On multi-loop four dimensional superstring amplitudes

Volume 246, number 3,4

PHYSICS LETTERS B

30 August 1990

On multi-loop four dimensional superstring amplitudes Vincent Rivasseau

L

Mathematics Department, Princeton University, Princeton, NJ 08544, USA

Received 16 May 1990

We apply the local non-renormalization results of Lechtenfeld and Parkes for superstrings amplitudes to the fermionic compactifications of Antoniadis et al. The local vanishing of the superstring integrand for the cosmological constant of N=4, 2 or 1 supersymmetric compactifications is checked up to respectively nine, five and three loops.

I. Introduction The results ofrefs. [ 1,2 ], although local in m o d u l i space, represent an i m p o r t a n t advance over previous work in the subject; on such issues as the local vanishing o f the cosmological constant they increase our understanding from two or three to eight loops. But these results are given in the case o f a type II superstring in ten dimensions. They clearly extend to the ten d i m e n s i o n a l heterotic superstring. However, it seems to us that their extension to four d i m e n s i o n a l superstrings is not totally obvious. It is the purpose o f this note (section 3) to explore such an extension, and hopefully to clarify also some o f the issues left pending in ref. [ 3 ]. In section 2 below, we remark also that the results o f refs. [ 1,2 ] extend to nine loops and we rewrite t h e m using the language of pfaffians and hafnians [4] in order to facilitate future efforts to extend them b e y o n d nine loops. In this note we advocate the same p o i n t o f view as in refs [ 1,2 ], namely we do not consider for the m o m e n t the subtle non-local p r o b l e m s o f p e r t u r b a t i v e string theory such as the non-splitness o f supermoduli space or issues related to its boundary. We are aware o f the p r o b l e m s related to a naive integration over supermoduli (see refs. [5,6 ] and references t h e r e i n ) and to the use o f a " u n i t a r y gauge" prescription such as the one o f ref. [ 1 ], or even o f ~-functions for the picture changing insertions [ 7 ]. As in refs. [ 1,2 ] we hope nevertheless that the local considerations in m o d u l i space will help in the long road to bring the p e r t u r b a t i o n theory o f superstrings to the same degree o f u n d e r s t a n d i n g that is now enjoyed by renormalized perturbation theory in the context o f q u a n t u m field theory. We recall first briefly the construction o f four d i m e n s i o n a l superstrings in the free fermion formulation o f ref. [8 ]. We will only consider the case o f four d i m e n s i o n a l compactification. The non-supersymmetric sector o f the right moving string excitations contains in a d d i t i o n to the four s p a c e - t i m e coordinates OeX u and their four fermionic partners ~7u a set of 44 extra real fermons 0 a. The supersymmetric left moving sector has in a d d i t i o n to the four s p a c e - t i m e coordinates 0zX u and their four fermionic partners ~ ' a set o f 18 real fermions Z i, yi and co', among which s u p e r s y m m e t r y is realized in a non-linear way. This means that the supercharge is G = Z ~u O.XU+ ~, xiyio) i .

( 1.1 )

i

F, the set o f the 66 free fermions ~u, Z i, yi, oY and 0" is a boolean algebra, with the symmetric difference as a d d i t i o n and the intersection as multiplication. The choice o f a particular four d i m e n s i o n a l compactification in the f o r m a l i s m o f ref. [ 8 ] is the same as the choice o f a particular subgroup Z of F considered as an additive 1 Permanent address after June 30th, 1990: Centre de Physique Th6orique, Ecole Polytechnique, F-91128 Palaiseau Cedex, France. 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

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group, plus a set of coefficients C(d) which take values _+ 1 for all one-loop spin structure assignments d. To define such a spin structure assignment, we fix a certain reference even spin structure which is noted as 0. Any other spin structure is a collection of 2g integers nj, mJ, j= 1, ..., g with values either 0 or 1, which specify a particular reduced theta characteristics. (Our conventions for theta functions and characteristics are those of ref. [9], hence slightly different from the notations of refs. [ 1,2] ). A spin structure assignment c~ is then a collection d], d~, j = 1, ..., g, of 2g subsets of ~. It corresponds to specifying, at each loop, the subset of those fermions of F which have periodic boundary conditions around the two corresponding loops in a homology basis of the Riemann surface. At one loop d is simply a set of two subsets 01, d2 of F, and the corresponding coefficients C(d) then determine which fermions are R a m o n d or Neveu-Schwarz and which set of GSO projections is performed on the theory. There is a certain set of consistency conditions that are required on ~, and the coefficients C which ensure in particular that the supercharge G has a well defined spin structure, and that the theory has correct factorization properties and one-loop modular invariance [ 8 ]. In the phenomenologically interesting case of a theory with space-time supersymmetry there is a particular set S in ~ made of the 10 left moving fermions ~,~,/2 = 1, ..., 4, and of the Z i, i = 1, ..., 6, such that at one loop the coefficient C(d) i s - 1 ifd~ =S, d2 = y for any subset y o f ~ w i t h S . y = 0 . The supersymmetry is of the type N = 4 , 2 or 1 depending on whether S has non-trivial intersections with other elements o f ~ . Assuming that supermoduli can be integrated out, the g-loop and N-point superstring amplitudes are locally given by an integral over moduli space which has been completely explicited in terms of theta functions and their derivatives in ref. [ 10 ]. Following the notations of ref. [ 1 ] and using the loop m o m e n t u m representation as in ref. [ 10], we have A~v(k,~)= f dpU f

lexp(inp~'g2p~), 2

Mg, N

N

3(g--l)

× 1-[ dyt dyz 1-I dm'dm'GJV( k, e;P~', m',Y~, zj)GN( k, 6;P ~', mi,.gl) , /=1

(1.2)

i=1

where G is the left piece of the integrand, an analytic function of the moduli which is a functional integral over the left handed conformal fields, and G, the right piece is an antianalytic function of the moduli. This piece is the complex conjugate of G in the type II superstring, but here it is different because the ten dimensional heterotic string or the four dimensional compactifications considered above have a different set of left and right handed conformal fields. In the left supersymmetric sector of the string we have

G~'(k,e;p~,m',yl, zj)=

(2(~i1 ) \

j=l

Y(zj)

3(g-- 1 )

N

i=i

l=1

I-[ (q, lb) l~ V(U,~t;Yz)

1

•

(1.3)

In the right sector of the string there is no supersymmetry and the integrand is therefore simpler: /3(g-- I )

N

\

GN(k'~;P~"m"'Fz)=~ ~=l (q'lb-) I-[ V(k',Et;)Tl)) . i=

l= 1

(1.4)

r

We refer to ref. [ 1 ] for the definition of the various ingredient in formulae ( 1.2)-(1.4). The vertex operators V(U, et; Yt) create external states of the string at locations y~ on the Riemann surface. The moduli m i, m s and the locations of the vertex operators Yt, Y~are integrated over the appropriate moduli space Mg,ufor the surface of genus g with N punctures. The brackets ( , )~ and ( , )r denote the path integral over the relevant left and right set of chiral free conformal field theories on the Riemann surface, which in our case includes the four string coordinates X ¢', their supersymmetric partners ~,u and the fermionic conjugate ghosts b and c (in both chiral sectors), the super ghosts fl and y (in the left sector only), and the 18 or 44 remaining fermions. The zero modes 406

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are cured via the insertion o f 3 g - 3 antighosts insertions (~/il b) and the 2 g - 2 insertions of the operators az 0~(z) O(~/ia Ibm" 17(z) = Yo + Y~ + Y2 + Om~-5

(1.5)

The ghost pieces of the operator Yare the same in the compactified theories as in the ten dimensional ones, and only the matter part of the integral Y~ is different. In four dimensions we have

Yl(Z)=-½ eO(~u OX~,+ i~=lZiYi(-Oi)

(1.6)

(as usually the superghosts have been bosonized via fl= e - ~ 3~ and y = e ° ~/).

2. T h e ten d i m e n s i o n a l case

With the standard choice of delta functions for the fermionic Beltrami differentials the insertions Y are made at specific points o f the surface, and these points are chosen so as to satisfy the condition 2g-- 2

Y~ zj=2A,,

(2.1)

j=l

where the vectors zj are obtained through the Jacobi map, and As is the vector of Riemann constants associated to a fixed spin structure o~. This is called a "unitary" gauge condition in ref. [ 1 ]. From now on we omit vector signs. The building blocks o f the superstring amplitudes can be computed in terms of theta functions. Then the vanishing o f the local integrand for e.g. the cosmological constant up to eight loops follows in ref. [ 1 ] from the Riemann identity: E

a

(OL [ ( ~ ) 0 a ( X 1 )Oo(x2)O~j(x3)Oo(x4)

t t ! = 2 g G ( x , t )Oot(x2)Oot(x3)Oo~(x4) ,

(2.2)

where the relative phase of spin structures c~ and ~ is (o~1~) = ( - 1 )x~=, ~w,.-~ +e6.-~,),

(2.3)

and the transformation of arguments is given by X] = 1 ( X 1 ..~_X2 .~_X3 .Ji_X 4 ) ,

X2 = I ( X 1 ..{_X2 __X 3 __X4) ,

x'3=½(xl-x2+x3-x4),

x'4=½(xl-x2-x3+x4).

(2.4)

Let us focus on the left sector contribution to the cosmological constant ( N = 0) where all the 2 g - 2 insertions are of type Y, (matter insertions). In ref. [ 1 ] the spin structure of all the fermions ~u and of the superghosts are equal and one has therefore a single sum to perform over spin structures 6 o f a contribution proportional to /2g--2

\ ql /2g--2

\ X /2g-- 2

\fly

where the sum is performed from 1 to 10 for each index #j. If for/1 = 1, ..., 10 we call A u the set of indices j such that #j=/t, the sets A u form an even partition of {1, ..., 2g--2} [i.e. a partition into sets which haven an even number IA u I - 2n (/t) of elements ]; if we group together the contributions corresponding to the same partitions the combinatorics of fermions generate pfaffians and the combinatorics o f bosons generate "hafnians" [4]. These objects are defined inductively, using the following notations for a sequence a~ < ... < azn: 407

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2n Pfaffa(al ..... a2n) = ~

k=2

( - 1 )k Pfaff6(al, as.) Pfaffa(a2, ..., 6k ..... a2n) ,

(2.6)

2n H a f n ( a l ..... a2n) = Y~ H a f n ( a l , a i) H a f n ( a 2 , ..., ak, ..., a2n) .

(2.7)

k=2

Using the results on the two point function for the q/and chiral X functional integrals we define the two point pfaffians and hafnians as Pfaffa(al, as) - Za(z,, -Zaj) ,

a a H a f n ( a l , aj) - OZa~lOz~ log E(z~,, z~j) ,

(2.8)

where E is the prime form and Z6 is defined as in ref. [ 1 ]:

9, with ZI some constant which plays no role in the following. Then using the unitary gauge condition to simplify the superghost integral as in ref. [ 1 ], one can write the left part o f the a m p l i t u d e as a spin structure i n d e p e n d e n t factor times a sum over spin structures c~ and even partitions o f { 1..... 2 g - 2} into sets Au=a~' < ... < a~,¢,) o f contributions

10 G~4 .......4~o = [ I Pfaffa(alu, -", a~,~u)) H a f n ( a l ' ..... a~,(u)) ( a l ~ ) Z a ( O )

5-g .

(2.10)

The factor Z a ( 0 ) 5 - g c o m e s from a factor Z a ( 0 ) s for the n o r m a l i z a t i o n of the system o f 10 free fermions [each giving Z a ( 0 ) 1/2], from the g - 1 W i c k contractions o f the 2 g - 2 fermionic insertions [this gives Z ~ ( 0 ) - Cg-i) ] and finally from one Z a ( 0 ) -~ which corresponds to the evaluation o f (eO(zl) ... e°(Zzg_2) )~Y using the unitary gauge condition. Fay's trisecant identity ( b o s o n - f e r m i o n equivalence) relates a d e t e r m i n a n t o f pairs o f Z factors to a single Z term with a sum o f arguments, it reads

( - 1 ) n ~ n - ' ) / 2 det Z ~ ( z ~ - w s ) = Z ~ ( O ) ~ - l Z ~ ( ~

z,-

y wj) .

(2.11)

i,J

We can write the d e v e l o p m e n t o f pfaffians in terms o f determinants:

( - 1 ) n ( n - - I )/2 Pfaff( 1.... 2n) =

2 ~- ]

1t={h . . . . . .

hn},K={k......

e(H, K) deta(hl ..... h,; k~, ..., k , ) .

kn}

(2.12)

hi = 1 , H r ~ K = O , I I ~ K = { I,...,2n}

This formula expresses a pfaffian with 2n arguments as a weighted sum o f ½( 2 n ) ! / n ! 2 determinants; e (H, K) is the sign o f the p e r m u t a t i o n which sends {1 ..... 2n} on {h~ ..... h~, kl ..... k~}. det~(h~ .... , h.; k.; .... k . ) is the d e t e r m i n a n t o f the n by n square matrix with entries a~j= Pfaffa(hs, ki)- W i t h this formula and Fay's identity ( 1.11 ) we can rewrite (2.10) as

10 G,~,A,...A,o=2 -

,,=~ [n(~)--ll+ 1--I

/'n(#)

~,

e ( H u, Ku)Za

It= 1 {HU,KU}

X... Hafn(a~' ..... aS, o ) ( c ~ [ O ) Z a ( O ) 5 - g + x Z l

/_z zhf,i

1

i,(It)-ll+ ,

n~)

)

zkf j= 1

(2.13)

where the symbol [n(/t) - 1 ] + means n(/t) - 1 if n(/t) ~> 1 and 0 if n(/z) = 0 . We have E ~ o n (/t) = g - 1. To use the R i e m a n n identity together with the R i e m a n n vanishing theorem one 408

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has to get rid of any factor Za(0) to a negative power. For g~< 8 it is shown in ref. [1 ] that this is possible. In fact we remark that the arguments of refs. [ 1,2] work also for g = 9 (the authors of refs. [1,2] realized this independently [ 11 ] ). Indeed at a given order g~< 11 the worst case for the factor ~ u= 1o l [ n (/~) - 1 ] + is when the numbers n(/t) are 0 or 1. Let us finish the arguments in this worst case at nine loops: then there is a product o f eight two point functions Z6 (z~ -z~j) and a factor Z 6 ( 0 ) - 4 in (2.13 ). Using Fay's identity together with some analysis of parity, the identity

~c~(Z1-Z2)Z6(z3-7-`4)=~Z~(~)[Z~5(Z1-Z2-~-z3-~4)-Z~(Z1-~CZ2-Z3-Z4)-J~(Z~-z2-z3+Z4)] for 8 even is established in ref. [ 1 ]. Applied to four pairs of factors Z6 (z~, --Za~) it allows to eliminate

(2.14)

the factor Z6 (0) -4. We are left with a sum over a certain number of terms, each of which up to a spin structure independent multiplicative factor is a sum over even spin structures of products o f four theta functions. The corresponding sum not restricted to even structures is identically zero by the Riemann identity (2.1) plus the Riemann vanishing theorem and the unitary condition (2.1). But using the symmetry properties of even and odd theta functions under exchange of e.g. {zl, z2} and {z3, z4} in (2.14) we can in fact conclude that both the even and the odd part of the full sum over spin structures vanish separately. This concludes the argument for the matter insertions. When ghosts insertions Yo or Y2 are involved one can always remove the non-trivial theta functions in the denominator by using the identity [2 ]

fl(x,) =

~,(yj) ~ j=l

6(fl(zD )

~Perm{Z~(x,-yi)}

Z~(0) -~-1 ,

(2.15)

k=l

where the z's have to satisfy the unitary condition (2.1), and Perm is a permanent, the symmetric analogue o f a determinant (a nice proof of (2.15 ) can be found in ref. [ 12 ] ). This identity is valid at any number of loops so that for the m o m e n t the limitations to nine loops come from the matter insertions. Since the ghost part of the picture changing insertions is the same in four dimensional or ten dimensional superstrings, to check that the results of refs. [ 1,2 ] extend to four dimensional compactifications we have only to consider the matter insertions, i.e. from now on we consider only the YI piece (1.6) of the operator Y in (1.5).

3. The four dimensional case

In this case we no longer have a single sum over one spin structure 8 to perform, but a sum over assignments 8 = (81, 82) in Z,g)<--g. These assignments are such that the supercharge ( 1.1 ) has a well defined spin structure, so the four fermions ~ ' have the same spin structure 6(~,u), which we abbreviate as 8 °, and we have for each i = 1, ..., 6 the condition 8(Z i) + 6 ( y ~) + 6 ( o J i) = 6 °. We put 6~=8(Z ~) for i = 1.... ,6. We organize the sum over assignments as in ref. [ 3 ], taking advantage of the presence of the set S in ,~ to average over all translates by S. This allows us to write the left piece of the matter part of the vacuum superstring amplitude, up to an unimportant multiplicative factor, as

E deZgxZg

Z

Ga~,,..~4,Sa,...,B6•

( 3.1 )

A1 ,...,M4,BI ,...,B6

The setsA~= {af, . . . , a~,(u)}andBi={b~ , . . . , b z 'n ( i 2 g - 2 products of factors Yl in (2.6). We have

)

} correspond

to a particular choice in the development of the

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6

Ga~,,...~4,8,,.,~6 = ~ H a f n ( a f , ..., affn(u)) I-I Pfaffa(yo(b'~, ..., b'zn(i~) /*=1

i=l

× Pfaff6(o,o (b~ . . . . . bl2n(i))... Hafn(b{, ..., bi~n(i))Z6(yo(O)-n(i)+l/2Z6(~o,)(O)

x

~ C(6+~,S)Faoa,~,A

A4,B,,,~6

-n(i)+l/2

,

I~O,ai,y,AI,...,A4,BI,...,B6 ={(~O"{-y) (OLl6°'-I-y>Zao+7(O) -1 , 4

(3.2)

6

1-[ Zao+y(O) -"~u)+~/2 Pfaffao+r(a~ ..... a~',~u)) 1-I Za,+~(O) -"~°+~/2 Pfaffa,+y(b], ..., b~,,o~) . i=1

It=l

(3.3)

The sign E(6 ° + 7) = ( - 1 ) xT=,a0j+ yq + ao.,+~ is necessary for consistency with the conventions of refs. [ 8,3 ] for the coefficients C [which for g = 1 depend implicitly on the choce c~= (1, 1 ), the single particular odd spin structure at one loop ]. The index 7 is summed over all characteristics in {0, 1}2% and represents the average over translating by the set S ~ . Every fermion not in S does not feel this translation, and all the corresponding terms independent of 7 have been grouped in front of the 7 dependent terms, which are collected in F. We fix now the assignment 6 and perform the sum over 7. The result would be 0 exactly as in the ten dimensional case if we had 6 ' = 6 °. This is always the case for the theories with N = 4 gravitinos, where the set S has trivial intersections with the other elements of~,. But when N = 2 or N = 1, we can have 6 ' = 6 ( Z ~) # 6 ° = 6 ( ~ u ) . Even in this case, the set Sis broken in such a way that (up to some relabeling) its intersection with other sets in 2 i s either 0, S, $1 = {~'~, Z~, Z4}, or $2= {q/u, Z> Zs} [3,8 ]. Therefore we have at most three different spin structures 6~; for i = 1, 2, 3, we have 6~+3= 6~; furthermore 5] 3 , 6 ' = 6 ° (addition of characteristics is performed modulo 2). We can expand pfaffians into determinants as before and apply Fay's identity. The worst case occurs again when pairs of arguments fall into different sets A or B; furthermore the worst case for pairing together theta functions with the reduction identity (2.14) occurs when the pairs fall into sets of the A type. In particular for N = 1 and different spin structures 6o, 6~ and 62 at four loops already we can have the three pairs falling into A~, A2 and A3, which leads to contributions for the fermions in S and the fl-Y system such as 0[(~ 0 ] ( 0 ) - - 2 0 [ a 0 ] (U, --U, ) 0 [ 6 0 ] (U 2 --U2)0160 ] (U 3 - - u 3 ) O [ a ' ] ( O ) O [ a

2 ] (0)0[63 ] (0) .

(3.4)

In this case we can only pair together two 0 [ 6 o ] hence one 0 [ 6 o ] (0) remains in the denominator and we cannot apply the Riemann identity. Similarly for N = 2 space-time supersymmetry we can have at most two different spin structures and the removal of theta denominators by the method of ref. [ 1 ] works only up to five loops (included). Many contributions where the pairs fall into convenient sets are zero, however, beyond three or five loops. The Riemann identity involved is more general than (2.1) since it has to involve different characteristics. It reads (see p. 67 ofref. [9] ): Y~ ( a l

e> ( - 1

)P~+*'~'~O.(x,)O~+~(x2)O~+~,(x3)O~+a+,,(x4)

=2g( -- 1 )~a+u)"0~ (x])O,~+a(x'2)O,~+u(x'3)O,+.~+,,(x'4 ) ,

(3.5)

where we define the symbol P(c~, fl) - Z~g=~cdiflS. To apply (3.5) one has to check that the rules of refs. [ 3.8 ] to compute the signs corresponding to the coefficients C ( 6 + ),S) give the correct signs. This analysis is tedious and similar to the one explicitly performed in ref. [ 13 ]. One has to apply the transformations rules of ref. [ 8 ] for the one loop coefficients C, the fact that C(c~) = - 1 for cq =S, o~2'S= ~ , the facts that C ( a ) = 1 for cq = 0 or a2=O, etc. Factors such as C(c~) with ~ =S, O ~ 2 = S o r S, or $2 occur in intermediate stages of the computation but because each fermion has to have an even spin 410

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structure they occur always an even n u m b e r o f times hence cancel out. Ultimately the curious signs o f (3.5) are left. We c o m p u t e d a typical contribution at nine loops for the N = 1 theory, in which pairs fall into At, A2, Bb ..., B 6. In this case we end up with a sum 3

( -- 1 )"((~'+~2>,')(a 7 even

I~°+ ~>0[~°+y] (to+Vo-Uo -Wo) I-I o [ ~ i + y ] (vi+vi+3 -wi-wi+3) i= 1

for some labeling o f the 2 g - 2 = 16 arguments zj as {to, vo, ..., v6, Uo, wo..... w6}. We write ~ = ~ o + 1 3 i and apply (3.5) with 2 = f l 1, It=fl2 and e=c~°+y (since 52i ~ i = ~ o we have automatically /l + I t = 133). C o m b i n i n g the R i e m a n n vanishing t h e o r e m with condition (2.1) we obtain again that the sum (3.6) over all spin structures 7 vanishes; with some parity analysis the even and o d d parts o f the sum both vanish. This kind o f argument shows that m a n y contributions in (3.1) vanish up to nine loops; nevertheless as r e m a r k e d above, it is only up to three loops for N = 1 and five loops for N = 2 that this argument applies to every contribution. The conclusion o f this analysis is that the m e t h o d s o f refs. [ 1,2] for the ten d i m e n s i o n a l superstring do not generalize so easily to four dimensions. Although there is no p r o b l e m with the R i e m a n n identity and vanishing theorem, it might be difficult to find analogues o f (2.14 ) which c o m b i n e together different spin structures in o r d e r to make further progress b e y o n d three loops for d = 4, N = 1 superstrings. An other m o r e promising possibility would be to find higher order identities which take into account the structure o f the bosonic hafnians which multiply the pfaffians in (2.13) or (3.2). Hopefully such identities might lead to the extension o f the local vanishing theorems to any order g. Let us also discuss briefly some previous related work. In ref. [ 3 ] a certain conjecture was m a d e which i m p l i e d the vanishing o f the superstring cosmological constant integrand and o f one, two and three point functions [ 13 ]. We d i d not realize at that time that the insertion o f the operators Y~ would give non-zero arguments to some theta functions o f the theory, hence the formulae o f ref. [ 3 ] are not completely correct. Nevertheless, because m o d u l a r transformations, which play the key role in ref. [ 3 ], do not act on the argument z o f a theta function, the m o d u l a r analysis in refs. [3,13 ] r e m a i n s basically correct. However, although the explicit prescriptions (2.1) and ( 3 . 1 ) - (3.3) given above are valid in local patches o f m o d u l i space it is still not clear to us how they extend to the full Teichmiiller space in an explicitly m o d u l a r invariant way.

References [ 1] O. Lechtenfeld and A. Parkes, Nucl. Phys. B 332 (1990) 39. [2] O. Lechtenfeld, Phys. Lett. B 232 (1989) 193. [3] D. Arnaudon, C. Bachas, V. Rivasseau and P. V6greville, Phys. Lett. B 195 1987) 167. [4 ] E. Caianiello, Combinatorics and renormalization in quantum field theory (Benjamin, New York, 1973 ). [5] E. D'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917. [6] J. Atick, G. Moore and A. Sen, Nucl. Phys. B 308 (1988) 1. [7 ] G. Falqui and C. Reina, Commun. Math. Phys. 128 (1990) 247. [ 8 ] I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 ( 1987 ) 589. [9 ] H. Rauch and H. Farkas, Theta functions with applications to Riemann surfaces (William and Wilkins, Baltimore, 1974). [ 10] E. Verlinde and H. Verlinde, Phys. Lett. B 192 ( 1987 ) 95. [ 11 ] O. Lechtenfeld, private communication. [ 12 ] A. Morozov, Phys. Len. B 234 (1990) 15. [ 13 ] D. Arnaudon, preprint Ecole Polytechnique A806.1287; Th6se de Doctoral, Universit6 Pierre et Marie Curie (Paris VI ) ( 1989 ).

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