Correlation functions for the βγ system from the sewing technique

Volume 248, number 3,4 PHYSICS LETTERS B 4 October 1990

Correlation functions for the fly system from the sewing technique

P. Di Vecchia NORDITA, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark

Received 20 June 1990

Starting from the vertices Vs;g for a free scalar theory and for bosonic and fermionic bc systems obtained through the sewing technique we compute the correlation functions for a bosonic bc system and we find agreement with the result first obtained by Verlinde and Verlinde using the path integral technique.

The correlation functions for a bosonic bc system, called also fly system, have been computed by now using a variety of methods [ 1-3 ].

By using the sewing technique ~1 an N-point g- loop vertex VN;g for both a free scalar theory and bosonic and fermionic bc systems have been recently constructed [ 4,5 ] (see also ref. [ 6 ] ).

In this let ter we use those vert ices for comput ing the correlat ion funct ions for a fly system and we f ind com- plete agreement with previous calculations.

In the case o f a free scalar theory the following N-point g-loop vertex has been constructed [ 4 ]:

VN;g=(dett~o)-l/2i~=~(~n i(ni, Oal)f~(i~= g i - (g -1 )a )

X exp ( - ½ ~ ~ dz O~o(i) (z) (~ i ) -Q) log[ V;(z) ]) o

[ ×exp ½ i,~l ~dz f dyO~'i)(z)log k Vi(z)-Vj(y) 0 0

Xexp (½ i,~=l ~ dZ~ dYO~(i) (z) log[ V,(z)- ~(Y) ] O~O)(Y) ) i~:j 0 0

Vi(z) ot 1 N X (0(]~)([~i i~l ~dz 0q~(i)(z)( ff (.t)/.t)--Qz~° 1T)} 0 zO

Xexp(Qi~l ~ dz O~(')(z) loga[Vt(z) ] ) . 0

(I)

We have in t roduced the s tandard R iemann O-function with characteris t ics ot and fl, given by

~ For a list of references on the sewing technique see ref. [4].

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 329


O ( f l ) ( z l r ) = ~ exp[2~zi( ~ ½(nu+otu)zuv(n,,+ot,,)+ ~ (nu+otu)(zu+flu)) ] (2) {n~ } ,u,v= 1 /t= 1

with nu integer. The vertex VN;g is expressed in terms of geometrical objects as the period matrix %, the holomorphic differ-

entials o9 u, the vector of Riemann constants A~ °, the function a and the prime form E(u, v). Their explicit expression in terms of the Schottky parametrization of the Riemann surface can be found in ref. [ 4 ]. Finally the determinant is given in this parametrization by

(det t~o ) - I/2 - ~I' nI~I__ 1 1 (3) - = 1 - k ~ '

where the product is over the primary classes of the Schottky group. By saturating the vertex ( l ) with the states

Iqi)i- lim [V~-l'(z)]oi(qi+Q)/E:exp(qi~o[Vzl(z)]}: 10)i (4) 2~Zi

for i= 1 ..... Nwith qi = _+ 1 one gets the following correlation functions:

NI N2 (q=01 1-[ :exp[-~o(zi)]: I-I :exp[~0(%)]: Iq=0)g

i= l h=l

1 NI N ~h---1 tr(Wh) e ~(N~ -N2 "k (g-- 1 )Q) IIi,j=l;i<jE(2i, zj) I-[h.~=l;h<k E(Wh, Wk) N2 1-~'fi l-k,~ ~ t l i = l l - t h = l L ' k i , nJ = .=~ . ~ ~ - - ~ i - ~ - ~ - - ~ ) I-[~2~ a ( z , ) °

zi Wh

{ ( ~ ) ( I 1 ~ l ~ ~¢ou_QA~Olz)}, (5) × O - ~ni~=l ~ oJu+ ~-~h=~

ZO ZO

reproducing the result obtained with other methods [ 7 ]. Notice that both the vertex ( 1 ) and the correlation functions (5) involve arbitrary spin structures.

In the case of a bosonic and fermionic bc system one can also compute FN;g with the sewing procedure. The detailed calculations are explained in detail in ref. [ 5 ] and, apart from the exceptional cases of 2 = 1 and g= 1 for integer 2 that must be treated separately (see ref. [ 5 ] for details), the result is

N (2A--I , ( g - - 1 ) ( f ~ ) Vlv.g, = (det 01-2)- ' I-I (i(q=-Q[) I-I ~ duc(vi)(u)AA(u)

i=1 A=I i 1 zt

X:exp(_i,~=,~du~dvc~i)(u)G(u,v)b~)(v)): , (6) zi zj

where G(u, v) is the Green function of the (b, c)-system on a Riemann surface of genus g and (det 0~_a) gives the correct normalization. The subindex of J means that O acts on conformal fields with dimension equal to 1 - 2 . Moreover, we have the product of (22 - 1 ) ( g - 1 ) ~-functions containing the c-fields which saturate pre- cisely the (22- 1 ) ( g - 1 ) zero modes of the b-fields, parametrized by the A-differentials AA (U). This particular structure is clearly in agreement with the Riemann-Roch theorem for genus g.

The b- and c-fields appearing in (6) are the transported fields with conformal dimension equal to 2 and 1 - 2 respectively:

C ( T i ) ( u ) ~ - [ V ~ - l t ( u ) ] l - a c ( i ) [ V ~ - l ( u ) ] ' b(x~)(u)= [VF"(u)]ab(i)IVFl(u)]. (7)

The vacuum charge Q i s equal to Q = ¢ ( 1 - 22) with ¢ = 1 for a fermionic system and ¢ = - 1 for a bosonic system.

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For a bc system the calculation of Vu;g has been explicitly performed only for those spin structures that do not involve spin field exchanges in the loops. The explicit expression of the A differentials, of det tg~_x and of the Green function for those spin structures in the Schottky parametrization of the Riemann surface can be found in ref. [ 5 ]. Remember, however, that the form (6) of the vertex is the same also for the other spin structures, although the explicit form of the geometrical objects in the Schottky parametrization of the Riemann surface has not yet been computed.

The bosonization rules [ 8 ] for a bc system imply

c ( z ) = e x p [ ~ ( z ) ] , b ( z ) = e x p [ - q g ( z ) ] (8)

for a fermionic system and

c(z)=exp[Clg(z) ] q(z), b ( z ) = e x p [ - ~ ( z ) ] O~(z) (9)

for a bosonic system. Remember that the scalar field q~ (z) has the contraction ( q~ (z) q~ ( w ) ) = ¢ log ( z - w ) and ~q is a fermionic bc system with 2 = 1.

Because of the bosonization rules (8) the correlation functions (5) can also be computed in the fermionic theory described by the vertex ( 6 ). They are obtained by saturating (6) with the following states:

lim b(vi)(z)lq=O)i, i= l ,2 , . . . ,N1 , z~z i

lim C(h)(z)lq=O)h, h = l , 2, ..., N2. (10) Z~Zh

The result is

NI N2

( q = 0 l I-I b(z~) I-I c( wh) I q = 0 ) =det ff~_~ G(Zl, z: ..... zm I w~, w2, ..., WN:), (1 1 ) i=1 h = l

where G is the determinant of the following matrix:

A I ( z I ) AI(z2)

A2(z l ) A2(z2)

AN(Z1 ) AN(Z2) G(zl, wl ) G(z2, Wl )

: i

~G(zl, WN2 ) G(z2, WN2)

"'" A [ ( Z N I ) \

• . . A2( N,) . . . I G(zu!, Wl) I

I G(zm, wu2) /

(12)

and N=NI -N2 = ( 2 2 - I ) ( g - 1 ). Bosonization implies that (5) and ( 11 ) must be equal. They are certainly equal for 2= / as a consequence of

eq. (4.33 ) at p. 33 of Fay's book [ 9 ]. For arbitrary 2 we have not been able to check directly their identity, but we will assume it in the following. In the rest of this letter we will show that under this assumption the correlation functions for a fly system can be explicitly computed.

We want to compute the following correlation function for a fly system:

M + N m + n ( q = 0 l f i ~ ( X a ) f i ?](Yb) 1-I exp[q~(ui)] 1-I exp[--q~(vk)]lq=O). (13)

a = 0 b = 1 i= 1 k = 1

Because of the bosonization rules

O[fl(z)] =-exp[~(z) ] , 8 [ y ( z ) ] - = e x p [ - ~ ( z ) ] (14)

and

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~(z)mH[fl(z)] , q(z)mO~H[7(z)] (15)

where H is the Heaviside step function, one can first compute the correlation function

M+N Mq-n ( q = O I f i H [ f l ( X a ) l f i 7(Yb) 1-I O[fl(Ui)] I-I ~ [ r ( V k ) ] l q = O ) . ( 1 6 )

a=o b= I i= 1 k= 1

( 13 ) can then be obtained from (16) by taking the derivative with respect to Yb for b= l, 2 .... , n and the limit yb--~VM+b with b= l, 2, ..., n.

Although the starting point is somewhat different many steps of the derivation are the same as in the paper by Losev [ 3 ]. We think, however, that it is interesting to show that also the old fashioned sewing procedure devel- oped in the old days of dual theories reproduces in a very natural way the intricacies of the fl7 system.

In order to compute (16) we consider a V~;g for a fl~, system with a number of legs equal to ~r= 3n + 1 + 2M+ (22- 1 ) ( g - 1 ) and we saturate it with the following highest weight states:

fdPa -~paeXp(ipafl~a})lq=O), a = O . . . . . n,

~d2i -~-n exp (iXifl~)~ I q = 0 >, i=1 ..... M+N, (17)

and

1 0 exp(iqbTtb__)~) [q=O>, b = l ..... n, i Oqb qb=O

f ~ exp (i#kTtk__)a) I q= 0) , k= l, . . . ,M+ n, (18)

corresponding to the primary fields appearing in the correlation function (16). The part of the vertex (6) relevant for those correlation functions is given by

V,~;g= (det ffl_a) -1 I-I ( , ( q = - Q I ) :exp - ~ c~i)G(gi, zj)b~)_,~ : i= 1 i,j= 1

t2a-"tg-l) ~ dva ( ~ C ~ i ) A A ( z i ) v A ) (19) X AI~__1 -~-n exp ii 1

where we have used the integral representation for the ~ and the step functions and we have omitted all factors V;- 1, (0) both in the states ( 17 ) and ( 18 ) and in the vertex ( 19 ) because they cancel anyway when we saturate (19) with the states ( 17 ) and (18).

Saturating the vertex (19) with the highest weight states ( 17 ) and ( 18 ) after some calculation we obtain the following expression for the correlation function (16):

) n dn M+NO~ i (detO-l -a) -1 I-I ~ H - ~ Yb) "4" ~, 2 i G ( u i ' Y b )

a=O27t'lpa i=1 i=1

× k~=l ~a~=O p 'G(xa 'vk )+ ,=12 2iG(ui, vk) a=l a paAA(Xa)+ ,=IZ 2iAA(U,) • (20)

The last N + M + n ~-functions can be used to perform N + M + n integrations and we will be left with only one integration over Po. We have to solve the following system of equations:

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M + N ~ paAA(Xa)+ ~, 2iAA(ui)=--poAA(Xo), A = I , 2 , . . . , N ,

a = l i = 1

M + N PaG(Xa, Vk)+ E 2iG(ui, vk )=-poG(xo , v,), k = l , 2 , . . . , M + n . (21)

a = l i = 1

The previous set o f equations can be solved in terms of the determinant of the matr ix G in eq. (12) , that is directly related to the correlation function of a fermionic bc system, see eq. ( 11 ).

We obtain

G(xl .... , ~ . . . . . . xn, ul ..... uM+ulVl,. . . , vM+n) (22) Pa = --Po G(Xl, ..., x, , u~, ..., UM+NIv~ ..... VM+,)

and

2i = --Po G(xl ..... x , , Ul ..... fii ..... uM+ul Vl ..... vM+,) (23) G ( X I , . . . , X n , U l , . . . , U M + N I U 1 , . . . , U M + n ) '

where :~a and ~i means that we have to substitute them with Xo. We can then insert (22) and (23) into the integrand of eq. (20) getting

n 1 (a~__oPaG(Xa,Yb)+ N+M ) ~ G ( X o , X I , ..., Xn , Ul~_ ..., UM+Nl l . l l , ...~ VM__+n, Yb._...__~) bI~I__l--~b ~ , ~ i a ( u i , Y b ) = [1 b=l (24)

= i = 1 a = l G ( X l . . . . , -~ . . . . . . Xn , U 1 . . . . . UM+N[ UI, . . . m UM+n) "

One can then easily per form the integral over the fi-functions and one gets the final expression for the correlation functions ( 16 ):

n G ( X o , X l , . . . , X n , U 1 . . . . . U M + N I V l , . . . , F M + N , Yb ) (25) ( det i~l-a)-l I-[a=O fi(Xo, Xl, ...,Xa . . . . . Xn , Ul . . . . , UM+N]VI, ..., VM+N)

after having used f dPo/2nPo = 1 following Losev [ 3 ]. The tilde in the last formula means that the corresponding variable is absent.

Expression (25 ) for the correlation function ( 16 ) agrees with eq. (33 ) of ref. [ 1 ] i f we take M + N = 0. Since (5) and ( 11 ) are equal we can compute G (zl .. . . , z , I w~, ..., win) in terms of O-functions, products of

pr ime forms and a-functions and we can substitute this expression in (25 ). In order to get then ( 13 ) f rom (25 ) we have to differentiate with respect to Yb for b = 1, 2, ..., n and then take the limit Yb~VM+b with b = 1, 2, ..., n. Since n of the pr ime forms have simple zeroes when Yb-*VM+b with b = 1, 2, ..., n we need only to differentiate these terms and per form the limit directly in the rest o f the expression. In so doing after some calculation we obtain the following expression for the correlation function ( 13 ):

M + N M

( q = 0 1 f i ~(Xa) f i q(Yb) I-I e x p [ O ( u i ) ] 1--I e x p [ - - O ( V k ) ] I q = 0 ) a = 0 b = l i = 1 k = l

n n ~" M + N n zo - - z..~ i = 1 = (det 0-o)_,/z 1-Ib=l [O('~)([--Ea=oXa U~+ Zff=~ Vk+ ~ = 1 Y ~ + Y b + (2~,-- l ) d u ] l z ) ]

yM+N u + M l~=o[O(°~)([X~,--E7,=oXb - i=l i Zk=~Vk+Z~,=1Yb+(22--1)A~u°]Ir)]

I-[ if= 1 [a(Vk) ]2a--1 H ~,b'= 1,b<b' E(yo, Yb') I]~,a' = 0 ; a < a ' E(Xa, Xa" ) 1-I N+Mi=I mk=lVrM E(ui, Vk) × (26) - [ M + N ] 22-- 1 M + N n n ' ,=1 [ a ( U , ) M I-[k,k'=l;k<k' E(Vk, Vk') ]-[i, i '=l;i<i' E(ui, ui,) 1 " - [ 6 = 1 I-~a=oE(xa, Yb)

which is equal to the expression (36) found by Verlinde and Verlinde in ref. [ 1 ]. We have used the following shorthand notation:

1 i x = ~ni ¢°u" (27) z0

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We thank A. Losev for many enlightening discussions on the sewing technique and the path integral formalism.

References

[ 1 ] E. Verlinde and H. Verlinde, Phys. Lett. B 192 (1987) 95. [ 2 ] J. Atick and A. Sen, Nucl. Phys. B 309 (1988) 361;

A. Morozov, Nucl. Phys. B 303 (1988) 343; A.M. Semikhatov, Phys. Lett. B 220 (1989) 406; U. Carow-Watamura, Z.F. Ezawa, K. Harada, A. Tezuka and S. Watamura, Phys. Lett. B 227 (1989) 73; L. Bonora, M. Matone, F. Toppan and K. Wu, Nucl. Phys. B 334 (1990) 717.

[3] A. Losev, Phys. Lett. B 226 (1989) 62. [ 4 ] P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, Nucl. Phys. B 322 (1989) 317. [ 5 ] P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, Nucl. Phys. B 333 (1990) 635. [ 6] P. Di Vecchia, in: Superstring and particle theory, eds. L. Clavelli and B. Harms (World Scientific, Singapore ) p. 136. [7] E. Verlinde and H. Verlinde, Nucl. Phys. B 288 (1987) 357;

V.G. Knizhnik, Phys. Lett. B 180 (1986) 247; L. Alvarez-Gaum6, J.B. Bost, G. Moore, P.Nelson and C. Vafa, Phys. Lett. B 178 (1986) 41; Commun. Math. Phys. 112 (1987 ) 503.

[ 8] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 9 ] J.D. Fay, Theta functions on Riemann surfaces (Springer, Berlin ).

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Correlation functions for the βγ system from the sewing technique