Once more on β, γ systems

Volume 226, number 1,2

PHYSICS LETTERS B

3 August 1989

O N C E M O R E O N fl, y S Y S T E M S A. LOSEV ITEP, 117 259 Moscow. USSR Received 22 April 1989

Multiloop correlators of free bosonic fl, 7 fields on arbitrary closed Riemann surfaces are deduced by direct computation of a functional integral.

1. One may hope to find explicit answers for any quantity in perturbative theory o f strings if it is somehow expressed through some correlators of flee fields on Riemann surfaces. These fields are free in the sense that their lagrangian is quadratic, but still not trivial because of an external two-dimensional gravitational field and a non-trivial topology of two-dimensional spacetime (Riemann surface). The simplest example o f such a free field is scalar 0, which takes on values in a circle (its radius may be infinite and is defined by the action

I

tN

"o

a~v t,0+OloR~) d2z,

(1)

R being a two-dimensional curvature scalar. Arbitrary correlators of 0 on a closed surface of genus p may be expressed in terms of certain special functions called theta functions for genus p (see for example refs. [ 1-3] about correlators of 0 and ref. [4] for the definition of theta functions). However, ( 1 ) does not exhaust all interesting free fields on Riemann surfaces there are two other systems of free fields with first-order lagrangians: Lt,,~ = f b 0c, L/s,:. = f f l 07. These are essentially chiral b and fl having spins (j, 0); c and 7 have spins ( 1 - j , 0) (i.e. they are tensors of rank (j, 0) and (1 - j , 0), and rank (j, k) for f(z, Z) dz j dzzk), 0 = 0/OZ, and the only difference between these two systems is statistics: b, c are fermions and fl, 7 are bosons. This, however, leads to a drastic difference between the correlators of b, c and fl, 7 systems at the multiloop level; b, c systems have been studied extensively [ 5 ] ; in fact they may be related to scalar field ¢~ in ( 1 ) with c~o= ( 2 j - 1 )/2~//22 through the bosonization prescription b = e °, c = e - ° [ 1 ]. For our purposes we shall only need one set of results for the theory orb, c systems [ 1,3,5]: for j = ½ det 0 ~ / 2 ' ~ = 0 ( 0 ) ,

(2)

O(x-y) ( b ( x ) c ( y ) )o - G ( x , y ) - O( O ) E ( x , y ) '

(3)

( b( xi )...b( x n ) c ( y l )...c(yn) )o =- G( x~ .... , Xn [Yl , ..., Yn)

= 0 ( Z ' / , x i - ~ ",=l y i ) [ I i < j E ( x , , x j ) E ( y i , Yi) = d e t , , . k G ( x , , , Yk). 0 ( 0 ) l~,,, E ( x , yj)

(4)

The latter equation is actually the Wick theorem for correlators o f two-dimensional spinors. The theory of b, c systems is in fact the basis of the modern formalism developed to describe a free bosonic string. fl, 7 systems (with a certain value for spin j = ~ ) were originally introduced in ref. [6] as superghosts in the 62

Volume 226, number 1,2

PHYSICS LETTERS B

3 August 1989

NSR formalism for a superstring. They were studied in ref. [ 6 ] with the aid of the " b o s o n i z a t i o n " / ? = 0{ e ~°, = q e - ~o, 0 being a field like ( 1 ) with ao = ( 2 J - 1 ) / 2x/2, and ~, ~/being a b, c system with j = 0. These ~ and r/ in turn may be bosonized, and an operator algebra of the fl, 7 systems may be reproduced in terms of two free scalar fields. However, the global properties of the manifold, where these scalar fields take on values, are yet unknown; it is only clear that if one passes along any noncontractible cycle on a surface, the two scalar fields get intermixed; thus, O and ~, r/are not independent globally. The correlators of ~, r/, 0 were constructed in refs. [ 7,8 ] starting from their analytical properties, as has been done in ref. [ 5 ] for b, c systems. These correlators have poles whenever a zero-mode of a bosonic field/? or 7 arises, and this makes the formulas much more complicated than they would be in the case of b, c fields. In what follows we present a straightforward derivation of these formulas from the lagrangian formalism without appealing to the global analytical properties of the correlators. To the best of our understanding, the same method of calculation is implied in ref. [ 7 ] ; however, unfortunately, it is not presented in detail there. We believe that such a detailed unmysterious treatment offl, y systems should appear in the current literature because of their increasing role in generic string theory (note that fl, y systems arise at least in the free field representation of the W e s s - Z u m i n o - W i t t e n model [9]; therefore, they will be important in the study of any rational conformal theo r ' / a t the multi-loop level). 2. Let us begin with the simplest case j = ½. Here, we will call fl= ~,, y= g/. We define

F,,.,(x,y)- f D~Dg, exp(i f vT/&u)~exp[ip~(xa)] ~exp[iq,,~(y,,)]

G(x~,yj,)

and the Green function is exactly the same as for the fermionic b, c system with j = ½, i.e. it is given by (3). Now, it is straightforward to find correlators of vertex operators of the type 1 0 eiqq/(.~ )

~(x) = T ~

,

6(q/(x))

=

dq eiq~u(.v).

~=o

Fp.qin

These are given by differentiating and integrating

I 'SI ~ ( ~ ( x o ) ) (21 11 a ( ~ / ( y , , ) ) ) 1

h= 1

= I1

(5). For example,

f dp~dq'Fp-~. . . .

a,b

G(x,,, y~,) ] - ' = (det 8,/2 ) - l [ G(Xl,..., x,, lYl, ..., Y,, ) ] 0(0) I-[,./, E(x,, Yt,) (det 8~/z ) -~ O(Zx- Ey) 1-[.<.. E(x., xo.)Eb<~, E(y~,,y,, )

= (det 81/2 ) - l [det,,~, =

(6)

The zeros of the theta-function in the denominator defines "unphysical" poles of the correlator, which are not dictated by an operator algebra, but reflect the global conditions for zero-modes offl and 7 to occur on a Riemann surface. 3. However, (6) is not the whole story even in the case of j=½. The thing is that for many purposes the correlators of the (, r/, ~ fields are required. According to ref. [7],

~=H(g/), rl=O~"d(~u)=OH(~'), e~*=d(~,),

e

~=d(q/).

(7)

Here, H is the Heavyside step function, and 63

Volume 226, number 1,2

~=H(#)=

f

PHYSICS LETTERSB

dp

i(p+i0~

3 August 1989

ewe.

Therefore, we will also consider correlators of the form

=

H(~/(x.))

t=l

6(~(u~))

b=l

q/(Yb) 1-[ 6(~t(vk)) k=l

.

(8)

Below we will see that the quantities of the vertex operators with a ~,-field are unambiguously defined by those of the operators with ~-fields. This correlator may be calculated as follows

(det~,/2)_1 fi dp~, fid2,"~["dl4 fi ~qt exp[~ (i

X

6 k=l

paG(X,,Uk)+ a

2,G(u,,vk)

(deter/2) -~

t=l

(9)

A total of n +M, ~-functions is sufficient to integrate out all p~ and 2 , except Po. From the set of equations p~G(x,,

a=l

M v~) + ~ L a ( u , , i=1

vk) = -poG(xo,

Vk),

we find p . = ( - l)"po

G ( x o , x , , ..., ~a ..... x , , u~ ..... u~, Iv, .....

2, = ( - 1) ~+'+ ~po

v,,+,,)

G(x~,...,x,,, u~,..., u , ~ l v ~ . . . . , VM+,,) G ( X o . . . . . x , , u~, ..., ~, . . . . . u ~ , l v ~ ..... v , + , ) G ( x ~ . . . . . x,,, u, ..... u , ,

Iv, ..... v , + , )

(10)

Here, the sign ~ means that the corresponding argument is omitted. In order to check these formulas, one should use the Wick theorem (4), and expand the determinant G ( x o . . . . . x,,, u, . . . . . u , ~ , l v , . . . . , v , + , , , v k ) - O

(since Vkappears twice among the arguments) in elements of the last column. Thus, we should substitute (10) into the first product in (9) and apply the Wick theorem (4) again in order to get

f i PoG(Xo, ...,x,, ul ..... UMLVl, ..., v.w+,, y~,) h=

1

in the numerator. The product ofp~ given by (10) appears in the denominator in (8). One should also include a factor of [G(x,,

..., x,,, u~, ..., u ,

l v~ ....

vM+.)]-'

coming from the ~-function, and note that fdpo/(Po + iO) = i in order to get the answer for the correlator in ( 8 ): 64

Volume 226, number 1.2

PHYSICS LETTERS B

3 August 1989

]-[~I=, G(xo,...,x,,,ul ..... uMlv, ..... UM+,,,Yb)

(det~,/2)_l

17I~=o G ( x o ,

..., 2~. . . . . .

x , , , u~ . . . . . u ~ , I v~ . . . . .

(11)

vM+,,) "

There is exactly one extra G-function in the denominator. All G's may be substituted by G=G.O(O); then (det ~,/2 ) -~ in ( 11 ) should be substituted by (det 0~/2 ) - ~"0(0) = (det 0o ),/2. In order to find a correlator in terms of the q-fields, q=Oq/.d(~,), one should take y-derivatives and tend v~t+/,~y~,. Since the product in the numerator of (11 ) has simple zeros as v~+~,~yb, the only role of the differentiations is to remove these zeros, and we have in terms of theta-functions

~(X,) f i rl(y/, ) )

t)=t

e i0t"n 2=1

e

iO(t,,)

=(detOo)l/2

1=1

]~,<,, E ( x , , x , )F[~,<~, E(y~,, y~,.)I]ivE(u,, vl)I][;l=, 0 ( - y b + E ' d x - Z ~ ' y + E ~ + u - E ~ v ) × l-L,.,,E(x.,y,,)I],<,. E ( u . u,,)Fl/<,. E(v,, v r ) F [ ~ = o O ( - x . + E ~ { x - Z T y + g ~ u - E , ~ ' v )

"

(12)

These are exactly the same formulas as in refs. [ 7,8 ] for the correlators of the fl, 7 fields in the case j = ½. 4. In order to define the correlators for generic j, it is enough to do the following trick [10]. In this case, because o f anomalies, correlators depend on p coordinates or on the metric on the surface. Everything simplifies greatly if the metric is chosen to be Iv, 14= IO,.,(O)w,(z)l 2. Then one may relate arbitrary fl, 7 fields to those with spin ½: fl= v~~- ~~, 7 = v2-2~,. The only thing one should take care of are the zeros which v, (z) possesses at the points RI", ..., Rp ~ on a surface. Because of these zeros the measure is p-

1

Dfl D y = D ~ D~' H d(~(R*) )d' (~/(R*) )...d(2j-2)(~t(R *) ).

(13)

Insertions in the RHS imply that we integrate over fields ~, with ( 2 j - 1 )-fold zeros at all points R*. Therefore, any correlator of the fl, 7 fields is equal to the same correlator of the ~, ~, fields with additional insertions (13), which may be expressed in terms of the ~ fields according to 3 ( ~ , ) = e ~'. In this way for arbitraryj we have instead of ( 12 ) ~(x,,) )

=

r/(yt,) h=

1

e i~''') t= I

~(x,,)

q (Y/,) 1)= I

=(det~o)t/2

e -i'~(''~) k=

I

fl,7

e i°("'~ 1= 1

e k=

I

i+(,,~)

e - ( 2 j - i) o ( ~ )

~= I

q).~

]-[,<,. E(x,,x,.)l-[t,
F[51=~O(-y,, + Y ~ x - Y T y+ 2~ ' u - Z ?'v- ( 2 j - 1)A,) × FI",=o O ( - x ~ + Z " x - Z T y + Y i " u - Z ~ ? v - ( 2 j - l ) A , )

(14)

An obvious limiting procedure is implied for ( 2 j - 1 )-fold coincident R*; M and N are related by the R i e m a n n Roch theorem

M-N=(2j-1)(p-1).

(15)

or, (z) stands for the standard contribution

.,(z) c r . ( z ) = F[",=~'E(z, R T ) ' 65

Volume 226, number 1,2 which is a F.(R*} =

PHYSICS LETTERS B

3 August 1989

p/2 differential w i t h o u t zeros a n d poles, )2 kfI]~<'--E(R*'---~R*')) ¢ 2v'(sR*) -'l-I,

To conclude, we have p r e s e n t e d a detailed l a g r a n g i a n d e r i v a t i o n (as suggested in ref. [7 ] ) o f a c o m p l i c a t e d expression for the correlators o f the/3, 7 fields, w h i c h essentially clarifies their physical m e a n i n g a n d m a y be c o n v e n i e n t for f u r t h e r a p p l i c a t i o n s . B o s o n i z a t i o n o f the ~, 7 system along the lines m e n t i o n e d at the e n d o f section 1 deserves i n v e s t i g a t i o n , as well. We are i n d e b t e d to A. G e r a s i m o v , A. M o r o z o v a n d K. T e r - M a r t i r o s i a n for v a l u a b l e discussions.

References [ 1 ] L. Alvarez-Gaum6, J.B. Bost, G. Moore, P.Nelson and C. Vafa, Commun. Math. Phys. 112 (1987) 503. [ 2 ] A. Gerasimov and A. Morozov, preprint ITEP-139-88. [ 3 ] P. Di Vecchia, M. Frau, K. Hornfeck, A. Lerda, F. Pezzella and S. Sciuto, Nordita preprint 88/47-P. [4] D. Mumford, Tara lectures on theta (Birkh~iuser, Basel, 1984). [5] V. Knizhnik, Phys. Len. B 180 (1986) 247. [6] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [7] E. Verlinde and H. Verlinde, Phys. Lett. B 192 (1987) 95. [8] J. Atick and A. Sen, preprint SLAC-PUB 4292 ( 1987); A. Semikhatov, Pisma JETP 49 ( 1989 ) 81 ; A. Morozov, Nucl. Phys. B 303 (1988) 343. [9 ] A. Morozov, preprint ITEP-43-89. [ 10] R. Iengo and D. Ivanov, Phys. Lett. B 203 (1988) 89.

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