Point-wise vanishing of two-loop contributions to 1,2,3-point functions in the NSR formalism

Nuclear Physics B318 (1989) 137-152 North-Holland, Amsterdam

P O I N T - W I S E VANISHING OF T W O - L O O P C O N T R I B U T I O N S T O 1, 2, 3-POINT F U N C T I O N S IN T H E NSR F O R M A L I S M A. MOROZOV

Institute of Theoretical and Experimental Physics, Moscow, USSR Received 2 November 1988

A point-wise vanishing on the module space of contributions to 1,2,3-point functions in superstring theory is demonstrated, provided that both odd moduli are located at one of the ramification points. An analysis of generalized Riemann identities, which are important for development of NSR formalism on hyperelliptic surfaces, is presented.

1. Introduction

There are general arguments, that in properly defined NSR superstring theory the one, two and three-point amplitudes (as well as partition functions) should vanish in all orders of perturbation theory [1]. However, we do not yet have any explicit construction of such a theory beyond one loop. It is widely believed that the proper theory arises from fermionic string after a special summation over spin s t r u c t u r e s - the analogue of GSO p r o j e c t i o n - is performed. Unfortunately this procedure appears not to be so simple. The most severe problem is that fermionic string theory, at least when it is considered "naively", seems to depend on the choice of odd moduli [2]. Arbitrary choice does not lead to anything similar to superstring. Moreover, even if one manages to get rid of the arbitrariness (which is in fact related to the choice of odd moduli near the boundary of module space, where the ideas of physical factorization are applicable), in a general coordinate system the integration over module space need not give rise to simple formulae (see ref. [3] for such a general analysis). Fortunately nothing still prevents one from believing that superstring formalism is simple enough, if some special coordinate system on the supermodule space of fermionic string theory is used. Moreover, in ref. [4] it was demonstrated, that the two-loop statistical sum corresponds to point-wise vanishing measure on the module space, provided one of the odd moduli is localized at the ramification point. Now a nice generalization of this coordinate system is known [5]: odd moduli should be located at zeroes of a holomorphic one-differential W, which is to be extended to v,2. (Similar arguments are also presented in ref. [6].) It is easy to check that for this choice of coordinates

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expressions for 1, 2, 3-point functions in two-loops are also point-wise vanishing on module space. This prescription seems also reasonable for higher genera. However, experience with two-loop partition functions implies a question: whether this prescription is too restrictive- for partition functions it appears sufficient to localize only one of the two odd moduli at the ramification point. We show below, that for point-wise vanishing of 1, 2, 3-point functions in this case it is necessary to tend the location of the second module to the same point, in accordance with the general prescription of ref. [5]. Moreover, some formulae, used in the present paper may appear useful for further development of calculus of Riemann surfaces of genus 2. We also give a more accurate formulation and proof of useful identities from the appendix in ref. [2], which appear to be a synthesis of Riemann identities and Riemann's vanishing theorem.

2. Basic formulae

All our calculations below are performed in hyperelliptic coordinates, and we use notation and results of ref. [7], see also refs. [6, 8]. Let us present here a list of the most important notations: the even spin structure (theta-characteristics) e corresponds to division of 6 ramification points a 1. . . . . a 6 into two sets: {A1, A 2, A3} and {B 1, B2, B3} ; 6

3

3

s2(z) - I--I(z-ai) = H (z-A,) i=1

a=l

2 - I~I__l(z-A~)

H (z-B~); a=l

(z-B~);

b/z

=

odd characteristics e, (or simply * ) correspond to a choice of single ramification point R , ~ ( a 1. . . . . a 6 ). Correlator of Grassmannian fields with the spin ~ is equal to [9]:

(~(x)~(y)>

Oe(0)

e

1 Ux+Uy ( d x d y ) 1/2

2

x-y

Correlator of Grassmannian fields with the spin 3 (if the two-dimensional metric is [v, 14;

v2,( z ) - O,,k(O)oak( Z ) = [C,(Z-- R , ) / s ( z )] d z , c, = (det o )1/2/7( a ) 1/4/S1/2 ( R , ) ,

A. Morozov / 1, 2, 3-point functions

139

see refs. [10, 7]) is equal to [11]: (~(x)~(y))

Fe(x' Y ) -

Oe(O)

e

1 ux-

- 2 ~

Uy ( d x d y ) 3/2 s(x)s(y)

We also need "generalized" Riemann identities [2]: Y'~( e , , e)Oe4 = O,

(0~ = 0~(0)) ;

(1)

e

Ze (e*' e)04 -uy

=0,

Vx, y;

(2)

Y'~(e., e)O~ uxuy d x d y d z d w e

ttzUw

(3) These identities are proved by the following reasoning. First, according to Thomae formula [7, 9,10] 3

04 = (det o ) 2 H ( A ) H ( B ) ,

H ( A ) - - V[ ( A , - A , ) . a
Sum over e with weights @,, e) is equivalent to antisymmetrization over five ramification points different from R,. Minimal power of antisymmetric polynomial II(a)/£Z(R,) is 10. In eq. (1) a 6-power polynomial appears, and the 1.h.s. vanishes, since 6 < 10. The same reasoning does not work so simply for eqs. (2) and (3), since polynomials of powers 12 and 18 respectively appear. However, antisymmetrization is easily performed in the case of eq. (2) and the answer is zero. In the case of eq. (3) it is reasonable to use eq. (2) in order to conclude that the 1.h.s. of eq. (3) vanishes when x = z or w and when y = z or w. Thus 1.h.s. of eq. (3) is proportional to ( x - z ) ( x - w ) ( y - z ) ( y - w). Moreover, if, for example, R , coincides with A1, then 1.h.s. of eq. (3) is obviously proportional to ( x - R , ) ( y - R , ) / s ( x ) s ( y ) ( x - R , ) ( y - R,). These remarks bring the problem to antisymmetrization of a polynomial of the power 12 = 18 - 4 - 2, which is again easy to perform. It may be a bit surprising that the 1.h.s. of eq. (3) is also proportional to (z - R,)(w - R,), since each item does not vanish if z = R , or w = R,. However, it is easy to check this statement, for example with the help of eq. (A.4) of ref. [2]: if z = R , , a factor of ( R , - A2)(R, - A3)FI3=x(Z - B~) - g2(R,) may be extracted from the 1.h.s. of (3), the remaining 7-power polynomial vanishes after antisym-

A. Morozov / 1, 2, 3-pointfunctions

140

metrization. Another way of reasoning uses the Riemann identity,

2 Pv2( Z1)p2( z2)V2( Z3)p2( Z4)

_i[( -7

E

(e,,e)O~(O)

even e

dztdz2dz3dz4

U2U4]] Ulb/3 + - Ul/23

U2U4

]J (Z1--Z2)(~2:Z3---~ :~4)(Z4--Z1)

Oe(Z12)Oe(Z23)Oe(Z34)Oe(Z41) E @,, e) E(zx ' z2)E(z2, z3)E(z3, z4)E(z4, even e

z1 )

,

from which we conclude, that

E

(e,,e)Oe4(O

even e

( UxNy UzUw - ( x - R , ) ( y - R , ) ( z - R , ) ( w - R,). + -UzUw UxUy

The separate items in the 1.h.s seem to have poles at z = R,, which are certainly absent in the r.h.s. Thus the poles drop out of the sum. One more way to derive eqs. (2, 3) is presented in appendix A.

3. Vanishing of 1, 2, 3-point functions In what follows we present only e-dependent structures and omit e-independent coefficients before them. There are two types of contributions to two-loop partition function of superstring. One comes from matter supercurrent and is proportional to

qse(a,b) /'e(a, b) 04(0)

(4)

(a, b are positions of odd moduli). Another one arises from ghost supercurrent; if b = R , , it is equal to

[~R/uolO4(0)

(s)

(uR - lim~0(UR+U/~))" If b = R,, the ratio 'Pe(a, R,)/Fe(a, R,) = s(a)g(R,)/ (a - R , ) d a d~ is independent of e. Therefore eqs. (4) and (5) vanish after summation over e due to eqs. (1) and (2) respectively. This was the argument of the last paper of ref. [4] in favor of taking b = R,. (Note, that while a 4: R , this choice disagrees with the prescription of ref. [5].) Let us proceed to calculate the 1, 2, 3-point functions.

One-point function."

• (a, x)~(~, b) F(a,b)

0? -~ ~uo- - +- - - ~ OJ

u~

.),~2~ O.

A. Morozov / 1, 2, 3-point functions

141

Two-point function." 1)

g'(a'x)q'(x'Y)g'(Y'b)

04

, (U~+Ux)(Ux+UY) Oe4

r(a, b) Ux

= 2)

u~ux Uv

1+--+

tlv

m+

bl a

~x)04

Ua

hV(a, b)[q*(x, y)]2

r(a, b)

oJ

(1),(2)

,0;

, [q,(x, y)12oj ,

+2+

04

O;

3) the contribution of ghost supercurrent: [ v l x ( x , Y ) ] 2o4uR-e- Ua

'

Ux + 2 + .

mUv Ux

UR O4e Ua

R,)Rv2,(x)v2,(y)

(3) ( x - y ) 2 ( a -

;,(a) da

Generically this expression does not vanish, however, it does, if a -* R,. Moreover, it is very essential, that a fifth order zero arises at

a-*R,: (a-R,)2-~4;

v2(a)-~2d~;

da-~d~.

This is important, because omitted e-independent coefficients before this structure may possess fourth-order poles at a ---, R,. To conclude, we see that for point-wise vanishing on module space of the correlator of two vertex operators @+(x)~p(y) with b = R , it is necessary to take a also equal to R,.

Three-point function:

q'(a.b)

1)

F(a, b) q'(x, y)'l"(y, z)q'(z, x)OJ -~ ( ~ + u,)(u, + u~)(u~ + Ux) oJ (')'(~ o UxUyU z

~'(a,x)~'(x,b) 2)

F ( a , b) --+--

,

[ g'(Y' z)]RoJ +2+

ua

° 04dxdydz uy da

+ UzU a

Oadxdydz Uybl a

da

(3) ~.2(x)~.2(y),.~(z),.2(a)s(a)(x_a)2(v_z)2" da

da

142

A. Morozov / 1, 2, 3-point functions

This expression is again vanishing only if a--* R., however this is only first order zero, and it is not enough to cancel the fourth order pole. This problem is really resolved by prescription of ref. [5] (the clever limiting procedure a, b ~ R . is necessary); note, however, that the structure under consideration does not really contribute to the two-point function of massless particles on shell because of kinematical reasons:

q(a,x)~P(x,b)(aX(a)aX(b))

comesfrom

((k~)({+)(x)S(a)S(b)),

and therefore is multiplied by (k{) = 0.

3)

qs(a, x)'~'(x, y)'tt(y, z)g,'(z, b) F(a, b) 04 IgxUz 04 UaUy

(3)

V2,(X)V2,(y)v2,(Z )

v2,(a) da

After the use of identity (3) we encounter the same problem, but now the kinematics does not save the situation. Instead one may carefully work out the complicated original correlator, making use of the Wick rule. It appears, that because of the symmetry in x, y, z of the expressions of type 3) at a ~ R,, various items in

([++~]"(a)[¢+~]~(b)[++~]'[q~+~]J(x) × [+ + gS]k[+ + 4T]'(y)[+ + if] ~[+ + gSl'(z)) cancel, and there are no contributions of type 3). 4) the contribution of ghost supercurrent:

(hR/u,,)q'(x, y)'t'(y, z)'l'(z, x)O4 ~. (ux + u,)(uy + Uz)(U: + u~) Ua

UxUyU z

, ,,,~(a)

(3) v2,(x)v2,(y)v2,(z ) ~ ( a

Again this is vanishing only if a ~ R..

Oe~

- R,)2[(x _ y ) 2 + (x - z) 2 + (y - z)2].

A. Morozov / 1, 2, 3-point functions

143

Note, that the problems encountered in the cases 2) and 3) for the three-point function are absent if the prescription of ref. [5] is carefully applied. This suggests that points a, b tend to R , in a specially adjusted manner (see appendix B for details). I am indebted to G. Moore for valuable discussions. I am grateful to the Aspen Centre, where the main part of this work was done.

Appendix A We give here a proof of identities (2) and (3) in terms of original Riemann identities and Riemann's vanishing theorem along with some related formulae. We believe, that this rather tedious calculation still enlightens a bit the sense of eqs. (2) and (3). Riemann identities, which we are going to use, state [12]:

E (e,, e)Oe(a)OAb)Oe(C)Oe(a) all e

2 (a+b+c+~ 1O,(a+b c ~t

= 2 PO,

(A.1) In eqs. (2), (3) only sums over even characteristics e appear. It is easy to find the corresponding identity by adding to eq. (A.1) the same identity with d replaced by - d . Since Oe(Z) are even (odd) functions of z for even (odd) characteristics e, the contribution of odd characteristics drop out of the answer:

E (e.,e)Oe(a)Oe(b)Oe(c)Oe(d even e

((a+b+c+d) = 2 p- 1 O, 2

O,

a+b-c-d) ~

O,

(a-b+c-d 2

(a b c+~) (a+b+c ~)(a+b c+~) X 0,

×0,

2

+ 0,

2

(a b+c+~)(a b c ~t/ 2

0,

~

0,

.

2

(A.2)

Our next task is to express combinations of ux,..., u w entering eqs. (2) and (3) through theta functions. For this purpose we use two different (but equal) represen-

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144

tations for pair correlators of ½ and 3-differentials on hyperelliptic surfaces of genus 2 [9,11,4]:

(~b(x)~(Y))e Oe(O)

(Ux+,y) ( d x d y ) I/2 2 U~U~ (X -- y)

(~(X)~(y))e

(U x -- U.v) ( d x d y )

-

O~(O)

Oe(X--y ) O~(O)E(x, y) '

3/2 .

2 u~Uy s ( x ) s ( y ) F, Oe(x + y - 2R, )

Oe(O)

E(x, y)o2,(x)o2,(y)F,.

(A.3)

(Note, that representations in terms of u~ and u~, are valid only in hyperelliptic coordinates, while those in terms of 0, E and o,(x) - v,(x)/E(x, R,) are universal. £2(R,) c,2d~6

F, = [ ~ ' , ( R , ) ] - ' -

g2(R,) i f , - d-------W-

and

are factors accounting for gravitational anomaly. ~ is local coordinate in the vicinity of the point R , , d R , = ~d~. F, (but not if,) has invariant definition in terms of

.,(z).) We rewrite eq. (A.3) for a while as

Ux+Uy

Oe(X--y ) -- Axy

Oe(O )

;

u x - uy Oe(X + y - uff~xU~ " - Bxy Oe(O )

2R,) (A.4)

Now we shall use this representation for calculation of eq. (2), applying at the first step eq. (1), which is a particular case of (A.2) or (A.1) with a = b = c = d = 0. Y'. (e,,e)~-v0e4(0) = • oveoe

1+ --

eveoe

Uv

04(0) = E

--~

04(0).

(A.5)

evene

Now we substitute eq. (A.4) into (A.5) (note, that U x ~ - - =

~((ux + Uy)/U~U~, +

(Ux - uy)/ ufd~-~uy)): (A.5) = ½A2xy Z even

(e,, e)O~(x -y)O2(O) e

+½AxyBxy E even

(e*,e)Oe(x-y)Oe(x+y-2R*)Oe2(O) " e

(A.6)

A. Morozov / 1, 2, 3-point functions

145

The first item vanishes directly by (A.2) (or by (A.1) - odd characteristics obviously do not contribute). The second item is proportional to 02,(x - R , ) O Z ( y - R , ) and this is zero by Riemann's vanishing theorem. Analogous calculation for eq. (3) is much more complicated. Doing the same trick and applying both (1) and (2), we obtain: £ even

UxU ,,

(e,,e)

* YOe4(O)

Uzldw

e

.+.(.. uy+.

(e,,e) ~ even

u.

e

~

~

= ±'42 .42 ~_~ ( e , , e ) O e 2 ( X - z ) O ~ ( y 4**xz'~yw even

04(0)

.

w)

e

1 2 + -gAx~AywBvw Y'~ ( e , , e ) O Z ( x - z ) O e ( y - W)Oe(y + W -- 2 R , ) even

e

1 2 + gAxzBxzAy w ~_, ( e , , e)Oe(x - Z)Oe(X + Z -- 2R,)Oe2(y - w ) even

+¼AxzBxzAywByw

e

~ @*,e)Oe(x-z)Oe(x+z-2R*) even

e

(A.7)

XOe(Y -- W)Oe(y + W -- 2 R , ) .

With the help of eq. (A.2) one may check that the first item in (A.7) is !'42 4"*xz A y2w \( _ 2 p 1 0 2 , ( x _ z ) o 2 , ( y _ w ) )

= - 2 p+' ( x - z ) Z ( y -

w)202,(x-z)O2,(y

E2(x, z)E2(y,

= --2p+l(x--z)2(y--w)

- w)

w)dxdydzdw

2

(A.8)

dxdydzdw

The second item in (A.7) is proportional to o.(x

-

z + y -

+O,(x

-

R,)O,(x

z + w -

-

~ - y + R,)O~,(w

R.)O,(x

-

z -

-

R,)

w + R,)O:,(y

-

R,)

and vanishes due to Riemann's vanishing theorem ( O , ( x - R , ) =- 0 for any point x

A. Morozov / 1, 2, 3-pointfunctions

146

on the Riemann surface.) The similar third term vanishes for the same reason. The fourth term in (A.7) is equal to

¼AxzBxZA.wBv y . w × 2P-l{ O , ( x - y ) O , ( x + y - 2 R , ) O , ( z - w)O,(z + w - 2 R , ) -O,(x-

w)O,(x + w - 2 R , ) O , ( y - z ) O , ( y + z - ZR,) } . (a.9)

In order to handle (A.9) we need the following identity to be proved in the last part of this appendix:

O,(x--y)O,(x +y-- 2 R , )

=

F~I(X) e2(x, R,)E2(y, R,)cdet[w2(x)

~I(Y)] .

¢02(y )

(A.10) In hyperelliptic coordinates

fOl(y ) ]__

cdet[%(x ) [w2(x)

¢o2(y)]

if, x--y F, s ( x ) s ( y ) d x d y

(this equation may also serve to define c). Thus for eq. (A.9) we have ( A . 9 ) = 2 t'+l ( x - z ) ( y - w ) E(x,z)e(y,w)

s(x)s(y)s(z)s(w) (dxdydzdw)2 E(x,z)E(y,w)

XO~,(X)d(y)od(z)od(w)( F,/&) 2

P, 12 ( x - y ) ( z - w) - ( x - w ) ( y - z) × ~,) s(x)s(y)s(z)s(w) dxdydzdw XE2(x, R,)E2(y, Now we substitute here o,2(x)

=

e,)E2(z,

R,)E2(w, R,).

v2,(x)/E2(x, R,)

to

obtain

(A.9) = 2p+l(x - z ) ( y - w ) [ ( x - y ) ( z - w) - (x - w ) ( y - z)] X

; (x),.~ (y),.~ (z) ~.~(w) dxdydzdw

(A.11)

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147

Combining (A.8) and (A.11), we have

Z

lgxUY 4
even e

z

---- - - 2 P + 2 ( X - - Z ) ( X - -

×

w ) ( y - - a ) ( y - - w)

,d (x),4 (y)..~ (~)..~ (w)

(A.12)

dxdydzdw

which is exactly eq. (3) of interest. As a check let us note, that four a priori possible combinations of signs before three different items contributing to (A.12) (one item is (A.8), two others come from (A.1I)), give rise to the following four answers for the polynomial factor in (A.12):

(x-z)(y-w)Cx-y)(z-w);

(X--Z)2(y--w)2;

0;

(x- z)(x- w)(y- z)(y- w). The first two candidate answers would not be symmetric in x, y or in z, w and would not vanish when x = w or y = z as the 1.h.s. of eq. (A.12) should. This reasoning gives an easy check that the signs are defined properly. In order to prove (A.10) we rely upon two results, concerning zeroes of the theta-function, which are due to Fay (Corollaries 2.17 and 2.18 from ref. [9]). First equation of Corollary 2.18 with A = a, B = R , and f = y - R , ( f belongs to the theta-divisor for 0,, since O,(y - R , ) = 0) gives

O,(x- a +y- R,)O,(x-y-

a + R,)

= O , ( - a +y)[O,,k(y - R,)~ok ( x ) ] E(x, a ) e ( x , R , ) / E ( a , R,). In the limit a -+ R , we have

O , ( x - y ) O , ( x + y - 2R,) =-E2(x,R,)[O, k(y-R,)%(x)][O,,k(y-R,)%(R,)].

(A.13)

One of the theta function derivatives in eq. (A.13) is very simple; from the definition of Prime bidifferential,

e(y, R,) =

0,(y- R,) p,(y).,(R.)

0, k(y- R,)~,k(R,) p,(y)~;(R,)

one easily concludes:

O,,k(y - R,)~%( R,) = E(y, R,)p,(y)u;( R,).

(A.14)

A. Morozov / l, 2, 3-point functions

148

Another derivative in (A.13) can be found from Fay's Corollary 2.17 [9]: with x 1 = x, x 2 = y and p---> x it gives

O*'k(Y--R*)~°k(x)

°al(Y) ] 2(y)

o,(y)°etl~0z~x~[ ~ )

(A.15)

Let us recall that o , ( y ) = u , ( y ) / E ( y , R,). Therefore

O,(x - y ) O , ( x + y - 2 R , ) = cE2(x, R , ) E 2 ( y , R , ) d e t

,o,(x)

,o,(y) ]

,o2(x) 2(y) ]

(A.16)

with c = - 8u~(R,). Going from canonical one-differentials ~01,~2 to v 1 = z dz/s(z), v2 = dz/s(z), vi = oij~0j we obtain d [OOl(X et[~02(x )

~ol(y) ] = det ° x-y oa2(y ) s(x)s(y) dxdy.

(A.17)

In order to find ~ and prove, that cdet o = £v,~(R,)det o = ff,/F, use once more eq. (A.3) and formula (2.17) from [9] with x l - - x , x 2 = y and p = R , . Then

( o , ( x ) = ~,(x)/E(x, R,))

O,(x+y-2R,)E(x,y)

=8det

~01(x)

~,(x)p,(y) °~l(Y) 1 = 8 d e t o

lim p~,

E(p, R,)

x-y

~,(p)

dxdy.

(A.18)

The limit limp~ ~, E(p, R , ) / u , ( p ) = 1/u~(R,). Now we make use of the analogue of eq. (A.3) for the correlator of g-differentials in the case of odd theta-characteristics: x--y

(~'(xlf(y)),-

s(xls(y) dxdyv,(x)v,(ylff, = O,(x +y - 2 R , ) E ( x , y)o2,(x)o2,(y)F,.

(A.19)

(Just as in eq. (A.3) the factor of deG,03/z(det 30) 1/2 is included into the definition of correlator. The factors if, and F, are the same as in eq. (A.3).) Comparing eqs. (A.18) and (A.19) we come to the relation of interest: c det o = ?u~(R,)det o = f f , / F , .

(A.20)

It is worth explaining, why u,(x)z,,(y) enter representation (A,19) in terms of hyperelliptic coordinates instead of a simple product { [ ( x - R , ) ( y - R , ) / s ( x ) s ( y ) ] d x d y } 1/2, i.e. how the moduli-dependent coefficient is fixed. The key requirement here is projective invariance with respect to the transformation z-~

A. Morozov / 1, 2, 3-pointfunctions

149

(az + b)/(cz + d). Under this transformation (x - y) --, (x - y ) / ( c x + d)(cy + d), dx --~ d x / ( c x + d ) 2 6

s(x)--*

] - 1/2

i~=l(cai+d)J

s(x)/(cx+d)3-N-1/2s(x)/(cx+d) 3

( a 1. . . . . a 6 are ramification points). Therefore [ ( x - y ) / s ( x ) s ( y ) ] d x d y

is multiplied by N +1 under the projective transformation. The residue transforms as

~( R , ) --) N-1/2~( R , ) / ( c R ,

+ d) 2

and the differential d R , - ~ d R , / ( c R , + d ) 2. However, d R , = ~ d ~ and what we really need is the transformation law for d~: recall that because of the gravitational anomaly the correlator (A.19) is the 4-differential at R , and behaves as d~ 4. Since ~2 transforms as the difference between positions of two points R , + ~2 and R,, ~2

~2

~2___~ ( c R * + d ) ( c R *

+c~2+d)

_ ( c R * + d ) 2'

or

~ - -cR, ~ -+ d

and thus d~ ~ d ~ / ( c R , + d). Therefore the factor

F, =- #2( R , ) / d ~ 4 ~ N-lff,. The whole correlator (A.19) is to be projective invariant, thus the product of two ½-differentials, staying at the place of v , ( x ) v , ( y ) should be invariant under projective transformations. On the other hand [ ( x - R , ) d x / s ( x ) ] 1/2 acquires a factor of N1/4(cR, + d)1/2 under this transformation. It cancels only after multiplication by c~,/2= (det o)X/2H(a)X/s/£(R,)l/4, which is exactly what is necessary to turn [ ( x - R , ) d x / / s ( x ) ] x/2 into v,(x): x-R. =

=

¢

.

-

-

s(x)

dx.

In order to specify the projective transformation of c, one should note that 6

H(a) -

I-I(a~- aj) i
~ N-SH(a)

while oij, which is the transformation matrix between ~0i and v~, v~ = oij~oj, acquires a factor of N +1/2 - this is the way vi transform - and therefore det o --, Ndet o;

cl,/2 ~ N a/4N - 5/8N1/8( c R . q- d )1/2 c 2 / 2 : N - 1 / 4 ( c R . q- d )1/2 c1./2,

150

A. Morozov / 1, 2, 3-point functions Appendix B

In this brief appendix we present the derivation of the statement about pointwise vanishing of the two-loop 0,1, 2, 3-point functions, calculated with the prescription of ref. [5]. In fact we restrict ourselves with contributions of matter supercurrents only; note that contributions 2) and 3) to the three-point function, which have not been discussed in a quite satisfactory way in the main text, belong to this class. Calculation of ghost supercurrent contributions with this formalism is more complicated: one should use the technique, presented in the appendix of ref. [5]. For vanishing of expressions under consideration it is enough to localize odd moduli at zeroes of any holomorphic one-differential (it is not even necessary to tend it to v2). In this case the following relation holds in Jacobian: a + b = 2A = 2R,,

and eq. (A.3) implies, that F~(a, b) = (~(a)~(b))JOe(O) - 1 does not depend on e. Additional requirement of ref. [5] is that ~(a + b ) = R , , and sum over even spin structures e is performed with the weights (e,, e), defined by the same odd characteristics *. Note that we need not use hyperelliptic coordinates the prescription of ref. [5] is not specific for genus 2. Let us proceed to calculation of supercurrent correlators with matter supercurrents.

Partition function (statistical sum): XlFe(a, b)O:(O) (A.3~ Oe(a

_

_

b)O?(O) (A.I~ 04,((a _ b ) / 2 ) = 04(a - R,) =- O.

We used here the Riemann identity (A.1). The last equation is implied by Riemann's vanishing theorem.

1-point function: ~P(a,x)q~(x,b)O 4

,Oe(a-x)Oe(b-x)O2(O)

(A.I, 0 2(x - - - a +2b ] o]z {*1~ 2 b

) = O 2 ( x - R*)O2*(a- R , ) = O.

2-point function:

1) g'(a, x)gz(x, y)'Ct"(y, b)O4

(A.3

Oe( a _ X )Oe( X __ y ) O e ( Y

(A.I~ 0, T

0, X - - - 2

= O,(a- R,)O,(x-

_

b

) ( __o+b)( O,

R,)O,(y-

y

2

R,)O,(y-

O, y - x +

__a-b) 2

x + a - R,) =- O;

0

~

~

÷

~

~÷

~

~

Ill

~

÷ ~

~

~

~

÷

w~~

~

~

~

~

÷

~m~

r

÷

~

~

~

~i~

~

~ 1~ ~ ~ H ~ ~°~

~

152

A. Morozov / 1, 2, 3-point functions

References [1] E. Martinec, Phys. Lett. B171 (1986) 189 [2] G. Moore et al., Nucl. Phys. B306 (1988) 387 [3] E. Verlinde and H. Verlinde, Phys. Lett. B192 (1987) 95; J. Atick, J. Rabin and A. Sen, Preprint IASSNS-HEP-87/45, to appear in Nucl. Phys. B; H. Verlinde, Preprint THU-87/26; J. Atick, G. Moore and A. Sen, Nucl. Phys. B307 (1988) 221, B308 (1988) 1 [4] V. Knizhnik, Phys. Lett. B196 (1987) 473; A. Perelomov et al., Phys. Lett. B197 (1987) 112; O. Lechtenfeld and A. Parkes, Phys. Lett. B199 (1987) 53; A. Morozov, Nucl. Phys. B303 (1988) 343 [5] A. Perelomov et al., Preprint ITEP-88/80, to appear in Int. J. Mod. Phys. A [6] E. Gava, R. Iengo and G. Sotkov, Preprint ICTP-88/51; M. Bershadsky, Phys. Lett. B201 (1988) 67 [7] D. Lebedev et al., Nucl. Phys. B302 (1988) 163 [8] V. Knizhnik, Comm. Math. Phys. 112 (1987) 567; M. Bershadsky and A. Radul, Int. J. Mod. Phys. A2 (1987) 165 [9] J. Fay, Theta Functions on Riemann Surfaces (Springer, Berlin, 1973) [10] E. Belokolos and V. Enolski, Funk. Anal. i ego Priloz. 21(1) (1987) 81 [11] V. Knizhnik, Phys. Lett. B180 (1986) 247 [12] D. Mumford, Tata Lectures on Theta (Birkh~iuser, Basel, Stuttgart, 1983, 1984)