Multiloop modular invariance of d = 10 type II superstring theory

Nuclear Physms B318 (1989) 397-416 North-Holland, Amsterdam

M U L T I L O O P MODULAR INVARIANCE OF d = 10 TYPE II S U P E R S T R I N G THEORY* Osamu YASUDA

Cahforma Insntute of Te~hnolo~', Pasadena, CA 9112.5, USA Received 14 October 1988

Using the pmture-changmg prescription for supermoduh and the bosomzation formula for the superconformal ghost determinant, it is shown exphotly that the higher loop amplitudes of the type II superstnng theory m ten-dimensional flat background are modular mvanant

1. Introduction

In superstring theories [1] modular lnvarlance plays a crucial role, since ~t allows us to restrict the integration region of the Teichmfiller parameters to the fundamental domain of the mapping class group, and therefore modular mvarlance ~s important to prove finiteness of the amphtudes. Although modular mvanance ~s well established at the one-loop level of superstring theories, only formal proof of multiloop modular mvanance of superstrmg or heterotic strmg theories has been given by several people [2-5], mainly because of the subtleties of the treatment of the superconformal ghosts. In this paper we show multiloop modular mvarlance of type II superstrlng theory m ten-&mensional flat background by using the picture-changing prescnpnon [6] for supermodull and the bosonizatlon formula [7-9] for the determinant of the superconformal ghosts. It turns out that the determinant of the superconformal ghosts is crucial to get the expression which is consistent with mult~loop modular invariance and the factorlzation property. When one uses the pmture-changlng prescription for supermoduh, it seems necessary [10] that the msernon points of the picture-changing operators depend on both the holomorptuc and antiholomorphlc coordinates of the moduh space in such a way that they satisfy correct factonzanon properties in the degeneration limit of the Raemann surface (see also ref. [11]). Throughout tiffs paper we will assume that such a good choice for the insertion points exasts for Rlemann surfaces with arbitrary genus. We will &scuss how these * This work supported m part by the U S Department of Energy under Contract No DE-AC0381ER40050 and by the Welngart Foundation 0550-3213/89/$03 50©Elsevmr Science Pubhshers B V (North-Holland Physics Pubhshmg Division)

0 Yasuda / Modular mvartance

398

insertion points of the picture-changing operators should transform under the modular transformation, and it turns out that the dependence of these points on the moduh coordinates is irrelevant. In sect. 2 we give a brief review of the multlloop modular transformations In sect. 3, using a simple example on a torus, we show how the abelian integral should transform under the modular transformations when the integration region depends on the modular parameters. In sect. 4 we exphcitly show the modular lnvarlance of the tugher genus superstring amplitudes. In sect. 5 we discuss the degeneration limit of the amplitudes, where the homologically trivial cycle of the Rlemann surface is pinched. In particular, we discuss the factorization property of the genus-two amplitudes m detail. In sect. 6 we give our conclusions.

2. The multiloop modular transformation Let us start with a brief discussion of the modular transformation for higher genus Riemann surfaces*. There are 3g - 1 generators of the mapping class group of the genus g Riemann surface [12], and they are represented by 2g x 2g matrices of the Siegel modular group Sp(2g, Z ) [13-15]. If we write the generator T of Sp(2g, Z ) as T= (D

C) ~ Sp(2g, Z ) , A

(1)

then g × g matrices A, B, C, D satisfy the following relations: tAG = tCA,

IBD = tDB,

t D A - tBC = 1 ,

(2)

where 1 is the g × g unit matrix and tA stands for a transpose of a matrix A. Here we use the following basis for the generators:

T

t ~

1

34, g N,_2e,

for t = 1, .., g,

1-3/1, g]

forl=g+l

. . . . . 2g,

for t = 2 g + 1 . . . . . 3 g - 1,

(3)

* The description of the modular transformation for higher genus Rlemann surfaces for physicists has been given in refs [4, 8] (see also references therein)

0 Yasuda / Modular mvanance

399

where

l<~l,k,14g

( M , ) , I = 6,, 8,l,

(4)

(N,), = (N,),+,,,+~ = 1, (N,),,,+I = (N,),+,,, = - 1

1~,~<~-1,

other components of N, = 0. U n d e r the modular transformation (1), the holomorphlc one-forms (l = 1 . . . . . g) and the period matrix r,j (t, i = 1 . . . . g) transform as

ga'='(Cr + D),jloo j ,

(5) ~o'

(6)

"T=(Ar + B)(Cr + D ) - '

(7)

In the &scusslons below, the transformanon property of the Rlemann theta function [13-15] is important. It is gwen by

Omo+a(f, ,7) = { d e t ( C r + D) 1/2 e'~[¢(')+:(c~+D)-*Cz]o,,(Z, ¢),

(8)

where

e='(c~ + D)-'z,

(9)

@(m) = m 0' ' m 0" - m' • m" + 2 m ~ - 8 " ,

m'o l = mS" /

1) -B m p

m

-~

m H

-C A

I

m') , m"

8' 8"

m{,m~ ....

[m{',

m 2.

(10) 1 { dlag(CtD)

1

m~

. . .m g

,...,

(11)

2 1 diag(A tB) ] '

02)

J

as the spin structure of the Raemann theta function O,,(z, r) and e is a phase whmh depends on A, B, C, D in a comphcated way and satasfies e s = 1. Fortunately, m string theories, { always appears m a mulnple of e s, so we do not have to worry about the phase of e. The other important transformatmn property is that of the Raemann theta constant, which is defined m terms of the divisor of O(z, r): ,1 = t ( C r + D)

aA + ,F6' + 8".

(13)

In the following analysis we will restrict ourselves to the modular transformations

400

o Yasuda / Modularmvarlance

generated by (3), where we have (14)

8'--0 and rather 8" = 0,

C 4= 0

(lSa)

or

0,

c = 0,

D = 1.

(15b)

Later we will need the transformation laws for the prime form [14,15] E(z, w) and a holomorphlc secuon o(z) of a trivial bundle [14]:

E ( z , w ) ~ e x p ,~rf w(c'r+D) \

oa E(z,w)

(16)

"W

where k z is the vector of the Rmmann theta constant with basepoint z defined by

a=kZ+(g-1)z,

(18)

and we have used the conditions (14) and (15) in eqs. (16) and (17).

3. The insertion points of the picture-changing operators Let us now turn to the discussion of the picture-changing prescription [6] for supermoduli. As has been &scussed in ref. [10] (see also ref. [11]), the insertion points of the picture-changing operators have to satisfy certain con&tlons m the degeneration limit of the Raemann surface. For example, m the degeneration hm~t of a genus-two Raemann surface, where the trivial homology cycle xs pinched, two picture-changing operators must lie on different tori to have a correct factonzatmn property (cf. fig. 1). This implies [10] that the insertion points are a complicated function of both the holomorpbac and antiholomorphlc modular coordinates, and a satisfactory choice for the inserhon points has been given in ref. [10] m the case of

{Q,~)

{Q,g}

Fig 1 A genus-two Raemann surface m the degeneration hnm, where two Dcture-changmg operators {Q, ( } he on different ton

401

0 Yasuda / Modular mvarlance

T

0

7-+1

1

Fag 2 The contour integral on a torus whmh ~s expressed m terms of the local coordinate z an a parallelogram

g e n u s - t w o R i e m a n n surfaces. In this p a p e r we assume that such a nice choice exists for a r b i t r a r y genus R i e m a n n surfaces so that they have a correct factortzation limit. Since the insertion p o i n t s d e p e n d on the m o d u h coordinates, it is a subtle issue w h e t h e r these insertion points receive extra t r a n s f o r m a t i o n in a d d i t i o n to those for c o n s t a n t points. To show that they do not receive such extra t r a n s f o r m a t i o n s , let us discuss a s i m p l e e x a m p l e on a torus. C o n s i d e r the following c o n t o u r integral on a t o r u s which is expressed in terms of the local c o o r d i n a t e z in a p a r a l l e l o g r a m (cf. fig. 2):

fB(r,

~) .

clz,

(19)

A(r,÷)

where

A(r,~)=

l+r ir]4 ,

B(r,~)=

l+r 'Ir ' I2 '

]r I > 1.

(20) (21)

N o w we p e r f o r m m o d u l a r t r a n s f o r m a t i o n which is given by

r --, 7r = - 1 / r .

(22)

A c c o r d i n g to the t r a n s f o r m a t i o n law (6), the holomorphac o n e - f o r m t r a n s f o r m s as 1

dz ~ --

dz. T

(23)

402

0 Yasuda / Modular mvartance

..x

-~-IlT -I/T

-1

0

Fig 3 The contour mtegral away from the parallelogram after the modular transformatmn (22) according to the prescnptmn (24)

If the two points A(~-, '7) and B(~', ,7) transform as

1 A ---, A ' =

- -A(-

1 / ~ ' , - 1 / ' 7 ) = '72(1 - T ) ,

,r

1 B --* B ' =

-

-B(

-

1/r,

'7"

-

'7 1/'7) = - (1 - r)

(24)

I"

and I~'1 >> 1, then these points would be no longer the same parallelogram, but be in different parallelograms far from the original one (cf. fig. 3). This ~mplies that the contour integral (17) goes around the nontrivial homology cycles several times (cf. fig. 4). However, since this modular transformation (22) is just the exchange of the a-cycle and the b-cycle, the prescription (24) is not correct. Instead, A and B should

-1-1/~ -1 Fig 4 The contour integral going around the homology cycles several times after the modular transformation (22) according to the prescription (24)

-lIT

0

Fig 5 The contour mtegral after the modular transformatmn (22) according to the correct prescription (25)

0 Yasuda / Modular mvarmnte

403

transform as

1 = - - A ( r , or)

A ~a'

l+r

1 B ~ B' = - - B ( r , ~)

l+r

(25)

so that the configuration of the parallelogram and the contour integral (17) remains unchanged (cf. fig. 5) Tins example shows that even though the points depend on the moduh parameters nontnvially, they transform in the same way as ordinary points do under the modular transformations. Hence an the following discussions of the modular transformation, we do not consider the transformation coming from the dependence of the insertion points on the moduli parameters.

4. The modular invarianee of the amplitude Let us now consider the Ingher genus amplitudes of type II superstring theory The N-point amplitude with the picture-changing prescription for supermoduh ~s given by ?v

A(1,...,N)g=

E C., C.,~fd'~ 'm'd'~-'m'I-I f d2< OI R , ~l L

x fDx. DX"DYCD~p"D~fDbDbDcDfDBDfiD~,D~ ' N

xe- olq l=1

2g-2

13

[{Q, ~ ( R , ) } +

O,~(R~)D,]

a=l

2g-2

X 1-I [ { ~ ) , ~ ( R . ) } + a=l

6g-6

O,~(R.)D,] I-I (~l,,b).

(26)

l~l

Here

So=

1 ~

f d2z[ax.ax. + 22t,x ~ + +,[%p,

+ tp ~ o~b, + ( b S c + fl-O7 + c.c.)]

(27) is the free action, a' is the Regge slope parameter, {Q, ~(R,)} is the pmture-changlng operator [6] defined by the contour integral of the BRST current Jz around the scalar ghost ~ at the insertion point R~ (a = 1. . . . 2 g - 2), and Z- is defined by Jz = - ~ c ( c 3 X . c 3 X + 2 ~ . ) t + O ~ . ~ ) - b c O c + ~ y t ~ . ( O X + l ) t ) + ~ y

8 D, = 3(rt,, b)

(l = 1 . . . . . 6g - 6)

1 2b.

(28)

(29)

404

0

Yasuda /

M o d u l a r znuarzance

is the antlcommuting operator introduced in ref. [10], and 2t", ~," (~t = 1 . . . . . 10) are the Lagrange multiplers [11] introduced to make the term ~°~b"~a~b~ hnear m the gravitmo X~ (a = z, Y) in d = 2 supergravity [16], Cm,, C,,n are the coefficient of spin structures m R for the right movers and m L for the left movers, m', ~ ' (z = 1 . . . . . 3 g - 3) are the holomorphic and antlholomorphlc moduh coordinates, and V(z,) is a certain vertex operator. Notice that there as no 3-function in the correlation function ((O~X~+ 12~)(0~X~+ z ~ ) due to 2~. Note also that the BRST current (28) is slightly different from the usual one m ref. [6] because of the Lagrange multiplers X",~,". However, the difference of the results wzth two BRST currents is proportional to M, X~, and these fields give us only contact terms which have been discussed m detail m ref [17], so that there should be no difference in the amplitudes. The amplitude (26) is expressed in terms of various correlation functions and the determinants of the 0-operators acting on various differentials. The correlation functions are given by

fw,~'(Im~),ji Im

( X ~ ( z ) X ~ ( w ) ) = - ~a'3 ~ In I E ( z , w)12 - 2~r Im

[

1

(X"(z)X~(w))=½a'6~'0~-0~ l n l E ( z , w ) 1 2 + g l n l o ( z

(+"(z)~k~(w)),~=-~

)

,~+ g 1l n l o ( w ) l 2] ,

0~(f~) 3~ a"~E'z,~ w)O~(O) ' '

< b ( z , ) . . . b( z3g_ 3)b( Xl) . . . b( xn)c( yl) . .

,

~ z=l

Z, +

liE(z,,

(x--y,)--3A

z
z=l

(32)

3

zj) zFI= l ~3(z,)

YI,
~(Xo)...~(x,)~7(Yl)..-~7(Y,) I~ n lOm(~_,2gl2Z = \ Y= J n 2g-2

(33)

\

I-I e +(-'')) t=l

ol,n

2

(31)

c( y.) ) 3g

=a'nO

~oJ , (30)

[

m

+ Xo+ E'j=i(xa-yl) - 2A --y,)

]-I,=o0,~(Ej= I z; + x 0 + E ln= l ( x j - y j ) - 2 A - x,)

n, < je(x,, xj)n, ~ ~E(y,, Yl) x n,,le(~,, yl)n,
(34)

where ~ and ~7 are the scalar fields which appear in the bosonization of the

0 Yasuda / Modular mvartance

405

superconformal ghost [6]: fl = e-OO~,

~, = e%/.

(35)

In eq. (31) o(z) and o(w) have been added so that this gives us the same answer when we integrate by parts. On the other hand, the determinant of the 0-operator acting on X-differentials is expressed through the combination Z x -= det' 0x (det' 01)1/2

(36)

for X = 1 (X"), X = 1 / 2 (~p~), X = 3 / 2 (fl, ~), X = 2 (b, c). For half-integer X, we have to specify the spin structure m. (Z3/2) m and Z 2 are essentially given by the case of n = 0 in the correlation functions (34) and (33), and Z I and (Zt/2),, are given by

o (~gt= lZt -- Z O - A ) I - I I . < , , J ~ g E ( Z , , Z 1=

-

z j )~I gt=lO( Z, )

~

act

,

% ( zs)Flg,=lE( z o, z , ) a ( Zo)

(zl/~) ~ = 0., (0).

(37)

(38)

Now let us discuss the modular transformation propertms of these correlation functions and the factors Z x. Using the transformation properties (6), (7), (16), and (17), it is easy to check that the propagators (30) and (31) of X" and )t",X~ are invariant under the modular transformations. As for the propagator of 4, ~, we have to use the transformation law:

=tdet(Cr + D)l/2explrr[-O(m) + z(Cr + D)-lCz]Omo+a(z,r), (39) where i, ~, m o, 8, and ~ ( m ) have been defined m eqs. (7), (9), (10), and (11), and we have used the fact that ~(m 0 + 8 ) = -¢~(m) for generators (3). Using eqs. (16) and (39), we observe that the propagator of ~p~ transforms as

E(~,w)Om(O)

E ( z , w)0mo+~(0)

(40) •

F o r the correlation function of the reparametnzatlon ghost, because of the transformation property (13), the theta function in (33) transforms as (with the restnctmns

0 Yasuda / Modular mvarmnce

406

(14) and (15))

O(z - 3A, ,1") --+0(~ - 3~, rr) =

+ D)-l(z-

= e det(Cr + D)l/2exp[lrrz(Cr + D) 1Cz]O(z- 3A, r),

(41)

where

z=

Y'. z,+ t=l

(x,-y,)

(42)

t=l

and z~ is given by eq. (13). The contribution z(C,r + D)-ICz is canceled by the similar terms coming from the prime forms E(z,, zj) and o3(z,) etc. in eq. (33), so that

( b( zl) . . . b( z3g_ 3)b( Xl) . .. b( x,,)c( yl) . . . c( y,,) ) --+e d e t ( C r + D)l/Z(b(Zl)... b(z3g 3)b(Xl)... b(xn)C(Yl)... C(Yn) ) . (43) Similarly, the transformation property of the determinant of the scalar fields can be obtained using the following:

O ( z - A, r) ~ e d e t ( C r + D ) t / 2 0 ( z - A, r), det ¢o,(zs) --+ det(C'r + D)

(44)

ldet w,(zs)

(45)

Thus Z 1 transforms as Z 1 ~ e det(C'r

+

(46)

D)3/2Z1 .

For the correlation function of the superconformal ghosts, the theta function in eq. (34) transforms as (with the restrictions (14) and (15))

O m ( z - 2 k , r ) ~ 0,.(.~- 2Z$, "~) = exp(-4rrtm'8")Om(t(C'r + D ) - i ( z - 2A ), q) = e det(Cr + D ) l / 2 e x p t r r [ - e o ( m ) - 4 r n ' 6 " +

XOmo+8(z-2A,r),

z ( C r + D)-ICz] (47)

407

0 Yasuda / Modular mt,artance

where

E

i=1

1=1

(x,-y,).

(48)

Notice that we have the extra factor exp(-47rtm'3") in addition to the factor exp[-t~rq,(rn)], and this factor plays the most ~mportant role to get the expression which is both modular lnvarlant and consistent with the factonzation property, as we will see later. Note also that this is the specml case of the general formula

(Z~),,--*exp,rr[eo(m)+2(2h-1)(m'o6"+m6'&)](Zx),~

for ?t ~ Z +

~ (49)

(modulo a phase proportional to the local anomaly) given in ref. [8]. Therefore we have the following transformation:

E-' det(Cl- + D) -1/2 +(Xo)...

~(Xn)rl(yl)...

2g-2 / ~(y,) I-I e<'("P . (50) t=l / mo+8

N o w we can get everything together to discuss the transformation property of the amplitude (27). It is easy to check that the factors ~ and det(C~" + D) cancel w~th each other in the contributions cormng from (Z1)-5, (Z~/2)], (det lm~')-5 and the correlation funcuons of the reparametrization and the superconformal ghosts. The correlation functions of X ", M, and ~" are mvanant under the modular transformation, the correlation functions of ~" are invarlant except the change in the spin structure (cf. eq. (40)), and the correlation functions of the reparametrlzation ghosts are mvariant up to the factor of ~ det(C~- + D) ~/2. Furthermore the only difference of the correlation functions of the superconformal ghosts after the modular transformation is the phase and the spin structure (cf. eq. (50)). Hence the only nontnvxal factor is the phase which depends on the spin structure, and we denote the amplitude symbolically as follows:

A(1. U).= f d3-3m'd3g-3m'fl f d2 , Cm. .... o,.

0

mL(O)5

(51)

w h e r e (Z3/2) m symbolically stands for the superconformal ghosts determinant, and

FmR(z,), which

comes from the correlation functions of ~b~, fl, y, depends only on spin structure m~ for the right movers, while FmL(Z,), which comes from +~', fi, ,~, depends only on m L. The factor G(z,) comes from the correlation functions of X ~

0 Yasuda / Modular tnvarlance

408

but this part is independent of spin structures m R and m L. Although the insertion point of the picture-changing operator is dependent on the moduli coordinates, according to the example that we have discussed above, the abelian integral should transform as

f AB(m"~')coJ ----)'t(C"i" + D ) j l [B('n" ~')¢Ok . (m', ~') UA(rn', ~')

(52)

So, the only change of F,,,R(z,) and F,,,(z,) is that of spin structures rap., m c. The transformation property of (51) under the modular transformation is

Om(O)5

F,,,(z,) (Z----~/2)-m ~

exp4wt[-q,(m) -

m'8"]

F,,o+~(z,)

0m°+8(0)5 (Z3/2)mo+ .

(53)

On the other hand, it is easy to see that the phase m eq. (53) is always + 1 for any generator in (3):

expn~rt[-ep(m)-rn'8"]

=1

for T' (t = 1 . . . . 3 g - 1).

(54)

Therefore, if we demand the modular invanance for the right and the left movers separately (this is in fact the condition of the type II superstring theory in ten-dimensional flat background), then ~t follows that

CmR = CmL : 1

for any ma, m L .

(55)

Only in this case is modular invariance of the type II superstrlng theory in ten-dimensional fiat background guaranteed.

5. The factorization property The conditxon (55) has been derived only by the reqmrement of multiloop modular mvariance, and we have to check that it indeed satisfies the factorization property. The factorization property of the multiloop amplitudes has been discussed in ref. [18]. In ref. [18] the degeneration limit "r,s --* drag(, h . . . . 'rg)

(t, j = 1 . . . . . g)

(56)

was considered, where sj ( j = 1 .... , g) is the modular parameter for each torus. In this hmit, the product of the factors Z x becomes [18]

z:o,.(o)

{

(57)

0 Yasuda / Modularmvarlame

409

where = exp47rt(m~ + mjt ! )

=

+1

for

11 ,

IS the coefficient of the GSO projection [19] for each torus Therefore the condition (55) actually satisfies a correct factorlzatlon property*. Thus we see that the condition (55) satisfies both multiloop modular lnvariance and the factonzatlon property. It should be stressed that the phase exp(-4~rtm'8") from the superconforreal ghosts is crucial to get the condmon (55). Had we had only the phase exp[-47rteo(m)] from we would not have arrived at the con&tion (55), and that would contradict with factorization property. Finally let us discuss the factorization property of the genus-two amphtude in more detail to examine the prescription in which the picture-changing operators are placed on the branching points of the genus-two Riemann surface [20]. For simplicity we consider only the contribution of the matter supercurrents e~p • OX in the genus-two partition function. It is given by

Om(O)5/(Z3/2)m,

Om(O)4Om(R1 -- R2) E (eq'(R')tP'OX(R1)eq'(R2ltp'OX(R2))-- ~ -~(Rll +-R-2--~) '

(59)

rn

where RI, R 2 are the insertion points of the picture-changing operators, and we have ignored the parts which do not depend on spin structures, since we are interested only in the terms which depend on spin structures here. Here we adopt the hyperelhptlc coordinates, since any genus-two Rlemann surface is hyperelliptic Rlemann surface [211. Let us consider a Raemann surface defined by 6 s2(z)

= I-[ (z - z(Q,))

(60)

t=l

winch is a two-sheeted covering of the Raemann sphere by the function z with ramification at the Weierstrass points Qa . . . . . Q6 (cf. fig. 6). We introduce the vectors k Q, (t = 1 . . . . . 6) of the R i e m a n n theta constants with basepoint Q, (t = 1 . . . . . 6). The vectors k Q, can be explicitly expressed in terms of the period matrix and the odd spin structures (in case of genus two) [14], and in our choice of * In the case of space-time boson amphtudes, we cannot determine the relative phase (+ 1 or -1) between even and odd spin structures by the factonzatlon property [3,28] In this discussion of modular mvarlance, we have exactly the same situation, and the difference m the relative phase corresponds to whether the theory is the type IIA or I1B [3]

410

0

Yasuda /

Modular mvartance

01

Q6

Fig 6 A hyperelhptic Raemann surface of genus two with branching points Q, (t = 1, (z = 1, 2) are the canonical homology basis

the homology basis, for example

k o,

= zr

k Q~

()0

+

1

and

2

1

1

1

given by

k Q4 a r e

=

,6) A , , B ,

2

,2+1

(61)

'

1 {r l + r l 2 + l

,+,21

(62)

Let us now consider the degeneration limit (63)

",s ~ diag(rD r2)' or m the hyperelllptic coordinates,

(64) We have to discuss two cases: the degeneration limit m which two picture-changing operators are on different ton (cf. fig. 1), and the degeneration limit m which they are on the same torus (cf. fig. 7). In the former case, let us suppose that R 1 is on one torus M 1 (with Wmerstrass points Qt, Q2, Q3, and Pl) and R 2 is on another toms M 2 (with Wemrstrass points Q4, Qs, Q6, and P2), where Pl, P2 are the posmons of two punctures on the tori (cf. fig. 8). It is convenient to choose Q1 and Q4 as basepomts of the vectors of the Riemann theta constants, and we get

R 1+

R 2 -

2A =

Rlo~+ Q1

-

k Q1

4 ~-

= f R ~ o + fR2

k Q4 -

1

1

(65)

0 Yasuda / Modularmvartame

411

{QA}

fQ,~} Fig 7 A genus-two Raemann surface m the degeneratmn hmlt, where two picture-changing operators {Q, f} he on the same torus

Fig 8 In the degeneratmn hmlt, the hyperelhptxc genus-two Raemann surface becomes a product of two ton with branching points Q~, Q> Q~, p~ and 04, Os, Q6, P2, respectively

In the hmlt (64), the h o l o m o r p h m one-forms behave [14,22] as f d I ( Z , t ) = C01(Z ) q" O ( t 2 )

ZE m 1

Col(z, t) = O ( t )

z ~ M2,

(66)

and the s a m e formulae with 1 ~ 2, where t = 2vrt'r12 -+ 0 is the pinching parameter, Co,(z, t) (t = 1,2) are the holomorphlc one-forms on the genus-two Raemann surface near the limit (63), and c%(z) (l = 1,2) are the holomorphlc one-forms on the ton respectively. So, in the hmit (64), two c o m p o n e n t s of (66) behave like

MI, M2,

fQtq

(65) 1 -

r~ + 2

~%

2

f, R~-co2

(65)2 -

Q4

'

2r2+1

(67)

2

On the other hand, in our choice of the homology bas~s, there is the following c o r r e s p o n d e n c e in the hmlt (63):

Zl(Pl ) ~---0,

coI = d z ~ , r1 Zl(Q1) = 7 '

r1+ l Z l ( O : ) = - -2 '

~2 = d z 2

(68)

z 2 ( P 2 ) --- 0

1 z2(O4)

1

z1(Q3) = 2 '

= 2

z2(Q5)

%+1 2

r2 z2(Q6 ) = ~-,

(69)

where zp z 2 are the local coordinates of the tori M~, M 2. Hence (67) can be written

-

65,2 --

2 /'02 --

2

/?2 W2

2,2+1 2

-r~

--

-1,

2 022 -- "/'2 -- 1

(7o)

0 Yasuda / Modularmvartance

412

Using the factorlzation formula [8] of the theta function in the limit (63)

Om( D ) - Om,( Dllrl)Om2( D21r2)

(71)

(m = [ml, m2] IS the genus-two spin structure, and D is the argument which splits into D 1 and D2) , we get

Om(RI+R2-2A)-Om,

1 °)1

-Zl-

l i t 1 0m2

=exp[--lqr('glq-'r2) q-R~rl(fpR1601q- fpRZw2)]

where Cm, (t = 1, 2) is the coefficient of the GSO projection defined in eq. (58). Also using the properties (66) and (71), we have

Om(O) - O,nl(Ol'rl)Om2(Ol'¢2) ,

(73)

0..(j;...,., t

,,4,

Therefore we get

[

(59) ~ m~,E.,2C"Cm2eXp -t~r('cl + "r2) + 2~,

(J;

lwl +

tl

2 °~2

'

(75)

so that we have a correct factorization property, since Cm~, C., 2 are the coefficients [19] which appear m the one-loop amplitudes, and the exponential factor in eq. (75) is canceled [18] by the contributions from Z { 5, Z 2 and o ( R 1 ) - z , o ( R 2 ) -2 in ( Z 3 / 2 ) m. On the other hand, when the p~cture-changing operators are on the same torus, the situation is different. In this case it is convenient to choose Q~ as basepolnt of the Rlemann theta constants, so that we have

R1 + R2 - 2A = J,~'lRlo~"~ "lR2(.O-Q1 JrQ1

2 k Q1

(76) QI

Q1

413

0 Yasuda / Modular mvartame

By the same calculation as before, we get

(S:' fR2"l]'7-') 0m2(7-21i7-2) 1 0")i

QI

X Orn1t p, Wi -'}-

Om(Ri-R.) -

, %17-10m.(OIr2).

o.(s;, ) .

(77)

~ollrI 0.,2(015),

(78)

f Q,wl = -7"1 -"

(79)

where we have used the fact

Px

2

Therefore we obtain

(59) - e x p [ - , ~ r (

7-1q- 'r2) q- 2¢r/( fpR1fOi ff"

lT-~) ]]C rn~Cm 2 O'x(f~"°'- f;,-,< "~ R'~

1-~a'~iIJ

Oml(fpRll~,Ol+J;l-~O,]7-,)

(80) where we have the correct coefficient (the exponential factor is again canceled by the contributions from Zt 5, Zz, o(R1)-2,~(R2) 2), but we also have the factor which depends on Ri, R 2 on the torus M v It should be noted that as far as the modular mvariance is concerned, there is nothing wrong with (80), since the arguments of the theta functions both in the numerator and in the denominator transform according to the law (6). Tbas disagrees with the claim in ref. [10]. As an example, let us place R 2 on one of the Weierstrass points on the torus M 1 say zl(Q2) = 0"i + 1)/2. In this case one is tempted to argue that

C . . ) ,8,,

414

0 Yasuda / Modular mvarzance

because of the fact

=

(82)

+ l.

Pl

However, one should not do this, since identification (82) is n o t consistent w~th modular transformation rule (6). Namely, (82) should transform as

f

2 Q2wl --* Pl

2

Wl

CTI + ~

(83)

t

instead of

~°1= ~1 + 1 ~ -a y l-+ b

2fpQ 2

c~"1 + d

+ 1

(84)

and the modular invariance on the torus M 1 is not ruined. Eq. (84) Is exactly the same confusion as the one which we have discussed m the example m sect. 3 The only difficulty in eq. (80) is the dependence on the points R1, R 2. As was explained in ref. [11], in the degeneration hrmt (63), the dependence on these positions should disappear, because these positions become those of zero modes of the scalar ghost after deforming the contour integral of the BRST current onto the annulus winch connects two tori. In fact, this dependence on R 1, R 2 does not lead to vanishing of the cosmological constant at one loop (1.e. the one-loop tadpole diagram gives a divergence). The remark above also applies to the original genus-two expression (59), where one can place R I on Q1 and R 2 on a certain point winch must be on the torus M 2 in the degeneration hmit (63), but should be away from any of the Weierstrass points*. In tins case k Qa ~s gwen by eq. (61) but should transform according to the transformation law of A (cf. eq. (13)), and we have the modular lnvarlance on the genus-two Riemann surface also in tins case. Thus we see that the prescription of refs. [20] does not rum the modular mvarIance, but some of their ans~itze do not gwe a correct factonzatlon property as we have dependence on the insertion points (cf. eq. (80)). When one considers the modular transformation, one should not mix up the local coordinates of the insertion points of the picture-changing operators with the vectors of the Rlemann theta constants. * The reason that R 2 has to be away from the Welerstrass points is because the argument of the theta functions would become half period (1 e, rm6 + m6' for certmn spin structure m~)) m that case and the theta functions in the denominator and the numerator would vamsh for some m Therefore some of the terms ru (59) would become 0 / 0 and the answer would be ambiguous

0 Yasuda / Modular mt,artance

415

6. Conclusions I n this p a p e r we have shown the m u l t i l o o p m o d u l a r i n v a n a n c e explicitly using the p i c t u r e - c h a n g i n g prescription for s u p e r m o d u l i , a n d that the m o d u l a r m v a r i a n c e r e q u i r e s the coefficient for spin structure sum to be identically one in case of the t y p e II s u p e r s t r l n g theory in ten dimensions. The c o n t r i b u t i o n of the superconform a l ghost d e t e r m i n a n t is crucial to get the a m p l i t u d e which satisfies b o t h the m u l t l l o o p m o d u l a r m v a r i a n c e a n d the correct factorizatlon p r o p e r t y . By using a s i m p l e e x a m p l e on a torus, we have clarified how the a b e h a n integral whose i n t e g r a t i o n region d e p e n d s on the m o d u l a r p a r a m e t e r should t r a n s f o r m u n d e r the m o d u l a r t r a n s f o r m a t i o n : The abelian integral should t r a n s f o r m a c c o r d i n g to the rule (6) irrespective of whether the integration region d e p e n d s on the m o d u l a r p a r a m e t e r s o r not. Thus we have the m u l t i l o o p m o d u l a r m v a r m n c e lrrespecttve of h o w we c h o o s e the positions of the picture-changing operators. But ff we take a b a d choice, then we have the d e p e n d e n c e of the a m p l i t u d e on these insertion points, a n d get a w r o n g f a c t o n z a t i o n p r o p e r t y . T h e a n a l y s i s in this p a p e r was quite simple because the phase (54) which a p p e a r s m the m o d u l a r t r a n s f o r m a t i o n of O,,(O)5/(Z3/2)m is always + 1. In the case of lower d i m e n s i o n a l s u p e r or heterotlc string theories, the situation would b e c o m e m u c h more comphcated T h e a u t h o r w o u l d hke to t h a n k O. Lechtenfeld for discussions a n d the A s p e n C e n t e r for Physics for their h o s p l t a h t y d u r i n g p a r t of this work.

References [1] M B Green. J H Schwarz and E Wltten, Superstring Theory, 2 vols (Cambradge Umv Press, Cambridge, 1987) [2] E Wltten, m Proc Argonne Syrup on Anomahes, geometry and topology, cds W Bardecn and A R White (World Scientific, Singapore, 1985) [3] N Selberg and E Wltten. Nucl Phys B276 (1986) 272 [4] [. Alvarez-Gaumd, G Moore and C Vafa, Comm Math Phys 106 (1986)40 [5] A Parkes, Phys Lett B184 (1987) 19, I AntomadlS, C Bachas and C Kounnas, Nucl Phys B289 (1987) 87, F Ghozz~, Phys Lett B194 (1987) 30. Y Kikuchi and C Marzban, Phys Rev D36 (1987) 2583, S Ferrara, C Kounnas and M PorratL Phys Lett B197 (1987) 135, A N Schellekens, Phys Lett B199 (1987) 427. H Kawal, D C Lewellen, J A Schwartz and S-H H Tye, Nucl Phys B299 (i988) 431, E Gava. R Iengo and G Sotkov, Phys Lett B207 (1988) 283 [6] D Fnedan, E Martmec and S Shenker, Nucl Phys B271 (1986) 93 [7] L Alvarez-Gaumfi, G Moore, P Nelson, C Vafa and J B Bost, Phys Lett B178 (1986) 41. Comrn Math Phys 112 (1987) 503, V Kmzhmk, Phys Lett B180 (1986) 247, T Eguchl and H Oogurl, Phys Lett B187 (1987) 127, M Dugan and H Sonoda, Nucl Phys B289 (1987) 227 [8] E Verhnde and H Verllnde, Nucl Phys B288 (1987) 357

416 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

[21] [22]

0 Yasuda / Modular mvartance

E Verllnde and H Verhnde, Phys Lett B192 (1987) 95 J J Atlck, G Moore and A Sen, Nucl Phys B308 (1988) 1 O Yasuda, Phys Rev Lett 60 (1988) 1688, 61 (1988) 1678 (E) and m preparation J Barmann, Braxds, hnks, and mapping class groups, Annals of Mathematical Studies No 82 (Princeton Umv Press, Princeton, 1975) J Igusa, Theta Functions (Spnnger-Verlag, New York, 1972), E Freltag, Saegelsche Modulfunktionen (Sprlnger-Verlag, New York, 1983) J Fay, Theta Functions on Raemann Surfaces, Lecture Notes m Mathematics No 352 (SpringerVerlag, New York, 1973) D Mumford, Tata Lectures on Theta, 2 vols (Blrkhauser, Boston, 1983) L Brink, P DI Vecchla and P Howe, Phys Lett B65 (1976) 471, S Deser and B Zummo, Phys Lett B65 (1976) 369 M B Green and N Selberg, Nucl Phys B299 (1988) 559 O Yasuda, Phys Lett B215 (1988)299 F Ghozzl, J Scherk and D Olive, Nucl Phys B122 (1977) 253 V G Kmzhnlk, Phys Lett BI96 (1987) 173, A Morozov and A Perelomov, Phys Lett B197 (1987) 115, A Morozov, Nucl Phys B302 (1988) 163, B303 (1988) 343, O Lechtenfeld and A Parkes, Phys Lett B202 (1988) 75, G Moore and A Morozov, Nucl Phys B306 (1988) 387 See, e g, H M Farkas and I Kra, Rlemann Surfaces (Springer-Verlag, New York, 1980) A. Yamada, Kodm Math J 3 (1980) 114