The energy-momentum tensor on a Riemann surface

Nuclear Physics B281 (1987) 546-572 North-Holland, Amsterdam

T H E E N E R G Y - M O M E N T U M T E N S O R O N A R I E M A N N SURFACE Hidenori SONODA Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, USA

Received 13 May 1986 (Revised 18 July 1986)

The techniques of conformal field theory developed for a Riemann sphere by Belavin, Polyakov, and Zamolodchikovare generalized to any Riemann surface. We especiallyconsider the energy-momentum tensor in the bosonic string theory of Polyakov. The correlation functions of the energy-momentumtensor with local fields are studied in detail. As an application we compute the scattering amplitudes of null physical states explicitly to show that they vanish. 1. Introduction

Two-dimensional conformal field theory is a field theory in the presence of a two-dimensional background gravitational field (or a background metric) with two invariances. One is the invariance under coordinate transformations, i.e. the theory is invariant under a change of the background metric induced by a coordinate change. The other is the invariance under Weyl scalings, i.e. the theory is invariant under any rescaling of the background metric if we scale the local fields appropriately. In a remarkable paper [1], Belavin, Polyakov, and Zamolodchikov fully developed conformal field theory for a Riemann sphere. The energy-momentum tensor generates a change of the background metric and plays a very important role. One of the main results of ref. [1] is that the correlation functions of the energy-momentum tensor with any local fields are constructed from the correlation functions of the local fields only. Recently the techniques of conformal field theory have been successfully applied to string theory by Friedan [2], and Friedan, Martinec, and Shenker [3] in particular. The purposes of this paper are threefold. The first is to give a brief overview of the bosonic string theory of Polyakov [4] from the point of view of conformal field theory on a general Riemann surface. In ref. [3] some remarks are made for Riemann surfaces other than the Riemann sphere. We try to make the remarks more concrete. This constitutes sect. 2 and a part of sect. 4. The second purpose is to derive a formula for a correlation function of the energy-momentum tensor with any local fields on an arbitrary Riemann surface. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

H. Sonoda / Energy-momentum tensor

547

This is done in sect. 3. We also consider a change of correlation functions under a change of the background metric in sect. 4 and show how it is written in terms of the energy-momentum tensor. The last purpose of this paper is to compute the scattering amplitudes of a spurious physical state with other physical states in the bosonic string theory. This has been discussed in ref. [3], and the amplitudes are shown to vanish by a qualitative argument. Here, by using the formulas derived in sects. 3 and 4, we compute the scattering amplitudes explicitly and confirm the result of ref. [3].

2. Review of conformal field theory In this section we would like to give an overview of conformal field theory by explaining how to compute the scattering amplitudes in the bosonic string theory & la Polyakov [4]. In the case of a Riemann sphere, most of the materials described here can be found in the previous literature. (See, for example, refs. [1], [3], and [5].) Let M be a Riemann surface. M consists of many coordinate patches with a local complex coordinate on each patch. The coordinate transformation from one patch to another is holomorphic. Let z be a local complex coordinate. It is an isothermal coordinate, i.e. the metric tensor has only gz~ component. In the bosonic string theory the only dynamical field is the coordinate xJ~(z, ~) (# = 0,1 .... , D - 1) of the string in the D-dimensional Minkowski space. The first step in the computation of the scattering amplitudes is to calculate the correlations of various local fields made of x ~ in the background metric gz~. The theory is free, and we can compute any correlation function from the propagator

(X/t(Z1, where ~ =

(-1

lO_i

Z1)Xv(Z2,

Z2) ) = ~/"~G(1,2),

(1)

The Green function G is defined by "

1

--G(1,2)= 2¢r

1 Z--x,(1)x,(2), n¢O t~,,

(2)

where xn(z, £) is the real eigenmode of the laplacian with the eigenvalue/~n: --8

~a z . { z ,

s)

=

(3)

For n = 0, the eigenvalue is zero, corresponding to a constant function. For n > 0, the eigenvalue is positive. In (2), the eigenmodes are normalized by /Md2Z~/g- x , ( z, £)xm( z, ~) = 8~,,,.

(4)

H. Sonoda / Energy-momentum tensor

548

Then G satisfies the equation 1

1

- g " a ~ o z ~ G ( z, e; w, ~) = _~<2~( z, w) - x~,

(5)

and the propagator has the logarithmic singularity as z 1 --, z2: (x"(1)x~(2)) - - ~ / ' q n l z I - z2l 2

(6)

A local field operator corresponding to a tachyon with momentum p" is given by e~:~.. The correlation function of tachyons is calculated as (exp(ipxx(z~, ~ ) ) . . . exp( ipNx ( z N,

£U)))

=exp[- ~,p~pjG(i, j) - ½

(7)

where N

Ep~=0. i=1

G(i, i) is divergent due to (6), and we regularize it by point splitting: C~(z, ~) -= a ( z , ~; z + ~z, z + ~ ) ,

(8)

where gzel~z[ 2 = ~. e is a fixed positive constant to be taken as zero at the end of computation. Let us consider a more general correlation function than (7): f¢= (O~XX~...

O~ X x~ O~x ~"... 0 ~, x~Mexp(iplx(Zl, Zl))...exp(ipNX(ZN, ZN))) (9)

We note that N is meromorphic in wi, since the two-point functions N

Epi~(O~x" x,(i)),(OzX~'O~x ),(3~x 3~x ) i=l

are meromorphic in z. Likewise f¢ is meromorphic w.r.t, ffi- Due to this meromor-

H. Sonoda / Energy-momentum tensor

549

phicity it is possible to define the contour integral: dp = a~_'ml

. . . a~'M ff1'1 . . , dP~lve ipx(z'~') ___mM_._nl

=~ dZl

~ dZM~ awl

dwu

- .r'C~2~ri "'" YCM 2~ri YC~+M27ri "'" ~C.+M 2~ri ×

1

1

( Z 1 -- Z ) m l ' ' "

1

( Z M -- Z ) m M ' ' "

1

(1~ 1 -- z~)nl " ' " ( W N -- z~)Y/N

X O z X ~ . . . Oz x ~MO~,x~... OwNx~e ipx(~' ~)

(10)

Here the loop C i is an anticlockwise contour around z, and C i is placed outside Ci+ 1- From the singularity (6), we can compute the commutation relations [5]: [a~m, a;] = [d~, ff,~] = -m~"~rm+n,0, [a~, a;] = 0.

(11)

One of the remarkable features of conformal field theory is that all local operators can be obtained by contour integrals like (10). Namely, any operator product expansion of two operators of type (10) contains only operators of the same type. In string theory the local operators thus created are called vertex operators. In the rest of this section we review the energy-momentum tensor T~. It is defined by T~=-½w~lim

(

OwX~'OzX~,+ D ( w

.

Since Ozx ~' is a meromorphic field, T= is also meromorphic. Due to 1 / ( w - z) 2 in (12), Tzz is not really a rank-two tensor. Under a holomorphic coordinate transformation z ---, w = w(z), it is transformed as [1]

T~w= -~w

T~+~D

Oz/Ow

2

(Oz/Ow) 2 J"

(13)

However, T~z can be made covariant by a local counterterm:

Lz=rzz+

D(( g azgze ),2 - 2 O z ( g

O~gze)}.

(14)

iPz is a rank-two tensor. But it is not meromorphic any more; for example, the expectation value of Tzz satisfies

V (~)

= ~DO~R,

(15)

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H. Sonoda /

Energy-momentum tensor

where R is the curvature scalar. (See appendix A for a summary of the complex tensor analysis.) The expectation value of 7"= gives a change of the determinant of the laplacian under a change of the background metric: - D/2

1

.

6

{ det'-g~eOe0~

-4-~(Tzz) = 8gZ--~ln

(16)

f d2z

where the prime in the determinant means the omission of the constant mode. Let ff be an arbitrary local field of type (10). Then we define the operators L. by the following Laurent expansion [1]: 1

V'z~(W' ~) = ~,, (z - w) "+~ ( L.~ )(w, ~ ) .

(17)

L 1 is always given by

L 1,~-(gz£) p2/2 az((gze)p2/2~).

(18)

The operator product expansions 1

T~OwX~'

I

(z-w) 2owx*'+-o2x"+finite'z-w 1

T~T~w = ~ D (- z - -w ) 4

2

1

+ -( z --w ) 2 Tww + -z-- w

OwTw., + finite

(19)

give the commutation relations

[ L., a~] = - r n a ~ + . , [L~, Lm] = (n - rn)L.+ m + ~2D(n 3 - n)8.+m, o.

(20)

It is obvious that in a similar way, we can build ~z and /~.'s out of Oex ~. In this paper we mainly consider Tzz and L.'s, and we do not discuss Tee and f . ' s except in a few places. However, the reader should keep in mind that every statement on T= and L n' s has a corresponding statement on Tee and L- . s' . Usually we only have to change z to ~. Let us consider an operator q~ constructed from e ipx by multiplying a~. (n > 0) operators. Since the meaning of the contour integral which defines a" . depends on

H. Sonoda / Energy-momentum tensor

551

the coordinate system z, the operator 0 transforms nontrivially under an infinitesimal holomorphic coordinate transformation:

z-~z'=z+

~ e,z "+1.

(21)

n~O

The corresponding infinitesimal change of ~ is given by

3q~(z = O) = - ~ e,L,q~ +l~0pZep.

(22)

n=O

In order that q~ transform as a tensor, the Virasoro conditions L,~=0

(n > 0 )

(23)

have to be satisfied. Then the rank of the tensor field q~ is given by l~

- 21p 2 + L 0.

(24)

The local fields which satisfy (23) are called primary fields [1]. Now, an important remark is in order. In string theory a local field q~ corresponds to a field of a particle. It is obvious that the definition of ~ must not depend on the complex coordinate z of the Riemann surface. As was mentioned at the beginning of this section, the Riemann surface consists of many coordinate patches which are connected to one another by holomorphic coordinate transformations. Therefore, only the local operators satisfying the Virasoro conditions (23) are defined globally on the entire Riemann surface, and these are the only operators which admit interpretation as particle fields. Let ~k (k = 1. . . . . N) be a primary field, i.e. a tensor of rank (lk,/k) (where lk = _ ½p2 + L0 = _ lp2 + L0)" Then N

~=

YI d2zk g ~ ) ( g Z * ~ k ) t k ( ~ l ' ' ' e ~ N ) g k=l

is invariant under holomorphic coordinate transformations, thereby it is defined globally on the Riemann surface. Here we see why the ranks of the tensor 4k w.r.t, z and £ have to be equal; we have only the metric tensor gZe to contract with. We continue reviewing the prescription for calculating the scattering amplitudes in sect. 4. In the next section we examine the energy-momentum tensor more closely.

552

H. Sonoda /

E n e r g y - m o m e n t u m tensor

3. Correlation functions of the energy-momentum tensor

The goal of this section is to obtain a generalization of the formula (3.4) of ref. [1]":

(Tji""'*N>=

U 1 k=lE ,,=~-'-a (z _ z ~ ) m + 2 ( e ; , , . . . L , , * k . . . * N

)

(25a)

for a Riemann sphere to an arbitrary Riemann surface M. Especially when all q~k's are primary with L 0 = a k, eq. (25a) reduces to

(r=<...

=

N ( A%k)2(~1 k=l ( Z --

+

-

-1

Oz~(q~... ~N)}-

(25b)

Z -- Z k

We note that the correlation of Tzz with the primary fields is completely determined by the correlation function of the primary fields only. Let us first consider an arbitrary meromorphic vector field v w on the Riemann surface M. For simplicity, we assume that vw has only simple poles at w = t 1. . . . . t M. Near w = ts, v ~ is given by 1 v w - A,--. w - t,

(26)

Since T w w - (Two) is a rank-two meromorphic tensor,

f ~ - d w v W ( ( Tww - (Tw~) )eOl... (aN } is

a

meromorphic one-form with respect

to

w;

N

(27) has poles

at

w=

ZI~ " " " ~ Z N , /1, • " " ' IM"

The sum of the residues of f¢ vanishes: N

M

This is because we can deform the contour around z 1 to the sum of the contours going in the other direction around zk's (k ¢ 1) and t~'s. From (17) and (26), we can evaluate (28). We find M s=l N

+ kE= l m =

1

1

(m + 1)!

* T h e formula is rewritten in our notation.

oq m + l V z k ( d p l . . .

LmdPk...

~bN) = 0 .

(29)

H. Sonoda / Energy-momentumtensor

553

N o w we use this formula to obtain the generalization of (25). For this purpose we need a m e r o m o r p h i c vector field v w with a singularity at z, i.e. t 1 = z. Let A 1 = 1. T h e n (24) is written as M

((Tzz-(Tzz))~J1-..~JN) = -

E ~s((Ttjs-- (Tt~ts))~Jl'"~N) s=2

-

N

E

k=l

~

1

(m + 1)!

m=-!.

Om+lvzk (q)l''" Lm~k"" ~N)" z*

(30) This gives the generalization of (25), once we know h o w to construct v w. W h a t is i m p o r t a n t a b o u t v w is its singularity at z. If there are two m e r o m o r p h i c vectors v~, v~ with the same singularity at z, then v~ - v~ is a m e r o m o r p h i c vector which is regular at z. A p p l y i n g (29) for v~ - v~, we find that the r.h.s, of (30) is the same for w and O2w• U1 A c o n s t r u c t i o n of v w is described in appendix C. The vector is given as follows: w _= Kzzw + 2 ~ Z hiz~v w, Vzz

( 31)

i where we a d d e d the z indices to show explicitly that v w, thus constructed, is a r a n k - t w o tensor with respect to z. In (31), K~w is the ghost p r o p a g a t o r which satisfies

1

l V"~Kz~ = - ~ 8 ( 2 ) ( z ,

w ) - Ei h --we i hizz,

(32)

where ( h i ~ } is an orthonormal basis of the holomorphic rank-two tensors ( i = 1 , . . . , m a x ( l , 3 ( p - 1)); p is the genus of M). (For Kzw, see appendix B.) The vector v w has the m e r o m o r p h i c singularities:

1

1

v~ ~ ~---A,, s W

ts

as

w ~ ts,

(33)

a n d it satisfies

V wViw =h-ww i •

(34)

(We ignore the delta function at ts, since we use v w only in the region where it is regular.) We explain how to choose t s and Ai, s in appendix C.

554

H. Sonoda / Energy-momentum tensor

F r o m (32) and (34), with respect to w, v~w in (31) is a meromorphic vector field with singularities at w = z, t~'s: v~w ~

1 -

as

-

W--Z

w ~ z,

(35a)

1 v~--Y~Ai,shi~

as

~

W-- t s T

w~t,.

(35b)

By substituting Vzw into v w of (30), we get

<(Tzz-

(Tzz>)~l... d~N>-~-- ~ ~ihizzAi's( (rt't~-(rt~t~>)~I'"~N) -- EN

fi

1

c) m + l

z~

(36a) This is the generalization of (25) we have been looking for. If all ¢k'S are primary with L o = z3k, then we find

<(L~- )~ ~5 =

-- E E h i z z A i , s ( ( T t s t s

- )d~l...~N I

i

N l 0zS~'ak<,~l...q,N> +Vz~(gz'~) "k/2201. ~ o , ~j2, \ , ~ 1 . . 4 ~ 5 )/ .

- E

k=l

The first term is proportional to the holomorphic tensor fields hi=. Besides this term, the correlation of Tz~ with the primary fields is completely determined by the correlation of the primary fields only. When the Riemann surface is either a sphere or a toms, there is a holomorphic vector field kZ. By applying (29) for kZ we find N E ( m -~ 1)! k=l m = - 1

<~11"." Zm~)k"" (/)N> = 0.

zk

(37)

This is the generalization of the Ward identity (A.6) of ref. [1]. Certainly we can recover (25) from (36). For the Riemann sphere with g:= = ½, the ghost propagator is 1 gg

ww

~

_

_

Z--W

555

H. Sonoda / Energy-momentum tensor

and there is no holomorphic tensor field. Then eqs. (36) give eqs. (25). Finally we compute the correlation function of L_.q5 t for n >~ 2. This is obtained by looking at the Laurent expansion of (36a) with respect to z - Zl: (L-.~l"

~2"'" ON)

dz 1 = ~12qri ( z - - z , ) n-l(TzzgPl"''dpN) 1 (n-2)!

n12

0.

(TzIz15/(~)l

1 -

~

(m+l)!

X (L,,,~ 1 • ~ 2 " "

N E

-

~

, . . IJ~N 5

1 _ _ (n-2)!

O~,+l O;'-2

z

Uztzl

')

__

Z -- Z 1

Z~21

~N)

1 1 Om+l (m + 1)! (n - 2 ) ! za

k=2 m=-1

(

az~--2U~tz, "

(~1''" L m ~ k " " (~N5

1 (n-2)~ Oz~-2h'zmY"Ai's((T',',-(T',',})eOl""¢'N)

-E i

"

"

(38)

s

This formula will be used in sect. 5.

4. Change of correlation functions under a change of metric

Let us resume the review of the prescription for the bosonic String theory. In sect. 3 we have explained how to compute correlation functions of local fields in a given background metric. We have seen that if the local fields satisfy the Virasoro conditions (23), then the correlation functions are defined globally on the Riemann surface. The p-loop contributions to the scattering amplitudes are obtained by integrating the correlation functions over all possible background metrics on the Riemann surface of genus p. Let us explain this more in detail. Let /ff be the space of all two-dimensional metrics on the Riemann surface M of genus p. The integral of a correlation function over ~ / diverges due to the gauge group ~ × ~/', where ~ is the group of coordinate transformations, and ~ is the group of Weyl transformations: gze ~ gze ez°(z" ~)"

(39)

H. Sonoda /

556

Energy-momentum tensor

The correct prescription for string theory is to take the integral of the correlation function over J g / ~ × ~ , the so-called moduli space. The dimensionality of the moduli space is the same as the number of linearly independent holomorphic rank-two tensors [6], since an infinitesimal nontrivial change of the metric tensor is given by a holomorphic tensor. The moduli space has a natural complex structure [7], and we can introduce local complex coordinates y( The coordinates y; specify an equivalent class of metric tensors. A gauge choice corresponds to a choice of the metric g(y, ~) as a representative of an equivalent class. Now, we can write down the final prescription for the computation of the scattering amplitudes:

amp=

f YI dyid37iO(Y, Y)

N

1-I d2Zkg~)(gZk~k)tk(e~l"''ePN)g~y,y),

(40)

k~l

where I2 is a measure in the moduli space [6-9]. We will describe it shortly. ~k is a tensor of rank (l k, lk). The coordinate z k is isothermal for the metric g(y, ~), thereby it depends on y, 37*. In order that the integral (40) make sense, the integrand must be invariant under N × ~ . The coordinate invariance is manifest. So we must check the invariance under (39). Under the Weyl transformation (39), the correlation function transforms as

(ff~'''¢N)---' exp - E

iPk'1 2 2o(k)

(~I..-~N)-

(41)

k=l

Since the rank of the tensor field Ck is given by (24), the integrand of (40) is invariant under Weyl transformations, if the operators ~k satisfy the mass-shell condition: L0¢k =/~0q~k = q~k

(k = 1 .... , N ) .

(42)

The operators satisfying the Virasoro conditions (23) and the mass-shell condition (42) are called physical vertex operators. To conclude the review we must give the measure I2. From now on till the end of the paper we discuss the case of genus p >/2, Then there is no holomorphic vector field, and the expression of I2 is simplified a bit. The measure ~2 is given by 1 I2(y, .~) = - - d e t ' det N~i

f d2zl/~ - ~7z~7}1)

det' - gZ~ 0~o3z

) D/2 "

(43)

In order to give the precise definition of N,7, let y be a local coordinate of the * In fact the complex structure of the moduli space implies that z k only depends on y but not on

[71

H. Sonoda / Energy-momentum tensor

557

moduli space around Y0- Let z be a complex coordinate which is isothermal for the metric g(Yo, Yo). Let ti(y, y) be a rank-two tensor which is holomorphic w.r.t, an isothermal coordinate for the metric g(y, ~). It is normalized by the condition

½f

(Oyig~.tSz+28yigze.tfe+Oyigee.tfe)=SU.

Here we use the metric by

Nij

g(y, fi)

(44)

to raise and lower the indices. Then Nij is defined

fdz~(ti 2

-

-zztj~z+2t-fetjze+ti- ~ tjee).

(45)

In fact, I2 given by (43) is Weyl invariant only for D = 26 [4, 6]. Therefore, we must restrict ourselves only to this case. Now that we have reviewed the prescription for the scattering amplitudes, we examine the y dependence of the correlation functions. Suppose we have a metric gze with an isothermal coordinate z. Let us consider an infinitesimal change of the metric 8g ~z. Locally any tensor of rank - 2 can be written as a derivative of a vector field:

6g ~= -2~TZu z.

(46)

u ~ is not necessarily defined globally. This is the case if 8g ~ corresponds to a change of the equivalent class of metrics. However, if we allow meromorphic singularities, any tensor of rank - 2 can be written as 8g'" = - 2 ~ 7 ' v ' .

(47)

The difference between vz and u z is meromorphic, and v z is a vector field on the entire Riemann surface M. (If (gz~)26gZ~ is antiholomorphic, then v z is given as a linear combination of vf's given by (C.4), (C.5).) First we consider a change of the correlation function of tachyons with arbitrary momenta. Let

fff=__{eilhXl... eipNxu) = exp [ --

as in (7).

8g ~

i
(48)

induces 1

8(x(1)x(2)) = - 4-~ f

d2zl/g- ( x ( 1 )

O,x)Sg~Z( O~xx(2)).

(49)

H. Sonoda / Energy-momentum tensor

558

(This can be obtained by first order perturbation theory.) Using the expression (47), we perform a partial integral in (49). Suppose v~ has a singularity ~z _

AS z - ts

_ _

(50)

at z = t~. Then using (5), we can evaluate (49) as

a
+ (vZ, Oz, + v--Oz2)
= - EA,(x(1)O,x)(o,.xx(2)) S

-xgf d2zgrg v"{ (3zxx(1))

+ ( 3zXX(2)) }.

(51)

F r o m this we find N

N

3f¢= E uzkC~Zk~-~- E k=l

+

~19kVz U 2-1 Zk.

k~l

,,,,,/x(,)o, xll

,

+SY'~pi(x(i)

o,x)(o,xx(,))]

(Y.

(52)

i

In order to derive this, we used the equation

(x(i) O~.x].=.,,+az,) =(x(i) Ozx(zi+Szi))_g,,:z 32,g:,5,.

(53)

The terms proportional to x02 canceled due to the momentum conservation Ek P~ = 0. Recalling the definition of the energy-momentum tensor T= (12), we can write down (52) in a more compact form:

3 (e 'pl xl... e,p~,~N) N

N

k~l

k~l

y' vZ,(eimX,... L_leip,",...eip~.x~ ,) + ~ Ozv:*(eiP'x~... LoeiP*"* ...eiP ..... ) (54) Next we consider a general correlation function of the local fields ~k (k = 1 . . . . . N). In this case there is a complication which is absent for tachyons. Namely

H. Sonoda / Energy-momentumtensor

559

the local fields (10) are defined in terms of an isothermal coordinate z. Therefore, corresponding to an infinitesimal change of the metric we must introduce an infinitesimal coordinate transformation:

z - , z ' = z + u z,

(55)

where u ~ is given by (46). Under the new coordinate system z', the metric tensor is given by g z ' e ' - (1 - 3~u z - 3eu )&.e. (56) We can define the new local fields ~, using the new isothermal coordinate z~. This change of the definitions of ~k and z k gives an additional contribution to the change of the correlation function of ffk's. Fortunately, eq. (54) has a very suggestive form, and it is not very difficult to guess the correct answer for the change of the correlation function:

N

~

= k~" =l m=-i

"1-

1

• m+luzk (~1"

( m + 1)!

~k

~sAs((Ttsts- (Tt.t~))~l...~N)

L,ndPk'."~N) "" N

t 2 CON) + E OzkUZk. ~pk(eOl.. k=l

-- E 0 m+l~/Zk k = l rn=O (D'I ~- "1)! zk (~1''"

Lm(Ok'''d~N).

(57a)

As a consistency check we note that (57a) is finite when z k approaches t s as it should be, since the singularity of v z~ is fictitious. We also note that (57a) is consistent with (22) under a modification of (55) by an additional infinitesimal holomorphic coordinate transformation. Let us consider, the physical vertex operators ~k (k = 1 . . . . , N), i.e., L~ > 0q~k= 0,

(58a)

(58b)

L0~k = 4~kThe same equations are satisfied for/~,,'s. In this case (57a) is simplified to N

= E

k=l N

E k=l where ~k is the tensor of rank l k.

/k £~zkuzk (d~l"''d~N)'

(578)

560

H. Sonoda/ Energy-momentumtensor

Before we finish this section we consider the y derivative of the measure ~2(y, 37) given by (43). For this purpose it is convenient to take a particular coordinate system of the moduli space [7]:

z--,g g,~(y, )7) = g°~(1 + ~?lz), = 2g~, g e e ( y ) = 2gze~, o ~

(59a)

~-=- E Y i--2 hi~.

(59b)

where

i

The coordinate z diagonalizes the metric for y = O. { h~zz} is an orthonormal basis of the holomorphic tensors for y = O:

f d2z~0 ]lZiZhjzz = ~ij"

(60)

We wish to compute Oyi~2at y = 0. Since the calculation is very straightforward, we describe it in appendix D. The final answer is given by

Oy# [ -

-

1 f d2z,~ hZZ__/~tot al\

(61)

where Zztz°tal is the sum of the energy-momentum tensors of the x ~ fields and the ghost fields (see appendix B). To summarize, we have computed the y dependence of the integrand of (40) given by (57b) and (61).

5. Decoupling of null physical states In sects. 2 and 4 we have reviewed how we compute the scattering amplitudes. The prescription makes sense only for the physical vertex operators satisfying (58)*. But some physical vertex operators are spurious, i.e. they can be obtained from a primary field by multiplication of L , ' s (n > 0). In the Hilbert space interpretation [2,10] the states corresponding to spurious physical vertex operators have zero norms, and they do not represent real particles. Therefore, the scattering amplitudes of a null physical state with physical states should vanish. * We only consider L,'s but not L,'s. The analogousthings can be done for E,'s also.

561

H. Sonoda / Energy-momentum tensor

Friedan, Martinec, and Shenker have examined this problem using the BRST formalism [3]. They show by a qualitative argument that the correlation function of a spurious vertex operator and physical vertex operators is a total derivative with respect to the two dimensional coordinates and the coordinates of the moduli space. (See the second paragraph on p. 23 of [3].) Therefore, the scattering amplitude, which is given as the integral of the correlation function, vanishes*. The purpose of this section is to compute the scattering amplitudes of spurious physical states explicitly using the results of sects. 3 and 4. First let us look at the simplest spurious physical vertex operator, given by ~P= L_lq~,

(62)

L0 = 5P 1 2 + l = 0.

(63)

where ~ is a tensor of rank l, and

Then ~p is a tensor of rank l + 1. From (18) we find (64)

( lp(Z1, e l ) ~ 2 . . . ~N) = ~7(['(~1~2""" ~NS,

w h e r e ~2 . . . . , q~N are arbitrary vertex operators. This is obviously a total derivative.

Next we consider the second simplest example: hb= (L_2 + 3L2 x)q~,

(65)

where 4~ is a tensor of rank (0, 2), and L0 = s p 1 2=-1.

(66)

Then q, is a physical vertex operator and a tensor of rank (2, 2). Let ~2 . . . . . ~N be arbitrary physical vertex operators. The calculation of the correlation function (~Pl~2 ... q~U) can be done straightforwardly using (18) and (38). But the hard part is to identify the result as a covariant total derivative. We summarize the calculation in appendix E and only state the result here: N H d2Zk~ k=l

(gZkY'k)lk(lPl~2"''~N)

N

= H d 2 z , ~ k ) ( e , z ~ ' ~ ) '~ k=l

x

gkt ~(~ oz, - ~z,)(,~,... ,~,,,5- E ~"~'-" (K~,~,(,~,. ~.

... ,,,,5

k'=2

-2=-Y.o.(gaH d2z~g(~d~z,.'~,~"t k•

J

/e 1 " " " q'N)

iz lzl\

(67)

* This is only formally true, since we ignore the complication due to the boundary of the moduli space.

562

H. Sonoda /

Energy-momentum tensor

where l 1 = 2, and l k (k >~ 2) is the rank of q~k- The covariant rank - 1 field ~ is defined in (E.3). K~w is the ghost propagator defined in (B.1). The coordinates yi and the tensors ti~z are defined by (59) and (44), respectively. The formula (67) is written in a manifestly covariant manner, and it is readily identified as a total derivative. Therefore, (67) vanishes upon integration. Now, we are ready to examine a more general case. We look at the following class of spurious physical vertex operators*. Let

+=

n--1 Y', L ( , _ j , ( ~ _ j ~ ) , j=o

(68)

where ~ is a tensor of rank l - n, and L0~ = (1 - n ) ~ .

(69)

~ _ j is a product of L_,, operators. ~ e j increases the eigenvalue of L 0 by j. We normalize £*°0 = 1. The operator ~p given by (68) trivially satisfies the mass-shell condition (58b). The Virasoro conditions (58a) are satisfied if

L1LP_jep + (2 + n - j ) , . ~ j+ldp -----0

( j = 1 . . . . . n - 1),

(70a)

L2,~jd p + (4 + n -j),~/9 j+2dp -----0

( j = 2 . . . . . n - 1).

(70b)

The conditions (70a, b) guarantee Ll~b = 0 and L2~ = 0, respectively. Due to the commutation relations (20), this is equivalent to the Virasoro conditions (58a). ~, thus defined, is a tensor of rank l. The reason we prefer to express a spurious physical vertex operator as (68) is that we can construct two covariant tensors out of £#_j~. The operator defined by

0=

n-2 1 Y~ ( n - 2 - j ) ] j=0

(Ozn-2-Jlzz)"~JdP

is a tensor of rank l for any second rank tensor

n2 X=~-("-~)~-

1

~ (n-2-j)! j=0

tzz,

(71)

and the operator

(

1)

lira 0~ -~-2 K z ~ - - -

w-~z

w -

z

~_j~

(72)

is a tensor of rank l - 1. The calculation of the correlation function of ~ defined * It is not clear whether or not this construction gives all possible spurious physical vertex operators.

563

H. Sonoda / Energy-momentum tensor

by (68) with the physical vertex operators 02 ..... ON can be calculated in the same way as for (65). The result is N

H d2zk~/g(k) (gzk~?k)/k(lPl02"''

ON)

k~l N

f2 1-I d2zk~g/~-J (g~ke~)'~ k=l

~-2

1

}

× ~-" (n - 2 - j ) t cg~-2-Jtiz'zS~'JO'02"" ON} j=0

N k~l N

n-2

k'=2

j=o

×

;,

1

o2...

(73)

Note the coordinate invariance of the right-hand side due to the covariance of (71) and (72). The density of the scattering amplitude (73) is a total derivative, and the amplitude vanishes as desired.

6. Conclusion

We conclude this paper by mentioning future problems. In ref. [3], the BRST formalism is discussed. The BRST charge can be explicitly constructed for a Riemann sphere, and it is generalized to any Riemann surface by invoking the principle of "analytic continuation". However, the precise description of how to formulate the BRST charge on an arbitrary Riemann surface has not been given at the level of explicitness we have attained for the energy-momentum tensor. One of the difficulties in proving the decoupling of null physical states as given by (73) is to find a total derivative in the moduli space. In order to say that something is a total derivative, we must pay a special attention to the covariance of the expression. This is the reason why we had to restrict ourselves to a class of null physical states (68). It is not clear how the BRST formalism is useful in this respect*. * The problems for the BRST formalism mentioned here have been solved recently in [11],

564

1t. Sonoda / Energy-momentum tensor

Finally the obvious thing to do next is to develop techniques of conformal field theory for fermionic strings. Progress in this direction has been already made in ref. [3]. I thank O. Alvarez, K. Bardakci, M. Dugan, N. Marcus, and P. Windey for discussions and help. This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under contract DE-AC03-76SF00098.

Appendix A COMPLEX TENSOR ANALYSIS The reader is referred to ref. [2]. Under a holomorphic coordinate transformation z ---, w = w ( z ) , a rank n tensor transforms as

t" .... =

t ......

(1.1)

where t has n indices. Similarly a rank - n tensor transforms as

t ...... =(Ow]"t Oz ]

..... .

(A.2)

In the same manner we can define tensors w.r.t. ~-. A rank (n, m) tensor is a tensor of rank n ( m ) w.r.t, z(Z). We can raise and lower the indices by the metric tensor gz~The covariant derivative of a rank n tensor is defined by V ( " ) t ..... = g z ~n O z ( ( g

z:~

) n t ..... ) ,

~7:,)t ..... = gZ~ O~t ..... .

(A.3) (1.4)

For a rank-n tensor the commutator of two covariant derivatives is given by [ V ' , V,] = -~nR,

(A.5)

where the curvature R is defined by R = -2gZ':O~(gz~O~gz~).

(A.6)

14.Sonoda/ Energy-momentumtensor

565

Appendix B FADDEEV-POPOV GHOSTS

The Faddeev-Popov ghost fields bz~, c z are defined by the propagator: g

l~ww (c bww> = 2,,

}

1 ~ .vwt.w,

iz

(B.I~

where tnz is the normalized eigenmode of the laplacian

- lyz~zt,z = ~.,t,z,

(B.2)

with the positive eigenvalue ~,. The ghost propagator satisfies the equation 1

1

2~r V Kww- 7r~ $~2~(z, w) -

i -zz

,

where { hi~~} is an orthonormal basis of the holomorphic tensors. Suppose there is a tensor field S~z whose derivative ~7:Sz~is known. Then we can recover s~ from ~7~s~ up to a holomorphic tensor:

s~ =_ f d2w¢~ -j-~ 1 l~zzV . . . . . Sww+ ~ hizzdd f 2wv~ h ~-~.,Sww.

(B.4)

The energy-momentum tensor of the ghosts is defined by T~-

~--,~lim c~' O~b~z+ 2 0~cW. b~z + ( z

"

(B.5)

The transformation property of Tzgh under a holomorphic coordinate transformation z ~ w = w(z) is given by

( Oz'2

13{33z/Ow3

Tw'~= -~w ) T ~ -

-~

cgz/Ow

3(32z/Ow2) 2 ) 2 ( 3z/Ow) 2 "

(B.6)

Therefore, from (13) the total energy-momentum tensor Zzt°tal =

T~z+ T~gh

is covariant for D = 26. If we define

Lzgh= Tzzgh_

13

z -7

2

(B.7)

566

H. Sonoda / Energy-momentum tensor

then 7~z~ is covariant. It satisfies : - 2 ~ r Y ' ~ h , V~hiz : i

(B.9)

~20zR,

if there is no conformal Killing vector (i.e. holomorphic vector). ( T ~ ) change of the determinant of - V ~ V } t) under 6gZ~: 1 4-'~-( i ~ )

8 = ~--2~21ndet' - ~7z~7.(1) . 6g

also gives a

(B.10)

Appendix C

CONSTRUCTION OF ~. First we solve the equation: Vwv~ = ~Tw+ delta functions.

(C.1)

Then, from (B.3), the vector v~w defined by (C.2)

Vzw =- Kz w + 2 ~ r ~ v ~ h i z z i

is m e r o m o r p h i c with respect to w with the correct singularity at w = y. T o solve (C.1) we multiply hlw w to (C.1) and get v

w

w

=

-ww htw w +

h i

delta functions.

(C.3)

Using the bosonic propagator ( x ( z , Y ) x ( w , ~ ) ) given by (1), we can solve (C.3). F o r i > 1, a solution is given by W

vihlww =-

j

1 dZufg

--

--uu

~-~ ( O w X ' X ( U , u ) ) h

i h i , u.

(C.4)

F o r i = 1, we have to pick an arbitrary point t o on the Riemann surface M. Then,

v hlw

-

1

1

-

--

--uu

(C.5)

where x o is the normalized constant mode of the bosonic field x. Using (5), we can easily check that (C.4) and (C.5) solve (C.3). Now, let t 1..... t 4 ( p _ l ) ( p = genus) be the zeros of h 1. . . . Then the vector v w has the following singularities: 1

1

vi -'* 2 ~ a i ' s w - t s

as

w ~ t~,

(C.6)

H. Sonoda/ Energy-momentumtensor

567

where A, 0 =

'

hltot°

,

(C.7)

Ai,s- Otshlt, [~il(OtsX.X(to))- f d2u ~ (Ot,X.X(Zl,~))(~UUhluu__~ilX2)]

(c.8) Let us consider the one-form hjwwV~" for j > 2. This one-form is not m e r o m o r phic. Its ~ derivative is given as follows:

VW( hjw~V w) = hjww ~ww + delta functions.

(C.9)

Since -jwwh ~w~i - - 3,jX2o is a finite function whose volume integral is zero, it can be written as a F-derivative of a finite one-form f j ~ , i.e., hjwj,

ww - aijx~o = v'%i~

.

(C.IO)

U p to a delta function, x 2 can be also obtained as a F-derivative:

x2 =

1 vw( G1 (O~x.x(,o)))+ ~-8(2)(w, to).

(C.11)

Therefore, 1

mj,~ - hjw~v w - fj,w - 8ji~-~ ( 3 ~ x . X ( t o ) )

(c.~2)

is a m e r o m o r p h i c one-form for j >/2, i > 1. Since the sum of the residues of m J t W vanishes, we find

4(p- 1)

E hjt, tAi, s= -~ij •

(C.13)

s~0

It is easy to check that (C.13) is also valid for j = 1. Using A,, s we can construct 1

vi,W__= 2~r ~Ai'sKt~t'"

(C.14)

s

This has the same singularities as o iw. F r o m (B.3) and (C.13), v i,w satisfies the same e q u a t i o n as v iiv.. --ww V"w vi, ' w _ - hi + delta functions.

(C.15)

H. Sonoda / Energy-momentum tensor

568

F o r p >/2, v; w is identical to vw, since v; ~ - vw is holomorphic, but there is no h o l o m o r p h i c vector for p >t 2.

Appendix D

CALCULATION

OF Oyi~/~ A T y = 0

W e c o m p u t e the y derivative of the measure ~2(y, Y) in the moduli space. F r o m (59), we find a,,g-~z'~ y = 0) = - 2 h ; z , Oy,gZe( y = O) = ay,gee( y = 0) = 0.

(D.1)

Then, c o m b i n i n g (16) and (B.10) we get

Oriln

f-d 2 " ~

det'-

=-fd2z

V z ~ t,

, ~-,

[ T z~ t°tal ,,. x>

(D.2)

Jy=0

T o first order in y, the second rank tensor t i defined by (44) is given b y

1

tiz~=hiz~_

z

f d2w -2~r- K

. . . . . Vw h iw•, z?l

z

tiz~. -- B~tlz~ ----Tlehizz , t i ~ _- rle~sti~ z z ~ ~- 0

(D.3)

Thus, to first order in y,

u,j= fd2zg~0 ( 1 -

2 -

--- 3,j.

.;2 (D.4)

This gives Oyidet N/jiy= o = 0.

(D.5)

F r o m eqs. (43), (D.2), and (D.5) we finally find OYi~2 y=o = 1 #~ f d2z r-h:-
(D.6)

H. Sonoda / Energy-momentum tensor

569

Appendix E C A L C U L A T I O N O F ( ( L _ 2 + ~L-1)~1 3 2

"~z..-~u)

Let us recall what we have. ~bI is a tensor field of rank (0, 2) with 2p ~ 2 = - 1 so that

+ = ( L 2+~L21)e~l

(E.1)

is a spurious physical vertex operator which is a tensor of rank (2, 2). The fields ~ 2 , - - . , q~N are physical, i.e. tensors of rank (•2, 12)..... (IN, IN)" U s i n g (18) and (38), the correlation function is given by < ~ ( Z 1 , Z 1 ) ~ 2 -- • ~bN>

= 3[Vz,

Oz,<~)l...d~N> AVgz,~., Oz, gZ'~l" Oz,<~kl...~)N> q-gzle , "~ oz,2g ZIZ, • (dpl

"

'

"

~N>]

N k=2

-- ~S ~i h i ' s h i z l Z l (

(Ttsts-

(

(Ttsts>)~l"''~NI

-~ < T z l z l > < ~ l ' ' ' ~ N >

1)

z-'*z~

Z -- Z 1

"+- lira O z OzZz, z~z 1

- Z -- Z 1

(E.2)

<~I'''~N>"

H e r e V~w is defined by (31). In the following we rewrite (E.2) in a manifestly covariant way. First we n o t e that

(

~z, =- lira Kz~ z

--

z1

1

)

(E.3)

"KT~ts+2hiz~,~7~.,Kt~t~),

(E.4)

+ ~g~ Ozg~e

Z -- Z 1

is a r a n k - o n e tensor. Second we note the equality:

(

1)

lim Oz o~,z,

z~z 1

=-O z

Z -- Z 1

lim

(

ozZz,

I z--*zl

1)

- -

Z --

Z1

EEA,.,(Vz h,

....

s

i

H. Sonoda / Energy.momentum tensor

570

where the energy-momentum tensor of the ghosts is defined by (B.5). Then using (E.3) and (E.4), the correlation function (E.2) can be written in a manifestly covariant way: (~1~2

" " " ~N)

N = 3~7z, Oz,(dOl"'" {~N) -k=2

N --~Tzl('zl(l~l.,.l~N) ) -- k~=2~Tzk(2~r~i hizlzlOZk(l~l...l~N))

-- E E A i , s(hizlzl( (Ttst s - (Ttst,>)q}l,..l#N) s i

+ (Zzlz 1 ) ( ' 1 . - "

dPN) -- t~Tzl 2,tr

hi&zlui ( ' 1 . . .

(E.5)

ON) ,

where v~ is defined by (C.4), (C.5). The first three terms are good honest total derivatives. According to [3], the last four terms should be a total derivative with respect to the coordinates of the moduli space. It is not very difficult to guess the answer. We try the following total derivative in the moduli space:

1 ~?

\,5 i

!

iz 3zlX

) 1''"

'

(E.6)

k=l

where z is an isothermal coordinate of g(y, fi). The tensor t i z z is holomorphic and dual to ayjg as in (44). (Note we are not using an isothermal coordinate in (44).) A is invariant under holomorphic coordinate transformations of the moduli space. We use the gauge slice given by (59) and evaluate (E.6) at y = 0. From eqs. (15) and (B.9) we find ~7~(T~7ta~) = - 2 7 r E h , --2Z v~ h ;~

.

(E.7)

i

As in (B.4), this is solved by

=

h, ( T ~ w ) +

d2wv/gKWS~f~ww~7 --Zzl..~ i " w h iww" i

H. Sonoda / Energy-momentum tensor

Therefore, at

we get from

y = 0

2~rY'ti~lzaOy igd = i ~'~

571

(D.6)

~/~ total ~ + x ZIZ1 /

(E9)

f d 2 z v l ~ K z l~~ , E - ~h i Wzhizz. i

Next we consider

B = Oyi

d2zk~((k) (g~kS~)t~(eO1...~m) •

(E.10)

From eqs. (57a) and (D.1), we get N

B = l-I d 2 z k ~

(gZke~)~

k=l

x (o7, o~,(e~,.... ,#,,,,.}- % v,,(,~,... ,~) N

+2O~,ui21(COl ..CON) + ~_,

~;Tz,(U~k(~l.-.*N) )

k=2

, + 2-;

)

-

(E,t

Here u[ is an infinitesimal coordinate change (55) associated with Oyig zz. There is a holomorphic ambiguity in u 7 but the ambiguity cancels in (E.11), i.e. B is invariant under holomorphic coordinate changes. Finally from (D.3) we find Oyitizlz ' =

-2hiz,z ' O~u~'-

f dZzv/g

~-~Kz, zlh~Zlp'zhiz z .

(E.12)

The first term is due to the coordinate change (55). Putting together eqs. (E.9), (E.11), and (E.12) we obtain N

A = I-I d 2 z k ~ k) (gZk~k) t~ k=l

N lYzk, ~-'hi~,~,v;k'(d)l"'"

a,rEh,:,:,(v, z1 0Zli

(~7:lU~l))(*l

total

• • - *N ) -~ (Tz1z1)(*,...

*~r)

]

. (E.13)

..1

572

H. Sonoda / Energy-momentum tensor

After a little algebra, it is easy to see that (E.13) is identical to the last four terms of (E.5). Therefore, we finally get (67).

References [1] A. Belavin, A. Polyakov and A. Zamolodchikov, Nucl. Phys. B241 (1984) 333 [2] D. Friedan, in 1982 Les Houches summer school, Recent advances in field theory and statistical mechanics, eds. J. Zuber and R. Stora (North-Holland 1984) [3] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B271 (1986) 93 [4] A. Polyakov, Phys. Lett. 103B (1981) 207 [5] V. Knizhnik and A. Zamolodchikov, Nucl. Phys. B247 (1984) 83 [6] O. Alvarez, Nucl. Phys. B216 (1983) 125 [7] A. Belavin and V. Knizhnik, Phys. Lett. 168B (1986) 201 [8] G. Moore and P. Nelson, Nucl. Phys. B266 (1986) 58 [9] E. D'Hoker and D.-H. Phong, Nucl. Phys. B269 (1986) 205 [10] P. Goddard and C. Thorn, Phys. Lett. 40B (1972) 235; R. Brower, Phys. Rev. D6 (1972) 1655 [11] H. Sonoda, preprint LBL-22172