Superconformal ghost correlators and picture changing

Volume 219, number 4

PHYSICS LETTERS B

23 March 1989

SUPERCONFORMAL GHOST CORRELATORS AND PICTURE CHANGING Marisa B O N I N I a,b, Roberto I E N G O a b c o

.... b

and Carmen N Q l q E Z a,d

International Centre for Theoretical Physics, 1-34100 Trieste, Italy lstituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy International School for Advanced Studies (ISAS), 1-34014 Trieste, Italy Consejo Nacional de Investigaciones Cientificas y Technicas, 1033 BuenosAires, Argentina

Received 2 December 1988

We compute the correlation functions for the system of superconformal ghosts fl, 7 (2= ~), including the corresponding spin fields, on arbitrary Riemann surfaces. Using fermionization, defined as a change of variables in the functional integration, we derive and generalize previous results obtained by bosonization. As an application we study the picture changing mechanism in the Ramond sector of the superstring.

An essential ingredient in the computation o f superstring fermionic multiloop scattering amplitudes is the correlator for the commuting superconformal ghosts [ 1 ]. Although great progress has been achieved during the last year [ 2 - 4 ] a complete understanding of this system is not yet available. Other than the commuting 2 = 3 ghosts of local supersymmetry (fl, 7), the system contains ghost spin fields Sg, which are necessary in order to have spacetime fermionic vertex operators with the correct conformal dimension and operators 6(fl) which are necessary to absorb the zero modes of the//-fields. Moreover, since a general scattering amplitude may contain spacetime bosonic vertex operators with non-vanishing ghost charge, operators ~(?) also have to be considered. In this paper we study the correlation functions for this system on arbitrary Riemann surfaces. Most of the literature deals with this problem through bosonization. This approach has certainly proved to be convenient and powerful for calculations. However, on higher genus surfaces there are some conceptual points in the definition of the bosonization prescription which are not completely clear yet. Namely the insertions of the auxiliary free fermion fields ~ and q, required by the neutrality o f the path integral measure, are not what one would naively expect from the index theorem for a ( l, 0) system. In order to avoid possible subtleties we present here an alternative method, by extending the procedure called fermionization in ref. [ 5 ]. By generalizing the calculation of the v a c u u m amplitude for a 2, 1 - 2 system to the case of external lines we are able to reduce the ghost correlators to the by now well-known correlators of the chiral fermionic system ( 2 = ½) [2,3 ]. Thus, the general ghost correlator is written in terms o f theta functions o f odd and even characteristic and abelian differentials. We then use these results to study the picture changing mechanism in the R a m o n d sector. As is well known, in the presence of a spin field S g (w) (S + ( w ) ) of ghost charge - ~ ( + ½) the field fl(z) ( 7 ( z ) ) behaves like ( z - w ) - 1/2. In general it is possible to restrict the functional integral over the f l ( z ) ( ~ ( z ) ) fields to those configurations behaving like ( z - w)" by introducing the product S~ = ~ (fl (w) ) ~ (fl' (w) )... X ~ ( f f ' - ~) ( w ) ) (similarly for 7)- Our strategy will be to derive expressions for correlation functions in the presence o f the previous constraints which hold for any integer n and then analytically continue them to n = - ~, since, as we shall see, this corresponds to correlators containing spin fields. Our starting point is the bosonic correlator 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

427

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PHYSICS LETTERS B

K(y,z;{w,,x,,ua})=

fl(y)y(z)

INI 2h-2 S}'(w,)S~'(x~) I - I i=l a=l

(~(#(Ua))

23 March 1989

/ B

(1)

"

This correlator has been constructed to give a non-vanishing result, i.e. the total ghost charge equals 2 h - 2 (h being the genus o f the surface). K(y, z; {w~, x~, u~} ) can be derived from the functional average over all the ghost fields (including the fl zero modes) o f the expression

0 0 f 1..NId f ] . . . d f T ' d g ~ . . . d g T ' I - 2h--2 [ d k a ( exp ( i p y ( z ) + i q f l ( y ) OpOq i=i a=l +i

i=l j= l

f{O~7~fl(w~)+i

i=l j=l

g~OJTly(xi) + i 2

kafl(Ua)

a=l

•

Decomposing fl into a sum of its zero modes and a part orthogonal to them, fl=fl± + 2~Ca(O~, we obtain 1

det M

(2)

K= -- det @ (det N) 2 ' where @ is the matrix of the zero modes, '~ab = (Oa(Ub) and the matrices M and N are given by

P(z,y)

/~(Xl,y)

Ox,P(x,y)

...

O~IP(xt,y)

... P(XN,y)

...

O~P(XN,y)"

P(z,w, ) 0,J~(Z, Wl )

M=

O~',P(z,w, )

N

,

ff ( Z, WN ) OnN~(Z, WN)

N=

nl ~ Ox,P(xl,Wl)

•.. P(XN, Wl)

o~,~P(xN,w, ) O"~Ow,P(x~¢,w,)

P ( x I ,w I )

Ox,ff(Xl,Wl )

Ow,P(xl,w~)

Ox,Ow,P(Xl,W, )

0w,P(Xl,Wl)

0Xl0nwll/~(Xl,W l )

nN nl ~ W ~) OxNOw~P(XN,

nN~ 0w~P(Xl ' W N)

•N ~ W~) O~Ow~P(Xl,

nN nN OWNOxNP(XN, WN)

• ..

...

and/~ is the (yfl) propagator in the presence of zero modes:

2h--2 /~(z,y)=

7(z)fl(y)

I~ cJ(fl(Ua) )

a=l

/ =P(z,y)--(~-')abP(Z,

Ua)q~b(y) •

P (z, y) is the propagator (~, (z)d± ( y ) ) . Notice that our normalization convention is such that ( I-[d(fl(ua) ) ) = d e t q~-l. The determinants on the RHS of eq. (2) suggest that it is possible to express the LHS as a ratio o f correlation functions of the fermionic fl-~, system. Indeed, a simple calculation shows that eq. ( 1 ) is equivalent to

K ( y , z ; { w i , xi, u~})= 428

F) 2 ' [< i = l S , / ( xnii ) S s ( w i "' ) E a2h-2

(3)

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PHYSICS LETTERS B

23 March 1989

where F indicates that the path integral is performed over anticommuting variables and S~(z) is equal to 1/(z) 1/'(z)... 1/~n-~)(z) since the arguments of the &functions are fermionic. Although this equality is defined up to a phase depending on ni, we follow the convention that it exactly holds. Phase conventions are also needed for the fermionic correlator since in particular we shall consider the continuation to half-integer ni. The latter will be fixed in the following. Similarly, bosonic correlators involving more 1/and y fields can be computed in terms of an effective propagator

Q(y, z;{wi, xi,

Ua} ) ~-- <~)(z)1/(y) UN=I S~'(xi)anBi(wi) ]-I 2h-2 ( F[/N=1 ,, ~, =h-2

(4)

S~ (x~)S~(w,)FI~=~ 1/(u~) >~

In fact, using Q to perform the Wick contractions we obtain

K(yl,...,yt, zl,...,zl;{wi, xi, ua}) =

i:1

e(yi)~,(zA

i=1

S~(w3G(x3 I-I 6(1/(Ua a=l

1

2h-2 1/(/'/a) F {h...IM} E Q(Y,,Za;{W, =
B

Xi, Ua})...Q(YM, Zt~;{Wi, X,, Ua})

•

(5)

Therefore, our initial problem has been reduced to the calculation of the 1t7 propagator in the presence of constraints for the fermionic system, i.e. the numerator ofeq. (4), to which we shall refer, from now on, as (~. The partition function for the fermionic 1/7 system has been discussed in ref. [ 5 ]. Here we generalize that procedure to the case of more general correlation functions. The fermionization is defined as the change of variables 1/= 09g-',

7=09-1~,

(6)

where ~Vand ~0are independent ½-differentials, i.e. chiral fermions, and o9= v 2 is a holomorphic abelian differential, Vo being a ½-differential with h - 1 zeroes corresponding to an arbitrary odd characteristic. As discussed in ref. [ 5 ], o969plays the role of a metric. The correlator (5) explicitly depends on Vo; however, due to conformal invariance, physical amplitudes will be independent of Vo. Performing this change of variables in the path integral we obtain O(Y, Z;{Wi' Xi' Ua}) =

<

X ~(z)W(y)

fi i=l

09(Y) UN=I (.oni(wi) ±-ta=l]-[2h--209(Ua) 09(Z) UN=I O)"i(.~i)l-lsh~-I 1 [ / ~ ; ( r s ) ] 4 h--I

2h--2

s=l

a=l

>

ni ni W 1-I ~)(r,)q)'(rs) 1-[ ~£l(Ua) S¢(x,)S~,(,)

,

(7)

where S~, = ~rap,... ~ t , - ~ ) . The locations o f the double zeroes of o9 have been called r~ and eq. (7) depends on these points through the jacobian o f the transformation [ 5 ]. The correlator in eq. (7) can be computed using [ 2,3 ]: zi < I ~ i V q i ( Z i ) > = H Ei
~i q i z i = ~ q. i f 0 9 ,

~iq ~ = 0 ,

(8)

P where qi = + 1 ( - 1 ) for T ( ~ ) , q~= n ( - n ) for S~, ( S ~ ) and q~= - 2 for each pair ~(r~) ~' (r~) as shown in ref. [ 5 ]. Eqs. (7) and (8) together with eq. (3) fix the phase convention for K. The denominator o f eq. (4) can be computed following the same steps. The bosonic correlation function ( 5 ) has then been completely determined. In particular n~= 1 reproduces the insertion of~(fl(w~) ) O(7(xi) ). Moreover, when n = - ½, S~ = S~- and Sr _ Ss and we obtain the correlator with ghost spin fields. From eq. ( 7 ) we see that in this case we can supplement the change o f variables (6) with 429

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(9)

S g = l ) o l S ~ 1/2 , S;__~l)oS~ 1/2 ,

where S~ 1/2 ( S ~ 1/2 ) are the matter spin fields usually denoted by S - (S + ). Notice that the conformal dimensions of these operators are - ] and + ], respectively, in agreement with the fermionization, i.e. their correlation functions have opposite conformal weight. Similarly, Off and ~? have conformal dimensions - 3 and ½, opposite to those of fl and y. It m a y be seen from eqs. (3), (8) that the usual operator product expansions,

fl(z)S +-(w) ~ ( z - w) -+1/2 :fl(w)S_+ ( w ) : ,

y ( z ) S +-(w) ~ ( z - w)* 1/2: ~,(w)S +_( w ) : ,

&(z)&(w) ~ (z-w),

are obtained. In conclusion, the most general correlator can be computed using fermionization, fl(x)y(y)

i=1

S+(z,)S;(wi)

j=l

~(fl(ts))6(~'(ss)) I~ d ( f l ( u , ) ) a=l

B

~o(x) f i co(wi) '/2 1UI ~o(ss) l~):~v'o(r,) 4 -- o ) ( y )

i=1 ~----~i) 1"~]=~

09(tj) ]-[2h----_--12O,)(Ua)

× ( ~ ( y ) ~ ( x ) I-[~, ~(si) ~(t,) I-IN=~ S + (zj)S- (wj)~]~52 ~(ua)Fihs-_? ~(r~) ~' ( r , ) ) ~M [ ( i=1 ~(Si)~(t,)~N=,S+(zj)S-(wj)l-[2ah=-~2~(Ua)l-[h--1 ~ t ( r , ) ~ ' ( r , ) ) ] 2 '

(10)

and eq. (8). [The generalization to a correlator with N ÷ (N - ) ghost spin fields S~- ( S g ) and M ÷ ( M - ) d(fl) ( d ( y ) ) satisfying ½( N ÷ - N - ) + M ÷ - M - = 0 is straightforward. ] As an example we explicitly give the result for the correlator fl(Z)~(y)

~-I Sg+ (wi)Sg (Xi) ~I a(fl(Ua) )

i=1

a=l

B

1 l 2h-2 Ua--2 ~s=l h-1 rs) O)(Z) f i Po(Xi) ~ hs-=ll Pfo(rs)40m(Z--Y--~Xi't-i~Wi-t-~a=l 1 1]o(Wi) ]7"2h-2 (/)(b/a) i " ~ ' ~ 2 h ~ ' 2 Z ' h ~ l '~.... l [Om(--'~Xi'~-~.Widl-~a=l Ua--2Zs=, -r,)] E(y,z)

-- o ) ( y ) i=1

E(z, wi)l/2E(y, xi)1/22~72 E(z, u~) × ~E(z,x,)~/~E(y, ~ 11 - - - •

×

w,)

I~ijE(wi' XJ) 1/4

a=~ E(y,

rs) 2 hill E ( y ,

Ua) ,=,

E(Z,

r~) 2

FI l - ' l E ( xUa) i ' 1/2

E(wi, rs)

]-~a,sE(Ua, rs) 2

I~i
(11)

It is possible to check that this equation reproduces the result given by Atick and Sen in ref. [ 3 ] in the particular case when the insertions ~ (fl( u~ ) ) are taken in pairs at the double zeroes of o9, i.e. setting u~ = r~+ ~, Uh- 1 + ~ = r~ - - e for a = 1, ..., h - 1 and taking the limit e ~ 0 . Now we want to use these correlators to relate the + 1 and - ½ ghost charge representation vertices of the massless R a m o n d states. In the R a m o n d sector, the vertex operator for the lowest energy state contains matter as well as ghost spin fields. There are in general two different representations of the vertex with opposite ghost charge ( + ½) [ 1 ]. As has been shown by Knizhnik [ 6 ] one can compute the vertex function from the path integral over an infinite cylinder attached to the surface. In this way the insertion of a R a m o n d state with ghost charge - ½ is obtained. The corresponding vertex operator is given by

V_ ~/2 = u'~S,~S; exp(ip.x) , where S , is the SO (10) spin field and the polarization u '~ is a Majorana spinor satisfying the Dirac equation /~u = 0. A scattering amplitude with external R a m o n d states is therefore given by a path integral over the ghost 430

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and matter fields on a surface with punctures of these V 1/2 vertices. As is well known on a genus h surface with N R a m o n d punctures there are 2 h - 2 + ½N zero modes of the fl field and therefore the measure must contain 2 h - 2 + ½N supercurrent and O(fl) insertions. We shall show that it is possible to remove the contribution to the measure due to the punctures by changing the representation of half of the vertices to the + ½picture and taking the path integral over a surface without punctures. Let us now see how the effect of one puncture can be removed. Due to the arbitrariness in the choice of the superbeltrami differentials [2 ], we take one of them to be ye(z) = ~ ( z , w) and then consider the limit w--,y, where y is the position of the puncture. In this case one has to compute the following correlation function:

/

1~ O((za, fl))O((XP, fl)) l--[ (lti, b) 1~ (Za, J)

a=l

i=1

a=l

[

(ZP,J)(ltP, b) + -~y,fl

c(y)V_~/2(y)...

/ (12)

where the dots stand for all the other external vertices V 1/2 and the rest of the measure associated to the other punctures. Z a, a = 1, ..., 2 h - 2, and Z P are the superbeltrami differentials and J is the supercurrent

J = ½0x ~ + ½by- c0/~- ~0c p. Following ref. [ 7 ] we associate to a puncture a Beltrami differential and therefore the measure will contain a factor (/t v, b). The Beltrami differential #P is generated by the vector field V, ,-e ,ez = rae -v, z where V is different from zero only in a neighborhood around the puncture, i.e. the position o f the vertex. In order to have the correct measure on the moduli space o f a surface with punctures we must also introduce a c field on the punctures. We have also considered the term coming from the dependence o f z e on the position of the puncture y. Z a are instead considered moduli independent (i.e. located at some moduli independent points ua on the Riemann surface). First we consider the terms containing (Z p, Jmatt), where Jmatt = ½ 0x'u ~[.Lu.The matter part o f the correlator ( 12 ) contains then

½u°'( ~,(w)Ox~'(w) So,(y) exp[ipx(y) ]...) , where the dots stand for the other supercurrent insertions according to the genus of the surface. In the limit w-+y this correlator is reproduced by [ 8,9 ]

fuSE(w, y) -1/2([y~aOxU(y) + ~iy~PpU~U(y) 7j,(y) ]Sa(y ) exp[ipx(y) ]...)

(13)

(more singular terms do not contribute by the condition on the polarization u~k= 0). This expression is still multiplied by a factor containing the ghost correlator, which can be computed using eq. (10), namely

(2 2

H (~(fl(Ua))(~(fl(w))Sg(y) ~Ii Vqt(Xi)

a=l

"

)

B

co(y),/2 yih_, s=~ v3(rD 4 h-I r~+ Z, qixi) S'2h-2 ~a=l Ua--2 Z s=l

= O) (W)]-I a= 2h--2 10)(Ua)Om(W--ly÷

xg(w'y)l/2

2h--2 a:lH

E(y, Ua)1/2 hill E(w, r~) 2 ~ E(y, Xi) q'/2 E(W, Ua) ~:, E(y, r~) lil ~ F,

where F is independent o f w and y. Va, represent the rest o f the measure and remaining vertices (for simplicity we have considered the case when they are only in the form of"constraints", i.e. Off, Oy or S + ). In the limit w ~ y the zero coming from the factor E(w, y) ~/2 cancels with the same factor in ( 13 ) and the rest is reproduced by the correlator ( 1-[~( ()~a fl) )S~l- ( y ) . . . ) B . Therefore, the insertion (Z e, Jm~tt) d( (Z P, fl) ) can be eliminated by replacing V ~/2 by 431

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½u~ygP( 0x~'+ ~ip ~ % u " ) S~ exp (ipx)S + .

(14)

(Notice that the insertion of (/t e, b) c (y) in (12) is equivalent to the insertion of the identity operator [7 ]. ) Next we consider the ghost part of the supercurrent, Jgh= ½by-cOil- 3 (OC)fl, inside (Z P, J). To remove some ambiguous singularities arising in the computation of ( 12 ), we choose a gaussian representation for the ~-functions Z~ a n d z e [ 10]: Za(z)=a -lexp(-a-llz-u~l

2),

XP(z)=a -lexp(-a-llz-w4

2).

Due to the bc ghost charge conservation the product 17I(Z a, J) must contain an odd number of ghost supercurrents. Let us consider for instance the case in which we have just one (Z a, Jgh) (the rest being matter supercurrents). Then the correlator (12) reduces to ½ J" dv, dv2 {[Z~(v2)zP(v, )-Z~(V, )zP(v2)l f dx O~VP(x)[0~,K(v,, v2; {y, ...} )R2(v2, X]Vl, y) + ~ K ( v I , V 2 , { y .... } ) O v t R 2 ( x ' v 2 ; V l ' y ) ] + z a ( v 2 )

OW Oy OZe(V, O ~ ) K(Vl, v2;{y, ...} )R~ (v2;y) }

(15)

where 3h--3

/

Rn(wl .... , Wn;Ul, ..., Un)= b(Wl)...b(w~)C(Ul)...C(U~) I~ (l ti, b) i=1

f 3h-3

=

i~=l dxilt'(xi)B(wl ..... W,,Ul,...,U~,Xl,...,X3h_3)

and

"H(wl,ul)

...

: B ( w~,...,Wn,Ul,...,Un,Xl,...,x3h_3 ) ~-~det H(Wn,Ul)

H(Wl,U,) i

... ...

n(X3h_3,Ul )

n(Wn,Un)

fl (Wl) " fl(Wn)

... f3h_3(Wl) : ... f3h_3(Wn)

H(X3h_3,Un)

fl (X3h-- 3)

... f3h_3(X3h_3)

H(w, u) and f are the ( b c ) propagator and the basis of the quadratic holomorphic differentials, respectively [111. In ( 15 ) K is the contribution of the fly system, namely K ( v ~ , v2; {y.... } ) = ( ~ ( (X P, J~) )~(Vl )Y(v2)Sg (Y)...>B =

f d t z e ( t ) (y(v2)fl(vl)fl(t)Sg (Y)..-)F [fdtze(t)(fl(t)S~(y)...)F]2 '

(16)

which generalizes the relation (3) between the bosonic and fermionic representations of the correlator when there is a ~( (Z e, fl) ) insertion. In the limit w-~y the term proportional to Z a (vz)ZP(v~) in (14) cancels with the last term after replacing OZe/ 0w= - (OZe/Ov~) and integrating by parts. (Notice that the integration over x sends R2 to --R 1. ) The remaining terms are

½ fdvl dv2/~a(v I )zP(v2) [OviK(Vl, VE;{y, ...} )RI (VE;Vl) -[- 3K(v~, v2; {y .... })Or,R1 (v2;vl) ] •

(17)

The integral over Za (v~) keeps Vl away from y. By using eqs. (10), ( 8 ) to express K in terms of prime forms the relevant part of (17) is 432

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m(t)o)(v~) ) Om(t_Y2 d_l)l_½y_{- ~ q~x~) ½f dt dv2 (zP(t)zP(v2)Rt(v2' v') ~(v2)vo(y E(v2, y)'/2 E(Vl, t) E(t, x~)qiE(v,, xi) q` _'~ × E(t, y)'/ZE(v2, t) E(v2, ~)-Eiv-~, y ) , / 2 E(v2, xi)qiE(y, xi) qi/21 ~ ) ×[ f d t z e ( t ) ~ O m ( t - ½ Y

+ ~xiqi)E(t,y)-l/2E(t,x~)qeE(y, xi)-q'/2F] -2 ,

where xi stands for u~ and rs, i.e. from the terms coming from the measure and the j a c o b i a n o f the t r a n s f o r m a t i o n ( 6 ) , ( 9 ) , and F i s a function i n d e p e n d e n t o f v2, y a n d t. R, (v2, v~ ) is the correlator. It is easy to see that in the limit w~y this expression is finite and is given by - ~R, (y, v~ ) °')(u I ) Om(Ul - uo(y) 3

½y+ Y~iqixi)E(v,, y)-I/2E(yl, xi)q'E(y, xi)-q'/zF [Om(½y+ •i q~x~)E(y, xi)qJ2F]2

(18)

The contribution to the limit from the terms in ( 17 ) is o b t a i n e d by taking the a p p r o p r i a t e c o m b i n a t i o n o f derivatives with respect to vt o f ( 18 ). By using the R H S o f eq. ( 11 ), one can see that the same result can be o b t a i n e d by c o m p u t i n g the correlator

( (Xa, Jgh)¼b(y) wS + ( y ) :

...7.

Therefore we have shown that the terms in eq. (12) due to the presence o f a puncture are r e p r o d u c e d by

2h--2

3h--3

2h--2

I-I s( (z o,/~) ) VI (u,, b) VI (z °, J )

a=l

i~l

/ v,/~(y)...

,

a=l

where V,/2 -- !,,-~. t~-r~'~:( 0 x ~ + ¼iP"TJ~ ~ ' ) S a +

½S, bT] exp ( i p x )

S~-: ,

(19)

which is precisely the vertex in the + ½ representation. Friedan, M a r t i n e c and Shenker [ t ] found the same expression by applying the BRST charge to V_ 1/2 and inserting a ~ factor which is necessary to absorb the zero m o d e o f the r/~ system, i n t r o d u c e d in the b o s o n i z a t i o n o f the ghost fields. Repeating this process for all the N punctures one obtains the scattering o f N / 2 fermionic states in the + ½ representation with N / 2 states in the - 1 picture 0n a surface without punctures. The same analysis can be followed in the N e v e u - S c h w a r z sector. Starting with the - 1 representation of, for instance, the massless vertex, i.e. W = E~~u exp ( i p x ) 5 (~), a n d removing the corresponding (Ze, J) insertion by taking the limit w--.y, we obtain the vertex with zero ghost charge, V= eu(OxU+ ip. ~ ~ ) exp ( i p x ) . One can see that in this case the ghost part o f the supercurrent does not contribute to the picture changing.

References [ 1] D. Friedan, S. Shenker and E. Martinec, Phys. Lett. B 160 ( 1985 ) 55; Nucl. Phys. B 271 ( 1986 ) 93. [2] E. Verlinde and H. Verlinde, Phys. Lett. B 192 (1987) 95. [3] J.J. Atick and A. Sen, Phys. Lett. B 186 (1987) 339. [4] C. Preitsehopf, Superconformal ghosts: chiral bosonization at genus 0 and 1, preprint UMD-EPP-88-206 (1988). [ 5 ] R. Iengo and B. Ivanov, Phys. Lett. B 203 ( 1988 ) 89. [6] V.G. Knizhnik, Phys. Len. B 160 (1985) 403. [7] S.B. Giddings and E. Martinet, Nucl. Phys. B 278 (1986) 91; E. Martinec, Nucl. Phys. B 281 (1987) 157; E. D'Hoker and S.B. Giddings, Nucl. Phys. B 291 ( 1987 ) 90. 433

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[8] J.J. Atick and A. Sen, Nucl. Phys. B 293 (1987) 317. [ 9 ] G. Aldazabal, M. Bonini and C. Nfifiez, Covariant superstring fermionic amplitudes, vertex operators and picture changing, preprint IC/88 (1988), Nucl. Phys. B, to be published. [ 10] E. Gava, R. Iengo and G. Sotkov, Phys. Lett. B 207 (1988) 283. [ 11 ] E. Verlinde and H. Verlinde, Nucl. Phys. B 288 (1987) 357; M. Bonini and R. Iengo, Intern. J. Mod. Phys. A 3 (1988) 841.

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