7
Volume 206, number 4 PHYSICS LETTERS B 2 June 1988 N-STRING, g-LOOP VERTEX FOR THE BOSONIC STRING P. DI VECCHIA, K. HORNFECK l Nordita, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark M. FRAU INFN, Sezione di Torino, Corso M. D'Azeglio 46, 1-10125 Turin, Italy and Niels Bohr Institute, Blegdamsvej 17, DK- 2100 Copenhagen 0, Denmark A. LERDA 2 Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA and S. SCIUTO 3 Dipartmento di Scienze Fisiche dell'Universit& di Napoli and INFN, Sezione di Napoli, Mostra d'Oltremare Pad 19, 1-80125 Naples, Italy Received 9 March 1988 We construct the N-string,g-loop vertex V~v.g for the orbital degrees of freedom of the bosonic string in terms of the first abelian differentials, the period matrix and the prime form. We also build the Ig) vacuum recently discussed by many people in the framework of an operator formalism on an arbitrary Riemann surface; our expression also contains the measure that takes into account the ghost contribution. The old operator formalism has been recently revived in different forms [ 1-6 ] ~ for computing the multiloop amplitudes in string theories. Following in particular the approach of ref. [ 1 ] the basic ingredients for construct- ing loop diagrams are the BRST invariant of the N-string vertex VN and the twisted propagator T. VN is con- structed in ref. [ 1 ] by sewing together BRST invariant three-string vertices [4,8 ]. As it has been shown in refs. [9,1,2] VN provides an off-shell extrapolation of the string scattering amplitudes, that on one hand reproduces the dual amplitudes at the tree level for on shell physical states and on the other hand can be used for computing multiloop diagrams. This approach allowed to write an explicit and simple expression [ 10,2 ] for the multiloop partition function and N-tachyon scattering amplitude ~2. At one loop it reproduces the well-known result. A detailed comparison of the two- and three-loop partition functions with known explicit result derived using geometrical methods [ 11 ] has been carried out by Petersen, Roland and Sidenius [ 12 ] finding complete agreement to "high" order in some moduli and exactly in others. A new operator formalism on an arbitrary Riemann surface has been recently introduced by many authors Also at University of Wuppertal, GauBstraBe20, D-5600 Wuppertal 1, Fed. Rep. Germany. 2 A Della Riccia fellow. 3 Work partially supported by the Italian Ministero dellla Pubblica Istruzione. ~ Seealso ref. [7]. ~2 It is not difficult to prove that eq. (29) of ref. [ 10 ] actually holds for any number of loops g and not only for g= 2. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division ) 643

N-string, g-loop vertex for the bosonic string

Embed Size (px)

Citation preview

Volume 206, number 4 PHYSICS LETTERS B 2 June 1988

N-STRING, g-LOOP VERTEX FOR THE BOSONIC STRING

P. DI VECCHIA, K. H O R N F E C K l Nordita, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark

M. FRAU INFN, Sezione di Torino, Corso M. D'Azeglio 46, 1-10125 Turin, Italy and Niels Bohr Institute, Blegdamsvej 17, DK- 2100 Copenhagen 0, Denmark

A. LERDA 2

Institute for Theoretical Physics, State University o f New York at Stony Brook, Stony Brook, NY 11794, USA

and

S. SCIUTO 3 Dipartmento di Scienze Fisiche dell'Universit& di Napoli and INFN, Sezione di Napoli, Mostra d'Oltremare Pad 19, 1-80125 Naples, Italy

Received 9 March 1988

We construct the N-string, g-loop vertex V~v.g for the orbital degrees of freedom of the bosonic string in terms of the first abelian differentials, the period matrix and the prime form. We also build the Ig) vacuum recently discussed by many people in the framework of an operator formalism on an arbitrary Riemann surface; our expression also contains the measure that takes into account the ghost contribution.

The old operator formalism has been recently revived in different forms [ 1-6 ] ~ for computing the multi loop ampli tudes in string theories. Following in particular the approach of ref. [ 1 ] the basic ingredients for construct- ing loop diagrams are the BRST invar iant of the N-string vertex VN and the twisted propagator T. VN is con- structed in ref. [ 1 ] by sewing together BRST invar iant three-string vertices [4,8 ]. As it has been shown in refs. [9,1,2] VN provides an off-shell extrapolation of the string scattering amplitudes, that on one hand reproduces the dual ampli tudes at the tree level for on shell physical states and on the other hand can be used for computing multi loop diagrams.

This approach allowed to write an explicit and simple expression [ 10,2 ] for the multi loop part i t ion function and N-tachyon scattering ampli tude ~2. At one loop it reproduces the well-known result. A detailed comparison of the two- and three-loop part i t ion functions with known explicit result derived using geometrical methods [ 11 ] has been carried out by Petersen, Roland and Sidenius [ 12 ] f inding complete agreement to "high" order in some moduli and exactly in others.

A new operator formalism on an arbitrary R iemann surface has been recently introduced by many authors

Also at University of Wuppertal, GauBstraBe 20, D-5600 Wuppertal 1, Fed. Rep. Germany. 2 A Della Riccia fellow. 3 Work partially supported by the Italian Ministero dellla Pubblica Istruzione. ~ Seealso ref. [7]. ~2 It is not difficult to prove that eq. (29) of ref. [ 10 ] actually holds for any number of loops g and not only for g= 2.

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Divis ion )

643

Volume 206, number 4 PHYSICS LETTERS B 2 June 1988

[ 13-16 ]. In this approach multiloop amplitudes are constructed by computing a product o f vertex operators between a normal vacuum and a new "vacuum", that is called the g-vacuum tg) . The g-vacuum for free fer- mions and bosons has been explicitly constructed [ 15,17,18 ] using the fact that it is annihilated by an infinite set of conserved charges.

Similar kinds of equations [ 4,5,19 ] are also satisfied by the N-string vertex. In this letter we construct the N-string, g-loop vertex Vs.g for the orbital degrees of freedom of the bosonic

string, that is the extension to an arbitrary number o f loops of the one written for the tree diagrams in refs. [ 20,9 ], in terms of the first abelian differentials, the period matrix and the prime form. It has the property of reproducing the multiloop scattering amplitudes involving any number of physical states. The analogous con- struction for free fermions has been performed by Pezzella [ 21 ].

Then we will discuss the relationship between the g-loop tadpole operator Vl.g and the g-vacuum discussed in detail in refs. [ 15,17,18 ].

The basic ingredients of our construction are the BRST invariant twisted propagator T and ( N + 2g)-string vertex (constructed by sewing together BRST invariant three-string vertices [ 3,8 ] ) discussed in sections 2 and 5 respectively of ref. [ 1 ]. The integrand of VN+ 2g is a function of N + 2g Koba-Nielsen variables zi. VN.g is then obtained by sewing together 2g legs of the VN+ 2g after having inserted a twisted propagator following the proce- dure explained in ref. [ 10 ] ~3. The ghost degrees of freedom corresponding to the N external states are then eliminated by saturating the vertex with N states I q = 1 ).

The integrand of Vn.g depends on the original ( N + 2 g ) Koba-Nielsen variables and on the g integration variables of the g propagators. The 2g Koba-Nielsen variables corresponding to 2g strings, that are sewn with each other, together with the g variables of the propagators give an explicit parametrization o f the moduli space, that is simply connected to the one in terms of the multipliers k~ and fixed points qu, ~u with/~ = 1, ..., g of the generators o f the Schottky group as explained in detail in ref. [ 10].

After performing the sewing procedure and saturating with the ghost states I q= 1 ) o f the external particles one gets the following expression for the N-string, g-loop vertex ~4:

1 E ~ Pi'a(~i)[Dno(U,)+Do~(Vi)] VN,g= dVi=I [V~(O) ] -1 ( .QI exp 2/=1,~=o

( 1~ ~ a~i)D,,,(Ui) ~ Dzs(To,)Ds,,,(Vj)a,~)) × e x p -2 ij . . . . o

V1 g (~n- -~0 0/(i)- Vi(z) f ( ~= ~ Ol(mi) m V~(;) ) 1 ×exp ~ ~ n! Oz~ t°u (2nlmz)~-~ m~ (1)

u . . . . l i=l z=o j l o ~ - . I Oy , = " y=0 zo zo

where

d V - 1-I u=, dz, f i / d k u d q u d ~ u ( 1 - ku)2"~ { ~ "~ - - ~ ~ - - S - S ~ H l q 11 (1 - k ~ ) -D f i (1 - k g ) 2 j (det Im z) -D/2 (2) d V.b~

and

<~'~1 = fii=l [i(x=O'Oal]t~(~Pi) " i , (3 ,

~3 See also refs. [22,23] for some details of the calculation. #4 Here and in the following we limit ourselves to consider the open string. The results for the open string can be easily extended to the

closed string by observing that the integrand of the N-string vertex for the closed string [ 24 ] and of the twisted propagator (see formula (5.31 ) of ref. [ 1 ] ) are the modulus square of the ones for the open string after of course having complexified the Koba-Nielsen variables and doubled the sets of oscillators apart from the momentum, that is common to the two sectors. Keeping this in mind we can see that the results given in the following refer also to the holomorphic part of the closed string.

644

Volume 206, number 4 PHYSICS LETTERS B 2 June 1988

The sum over a in the second line of eq. ( 1 ) is over all elements of the Schottky group. It does not include the identity if i =j. The sum over the indices l and s from 1 to oo is understood, r is the period matrix and o9 u are the g first abelian differentials. The projective transformations V, and U~=FV7 l are defined in ref. [ 1 ] together with the representations of the projective group Dnm (Y) corresponding to the orbital modes. We have also used the identifications cen = x/~an for n ~ 0 and a o = ao = p is the momentum. The volume d Vabc is given by

dV~ = dpadpbdpc (P, -Pb) (Pb -Pc) (P~ -Pc) ' (4)

where Pa,~,c are three out of the N + 2g fixed points t/u, ~u and Koba-Nielsen variables zi. The second exponential in eq. ( 1 ) is the "automorphized" extension of the tree diagram N-string vertex VN

as given in ref. [23]. The first two exponentials correspond to the contribution of the classical action in eq. (4.11) ofref. [2].

The last term in eq. ( 1 ) together with (det lm z) -D/2 in eq. (2) comes from the integration over the momenta circulating in the loops and is equal to the last term in eq. (4.11 ) of ref. [ 2 ].

In deriving eq. ( 1 ) we have also used the relation

Dnm(UtVj)= ~ Dn,(U~)D,n(Vj)+D~o(Ut)6,,o+Dom(V~)6o~ (5) l=1

that shows that the matrix D,m(~) is not quite a representation of the projective group. Using the following useful relations:

1 ~ (i) 1 u -.., ~ E a. D.,(U,)~ D,s(Tc,)Ds,.(Vj)a,~'+-~ ~_ 1 ~ p,.a(,/)[D,m(U,)+Do,dV,.)]

z, i~=j n,m=O -- n=O

lz = - - U n 2 i,~j . . . . 0 n! m! O'~ O" l°g E" V~'z'' ~'y" , lz=y=°a''')

1 ~ ~ 1 0 " G ( z ) l ~ = o , (6) p~ log V~(O)+ ~ p~.a~ ~' +~ i=1 i = 1 n = l Z i - - Z i _ 1

and

_ ! ~ ~ a(,/, D,a(U,) 2 D,,(T,x)D,m(V~)a~ ) 2 i= l n,m=O o ~ 1

2i~ll N ~ ^ ( i) 1 1 . m 1 E[ V,.(z), V,(y) ] ~=y=oOtg ) - n . . . . o o g '

it is possible to write VN, g in its final form:

VN'g: dV<ff2l f i [V}(O) ]P212-l ,~1 ~=,~P"a(""v/nn-~v. z,-z,_l /

( l ~ I I ) ×exp ~ ~ ^(i) logE[Vi(z), Vj(y)]lz=y=oa~)

ivaj n,m=O

i = l . . . . 0 n l m l 0 n 0 7 l o g ~ i ~ Z V i ( y )

V,(z) U(y)

[1 ~ ( ~ ~ a(~° f ) ( N ~ °t~) f o 9 " ) ] , (8) ×exp ~ T O ~ ogu (2zcimz)Gl =~lm~__O__~_.V O~ /~,v = I i = 1 n = 0 z = 0 J y = 0

zo zo

645

Volume 206, number 4 PHYSICS LETTERS B 2 June 1988

where the prime form is defined by

E ( z , y ) = ( z - y ) FI' [ z -T~(y) l [y-To~(Z)] (9) ,~#, [ z - Ta(z) ] [y - r,~(y) ] '

where H' means that T,~ and Tg i are counted only once. VN,g has the property of giving the g-loop scattering amplitude involving N physical states when its is saturated

with N transverse states [25 ]. In partieular if we saturate VN,, with N tachyon states IP;, Oa); for i= 1 ..... Nwe get

dVl~I [V;(0)] p2/2-' 1~ E(z,,zj) exp - ~ ~ o)u(2rcIm r) ; , ' o)" , (10) i = 1 i < j k /~,P= 1 zi t

that is equal to the corresponding expression computed in appendix B of ref. [ 2 ]. The part depending from the tachyon momenta agrees also with the expression given in refs. [ 17,26,27 ]. In particular in the approach ofrefs. [ 17,27 ] one finds the prime form defined in terms of the B-functions:

E(z, y)dz-l/2dy-1/2= O[c~] (f~,r_o I z') [¢°t'(z)3;,O[ °~ ] (01 z)] 1/2 [¢o~,(y)0;,O[ c~ ] ( 0 1 ~.) ] 1 /2 , (11)

where o~ is any odd spin structure on the Riemann surface. Since the two expressions (9) and ( 11 ) have the same analyticity and periodicity properties, with the same

normalization (E(z, y) ~ z - y for z~y) they must be equal. They provide two alternative expressions for the prime form.

In conclusion the basic object in our approach is the N-string, g-loop vertex; any string scattering amplitude can be easily deduced from it.

It is however interesting at the end of this letter to make a connection with the recently proposed operator formalism on an arbitrary Riemann surface and in particular to construct in our formalism the g-vacuum [ 15,17,18 ], that is the starting point of the new operator formalism [ 13-16 ]. To this aim it is useful to consider more carefully our vertex VN,g for N= 1.

V~,g corresponds to a Riemann surface of genus g with one puncture; it is given by an integral over 3 g - 3 moduli for g> 1 and one Koba-Nielsen variable z~. More generally, for g> 0 the integrand will depend on the g multipliers and 2g fixed points of the generators of the Schottky group and on Zl, but three out of these last (2g+ 1 ) variables must be fixed because of projective invariance; it is particularly convenient to choose the projective transformation VI (z) corresponding to the only external leg to be equal to the identity: V~ ( z ) = z. This can be achieved by fixing the Koba-Nielsen variable of the external leg to be z~ = 0 and the two neighboring variables to ov and 1 respectively. With this choice Vl,g takes the simple form ~5

rig, dV(0[exp .~= nQ., , ,--~-)exp~-~ ~ ~ A~(2nlm r ) ; 1 ~. ,,, J ° ~ m . \ = .-zTAm] , (12) n, 1 ,u ,v= 1 n = l m = l

where

1 1 1 . E ( z , y ) I ~ = ~ A~z"-idz' Q'm= 2 ( n - l ) ! ( m - 1 ) ! 0 ~ 0 ~ - - l o g ~ , (13,14)

n = l z ~ y = O

and dV is given by eq. (2) for N= 1 so that the integration runs over the g multipliers and ( 2 g - 2 ) fixed points The two exponential terms in (12) were obtained in the old days by Cremmer [ 29 ].

~5 A similar result has been independently obtained in ref. [ 28 ].

646

Volume 206, number 4 PHYSICS LETTERS B 2 June 1988

In par t icular for g = 1 eq. (12) reproduces the orbi ta l par t of the one-loop result [ 30,31 ] with the choice ~= co, r/= 1 / (1 - k ) and T,(z) =k~z+ ( 1 - k " ) / ( 1 - k ) .

Our Vl,g is a bra vector in the Fock space o f the orbi tal modes of a single string which reproduces physical ampl i tudes when sa tura ted with a t ransverse state ~¢6 (o f course only for the closed string the t a d p o l e - v a c u u m transi t ion ampl i tude is non- t r iv ia l ) .

The g-vacuum (g[ corresponds ins tead ~7 to a R iemann surface together with a point P and an analyt ic coor- d inate vanishing at P and gives the par t i t ion funct ion when saturated with the orbi tal ord inary vacuum [ 0 ) orb = [P = 0, Oa) and the scattering ampl i tude o f N strings with g loops when saturated with the vector

f i~___l [dwi ~i(Wi) ] lO)orb ( 1 5 )

where ~ , (m~) is the vertex opera tor [ 32 ] for the emission o f the state ai and is a conformal field with conformal weight 1 buil t by means o f the coordinate o f the string xu(w) and its derivatives. The integrals on the Koba -Nie l s en variables w~ run over all the R iemann surface for closed strings and on the boundary for open strings according to the rules of the old dua l theory ~8

In the BRST invar iant formula t ion o f the bosonic string there exist two kinds of states, that are annihi la ted by the BRST charge Q [ 33,34 ]: one is the state [q= 0)10)orb , that corresponds to the vacuum of the theory, while the others are o f the type I q = 1 ) I D D F ) , that ins tead correspond to the physical states o f the theory. We have seen that the physical scattering ampl i tudes are ob ta ined by saturat ing WN,g ~9 with physical states of the last type. This means that we must use those states every t ime we have a puncture. I f ins tead we saturate a leg of Vu, g with the vacuum I 0 ) orb, we e l iminate of course the harmonic oscil lators of the corresponding leg, but we are left with an extra Koba -Nie l s en var iable in the measure. We want to show now that instead, i f we start f rom a slightly modi f ied vertex and i f we saturate its leg with the vacuum state [q= 0 ) we get the vacuum ]g) , that does not represent, by itself, the emiss ion o f any state.

Let us consider an object VN.g, that is ob ta ined f rom the vertex Vu,g by inserting in some internal or external leg o f the Feynman- l ike d iagram a BRST invar iant Della Se lva-Sa i to vertex [ 8] together with a project ive t ransformat ion P(x)g2, that is a par t of the BRST invar iant twisted propagator [ 1 ]:

f P(x), P(X)=xL°g2(1--X) w, W=Lo-LI (16) dx

T=(bo-bl) x(1-x-----~

The insert ion of the addi t ional twist opera tor is needed in order to get a cyclically symmetr ic expression as one gets the vertex of Caneschi, Schwimmer and Veneziano from the asymmetr ic vertex [ 35 ]. The vertex so con- s tructed can be formally ob ta ined f rom WN+ t,g replacing one propagator with P(x) and saturat ing only with N ghost states [q= 1 ) .

I f we then saturate the leg o f the Del la Se lva-Sa i to vertex with the vacuum [ q = 0 ) b 0 ) orb and the other N legs with states [ D D F ) we get

N N ~-'N,g[ q:O) I 0)>orb H I D D F ) , = VN, g U ] D D F ) ~ . ( 1 7 )

i=1 i=1

The previous ident i ty follows f rom the fact that the Del la Se lva-Sa i to vertex gives 1 when its open leg is satu- ra ted with the vacuum and that the inser t ion of P(x)g2 is i r relevant when all external legs are sa turated with

~6 Remember that the single external leg has been already saturated with ] q= 1 ) in the ghost space. ~7 We must note, however, that in ref. [ 6 ] by the g-vacuum is meant just the g-loop tadpole operator, that is our Vj,g. ~8 "'Radial ordering" for the closed string is understood. ~9 By WN.g we mean the operator in the full Fock spaces (orbital + ghost) of the N strings, which gives VN,g when saturated with N states

Iq= l)gh.

647

Volume 206, number 4 PHYSICS LETTERS B 2 June 1988

states annihilated by the BRST charge ~o. In particular for N = 0 we get from (17) the partition function Vo.g:

l~O,g I q = 0 ) 1 0 ) o r b = Vo,g. (18)

Before saturating with I q = 0 )10 )orb our Vo,g still depends on the variable x introduced by the projective transformation P(x) (or equivalently by the Koba-Nielsen variable Zl of Vl,g), but all the Vo,g(Z~ ) belong to the same cohomology class, that is,

~'o,g(z, = a ) - Vo,u(zl = b ) = (21QBRST , (19)

where [2) is an arbitrary vector; then the z~ dependence is irrelevant. The g-vacuum ( g I is finally obtained by saturating Vo,g with I q = 0 ) so that Vo,g = ( g l 0 ) orb- The contribution

of the ghost-zero modes differs from the one of Vl,gl q= 1 ) for a factor that multiplied by x( 1 - x ) / d x cancels the factor dz~/V'I (0) in the expression (8) o f Vl,g.

In the cohomology class of the Vo,g it is convenient to choose the representative which has the simplest expres- sion, by choosing a suitable value ofz l depending on how three o f the 2g fixed points have been fixed by projec- tive invariance.

The most convenient one is again the one discussed before eq. (12). In this way we get the g-vacuum (gl = Vo,gl q = 0 ) exactly the same expression on the RHS of eq. ( 12 ), but now d Vis given by eq. (2) with N = 0 and we must integrate over the ( 3 g - 3 ) moduli.

We stress again that the two expressions have a different meaning because G,g represents the emission of a particle and then it depends on ( 3 g - 3 ) moduli and one puncture; (gl is instead a Bogoliubov transformation of the ordinary (tree-) vacuum; its dependence on the extra variable beyond the moduli, related to the choice of an analytic coordinate on the Riemann surface, disappears when (gl is saturated with 10) or with the vector ( 15 ) to give the partition function or an N-string amplitude respectively.

The two exponential terms in eq. (14) are equal to the holomorphic part of the g-vacuum of refs. [ 15,17,18 ]. Moreover our expression contains also the measure that takes correctly into account the ghost contribution in the loops.

One could independently prove that the g-vacuum (gl gives the N-string, g-loop amplitude when saturated with the vector ( 15 ), by using the factorization properties o f the vertex VN,g (see section 5 and appendix B of ref. [ 1 ] ) to write it as a product of Vo,g times N three-reggeon vertices. They can be put in the Della Selva-Saito form in order to transform any external transverse state in the corresponding emission vertex ~,. Finally, the LHS ofeq. (17) becomes

Vo,g [dw,~/£,(Wi)][q=O)lO)orb=(gll~[dwi~t/£,(Wi)]lO)orb, l - - i = l

(20)

while the RHS of eq. (20) is just the N-string, g-loop amplitude.

One of us (S.Sc.) thanks N O R D I T A for the kind hospitality extended to him. K.H. thanks the Deutsche Forschungsgemeinschaft for financial support. We thank G. Cristofano, F. Nicodemi and R. Pettorino for many discussions on the tadpole operator.

Note added. After the submission of this paper for publication we became aware that an N-string, g-loop vertex has also been proposed in ref. [ 36 ]. However, in that paper the integration measure over the moduli is not given.

~Cq'he conformal invariance of our BRST invariant formalism has been proved in section 5 of ref. [ 1 ].

648

Volume 206, number 4 PHYSICS LETTERS B 2 June 1988

References

[ 1 ] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Nucl. Phys. B 298 (1988) 527. [2] J.L. Petersen and J. Sidenius, NBI-preprint HE-87-35, Nucl Phys. B, to be published. [3] A. Neveu and P. West, Nucl. Phys. B 27 (1986) 601. [4] A. Neveu and P. West, Phys. Lett. B 179 (1986) 235; B 193 (1987) 187; B 194 (1987) 200, and CERN preprint TH-4757/87;

see also P. West, Lectures presented at Carg~se and the Johns Hopkins Workshop, CERN preprint TH-4819/87, and references therein.

[ 5 ] U. Carow-Watamura and S, Watamura, Nucl. Phys. B 288 ( 1987 ) 500; and Tohuko University preprint Tu/87/317. [6] A. LeClair, Nucl. Phys. B 297 (1988) 603. [7] G. Aldazabal, M. Bonini, R. Iengo and C. Ntifiez, Phys. Lett. B 199 (1987) 41; and ITCP preprint IC/87/414. [8] P. Di Vecchia, R. Nakayama, J.L. Petersen and S. Sciuto, Nucl. Phys. B 282 (1986) 189. [ 9 ] P. Di Vecchia, R. Nakayama, J.L. Petersen, J. Sidenius and S. Sciuto, Phys. Lett. B 182 ( 1986 ) 164; Nucl. Phys. B 287 ( 1987 ) 621.

[ 10 ] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 199 (1987) 49. [ 11 ] A.A. Belavin, V.G. Knizhnik, A. Morozov and A. Perelomov, Phys. Lett. B 177 ( 1986 ) 324;

A. Morozov, Phys. Lett. B 184 (1987) 171; A. Kato, Y. Matsuo and S. Odake, Phys. Lett. B 179 (1986) 241.

[ 12 ] J.L. Petersen, K.O. Roland and J.R. Sidenius, Niels Bohr Institute preprint NBI-HE-88-04. [ 13 ] L. Alvarez-Gaum6, C. Gomez and C. Reina, Phys. Lett. B 190 ( 1987 ) 55. [ 14] N. Ishibashi, Y. Matsuo and H. Ooguri, Mod. Phys. Lett. 2 (1987) 119. [ 15 ] C. Vafa, Phys. Lett. B 190 (1987) 47. [ 16 ] E. Witten, Princeton preprint PUPT- 1057 (May 1987 ). [ 17 ] S. Mukhi and S. Panda, Tata Institute preprint TIFR/TH87-51. [ 18 ] L. Alvarez-Gaum6, C. Gomez, P. Moore and C. Vafa, CERN preprint TH-4883/87. [ 19 ] P. Di Vecchia, K. Hornfeck and M. Yu, Phys. Lett. B 195 (1987) 557. [20] C. Lovelace, Phys. Lett. B 32 (1970) 490. [21 ] F. Pezzella, Nordita preprint, to be published. [ 22 ] V. Alessandrini, Nuovo Cimento 2A ( 1971 ) 321;

V. Alessandrini and D. Amati, Nuovo Cimento 4A ( 1971 ) 793. [23] C. Montonen, Nuovo Cimento 19A (1974) 69. [24] A. Clarizia and F. Pezzella, Nordita preprint 87/55 P, Nucl. Phys. B, to be published. [25] P. Di Vecchia, E. Del Giudice and S. Fubini, Ann. Phys. (NY) 70 (1972) 378. [26 ] S. Mandelstam, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore) p. 46; and talk Workshop on

String theories ( Santa Barbara, CA, December 1986 ), unpublished. [27] E. Verlinde and H. Verlinde, Nucl. Phys. B 288 (1987) 357;

see also D. Mumford, Tata Lectures on Theta, Vols. I and II (Birkhauser, Basel, 1983 ). [ 28 ] G. Cristofano, R. Musto, F. Nicodemi and R. Pettorino, Naples preprint ( 1988 ). [ 29] E. Cremmer, Nucl. Phys. B 31 (1971 ) 477. [30] D. Gross and J. Schwarz, Nucl. Phys. B 23 (1970) 333. [ 31 ] G. Cristofano, F. Nicodemi and R. Pettorino, Phys. Lett. B 200 ( 1988 ) 292; and Naples preprint ( 1988 ). [ 32 ] S. Fubini and G. Veneziano, Nuovo Cimento 67A ( 1970 ) 29; Ann. Phys. (NY) 63 ( 1971 ) 12;

P. Campagna, S. Fubini, E. Napolitano and S. Sciuto, Nuovo Cimento 2A ( 1971 ) 911. [33] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 34 ] M.D. Freeman and D. Olive, Phys. Lett. B 175 (1986) 151. [35] S. Sciuto, Lett. Nuovo Cimento 2 (1969) 411. [36] A. Neveu and P. West, Commun. Math. Phys. 114 (1988) 613.

6 4 9