phys. 92 (1984) 455. [ 5 ] V. Knizhnik and A. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [6] M. Halpern, Phys. Rev. D 4 (1971) 2398; P. Goddard, A. Kent and D. Olive, Phys. Lett. B 152 (1985) 88; G. Moore and N. Seiberg, IASSNS preprint (1989). [7] M. Wakimoto, Commun. Math. Phys. 104 (1986) 605; A. Zamolodchikov, unpublished; B. Feigin and E. Frenkel, Usp. Mat. Nauk 43 ( 1988 ) 227; V. Dotzenko and V. Fateev, ITP preprint ( 1988); A. Alekseev and S. Shatashvili, LOMI preprint LOMI E-16-88. [ 8 ] L. Alvarez-Gaum6, J. Bost, G. Moore, P. Nelson and C. Vafa, Commun. Math. Phys. 112 (1987) 503. [9] V. Knizhnik, Phys. Lett. B 180 (1986) 247. [ 10] E. Verlinde and H. Verlinde, Phys. Lett. B 192 (1987) 95; J. Atick and A. Sen, SLAC preprint SLAC-PUB-4292/88. A. Semikhatov, FIAN preprint (1989); A. Morozov, Nucl. Phys. B 303 (1988) 343. [ 11 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. 13271 (1986) 93. [ 12] M. Bershadsky and H. Ooguri, IASSNS preprint IASSNS-HEP-89/09. [ 13 ] B. Feigin and E. Frenkel, to appear; A. Gerasimov et al., ITEP preprints ITEP-89/64,70,72,74.
242