Quantized strings on Riemann surfaces and the string-field-theory vertex

PHYSICAL REVIEW D VOLUME 38, NUMBER 6 15 SEPTEMBER 1988

Quantized strings on Riemann surfaces and the string-field-theory vertex

Nobuyoshi Ohta Institute of Physics, College of General Education, Osaka University, Toyonaka 560, Japan

(Received 1 April 1988)

It is shown that there are four different methods to define an operator formulation of strings on Riemann surfaces. These methods are illustrated by simple examples which allow intuitive understanding of the meaning of the formalism. As an application, open-string field theory is regarded as the string theory on a disk, and the vertex and more generally N-string vertices which are SL(2)- invariant are constructed by these methods. For any (reasonable) conformal mapping, the formalism allows one to construct a corresponding string field theory. The symmetry properties of the vertices, in particular their Becchi-Rouet-Stora-Tyutin invariance are discussed in detail. An extension to a higher genus is also discussed.

I. INTRODUCTION

The current upsurge of interest in string theory as a unified model of the fundamental interactions has created a renewed interest in the formulation of string theory.' At present there are mainly two complete and successful formulations of the bosonic string theory. One is a tradi- tional approach of the field theory of ~ t r i n ~ s . ~ - ] ~ The other is the path-integral f~ rmula t ion . '~ - " In principle the first approach is complete in the sense that it gives all the necessary rules to compute the off-shell scattering amplitudes covariantly to arbitrary order, but it also has the disadvantage that the rules have technical difficulty in doing the calculations and some of the properties of string theory such as the modular invariance is somehow hidden. On the other hand, the path integral seems to provide the most powerful method to compute the amplitudes especially in higher genus though the origin of the gauge symmetry in string theory is not manifest.

One of the most important developments in string theory is the realization that the conformal symmetry plays a very important role in the formulation of string

It is thus desirable to develop a method to enable study of properties and symmetries of the conformal field theories in terms of which string theory is well described.

Recently a new approach called the operator formalism on Riemann surfaces has been proposed.2'-27 This may be regarded as an attempt to extend the conformal field theory formulated so far on the (tree-level)

to more general Riemann surfaces. In a sense this is an intermediate approach between the field theory and path integral, and would be most suitable for our purpose. In this paper we would like to investigate the meaning and methods of this formalism. In particular we show that there are four equivalent but different ways to define the formalism. As an important application, we show that these methods uniquely characterize the SL(2binvariant three vertex or more generally N- string vertices that appear in string-field theory. The formalism clarifies that we can always define a field theory corresponding to one conformal mapping from the con-

formal plane to the string world sheet.28 The paper is organized as follows. In the next section

we discuss the simplest examples of the world sheet with holes and cross caps29*30 in closed-bosonic-string theory. There, we will see that all the essential features of the formalism appear and the four methods may be intuitively understood. In Sec. I11 we apply the formalism to the field theory of open bosonic string^.^-'^ Starting from the "disk state" on the conformal plane corresponding to the scattering of N open strings, we determine the general N-string vertex which is defined by a conformal mapping. We show how the present formalism enables us to construct an SL(2)-invariant vertex in string field theory and demonstrate its Becchi-Rouet-Stora-Tyutin (BRST) invariance. These are all done on the SL(2)-invariant vacuum, and we will also clarify the relation to a formulation using a different v a c ~ u m . ~ * ~ ~ * ~ ' In Sec. IV we discuss the formalism for closed bosonic strings on the torus and its extension to the Riemann surfaces of higher genus.27 Our approach is a little different from that in Ref. 27 in that we mainly use the information contained in the Neumann functions on the Riemann surfaces rather than mathematical notions of line bundles and meromorphic section, etc. Of course the final answer should be the same, but we believe that the present exposition of vari- ous methods offers an intuitive understanding of the formalism and may also be useful in extending this to other theories. Section V is devoted to conclusions.

11. QUANTIZED STRINGS ON RIEMANN SURFACES

In this section we show that there are four methods to define the operator formalism on Riemann surfaces by taking the simple examples of Riemann surfaces of a disk or cross cap. The basic idea is that we perform a first quantization of strings imposing the boundary conditions appropriate for each Riemann surface and express the results in terms of the operators of strings quantized on the sphere.

In order to fix our notation, let us first list the mode expansion for the string on the sphere:

1835 @ 1988 The American Physical Society

1836 NOBUYOSHI OHTA

where

other (anti)commutators vanish and we have omitted the spacetime index p. On the conformal plane defined by

these become

and similarly for antiholomorphic parts, where the following conformal transformation properties are used

which are derived from the requirement that the BRST invariance be preserved.

Now our example is the following. Suppose that a closed string is created at the time T= - cc and propa- gates to a certain time r a t which it is annihilated into the vacuum. The world sheet which describes this process is depicted in Fig. 1. This sheet is mapped into the conformal plane z by (2.2) and is expressed as a disk in Fig. 2. Our problem is to quantize the string with these "boundary conditions" and to express it in terms of the mode operators on the sphere defined in (2.1) or (2.3).

A. Boundary operators

Our first method is the most naive one: namely, to solve the field equation by imposing the boundary condi-

FIG. 1. Closed string vanishing into the vacuum at time r.

FIG. 2. The same process as in Fig. 1 on the conformal plane.

tions. The conditions we must have in this case are

which means that no momentum flows at the edge of the string world sheet. This leads to the conditions on the mode operators in (2.1):

Instead of imposing these as operator relations, it is more convenient to introduce a "boundary operator" which imposes these conditions on the quantum fields when they are placed next to the operator.29s30 This is obvious- ly given by

The boundary conditions to be imposed on the ghost must be obtained by requiring that the total boundary operator be BRST invariant. Recall the form of the BRST charge

where xh and X u represent holomorphic and antiholomorphic parts of X, respectively. The boundary condition (2.5) then leads to

for the BRST invariance. In terms of modes in (2.1), these are rewritten as

The boundary operator for the ghosts is then obtained

QUANTIZED STRINGS ON RIEMANN SURFACES AND THE . . . 1837

where the ghost zero-mode vacuum is defined by

It is not difficult to check that the boundary operator given by the product of (2.7) and (2.11) is indeed BRST invariant,

by using the mode expansions. This method is the one used in Refs. 29 and 30 in a

different context, is very intuitive and clearly shows that what is done is to perform the (first) quantization of strings with appropriate boundary conditions. It does not seem to be easy, however, to apply the same method to more complicated cases to be discussed in the following sections. We now show that the same results may be obtained by different methods which are more tractable when applied to complicated surfaces.

B. Bogoliubov transformation

The most standard procedure to perform first quantization is to expand the field operator in terms of basis functions which satisfy the appropriate boundary conditions and define the mode operators as their coefficients. Let us do this for the string with the boundary condition (2.5) which, on the conformal plane, becomes

with r2=e2' . The basis functions are easily found and are given by

and their complex conjugates. So the mode expansion takes the form (omitting zero modes)

We can express these new mode operators Bn ,B, in terms of oscillators on the sphere as

Now since the string annihilates into the vacuum, the boundary may be regarded as the vacuum for the string

(2.16) and hence is annihilated by these mode operators. Thus, we obtain the same conditions as (2.6) and, hence, the same boundary operator (2.7) in this way.

The ghost part is similar. The boundary conditions are - z I C ( Z ) = -r - lC( r ) ,

z ~ B ( z ) = z ~ B ( F ) for Iz 1 2 = r 2 .

There are two kinds of bases satisfying (2.18). For C ,

For B,

As before, we expand the string coordinates and define mode operators. Since the boundary condition (2.18) mixes left- and right-moving sectors, the mode expansion is made for the sum of left- and right-moving sectors. These modes can again be expressed by those on the sphere (2.3) as

From these, we see that the vacuum of this system should be annihilated by

which uniquely determine the state (2.1 1). Thus we have another method which gives the same

boundary state. Note that what is done here is to expand the string field on new bases and express the modes defined on different bases. This is a well-known technique of Bogoliubov transformation in cosmology and has recently been considered in a different context of defining string field-theory vertex,32 which motivated the

1838 NOBUYOSHI OHTA - 3 8

present analysis. It is clear that this method is completely equivalent to the first one, but this is more powerful since we can always define Bogoliubov transformation once the basis functions are known.

C. Conserved charge method

This is the method considered by ~ a f a . ~ ~ We wish to show that this is a slightly different but almost same method as the Bogoliubov transformation. The idea is that in the path-integral formulation of strings, the state represented by a disc in Fig. 2 is a number obtained by the path integration performed on the disk. If one con- siders the variation 6 X which is a solution of the field equation and preserves the boundary condition, that gives a symmetry of the action which defines the path integral. We emphasize that if 6 X does not satisfy the boundary conditions, it gives rise to a boundary term and is not a symmetry. As a consequence, the state defined by the path integral has the symmetry under the shift X - + X + S X .

Now the boundary condition (2.14) on our disk is satisfied by the basis (2.15). So the variation

is a symmetry of the state and generates an infinite number of conserved currents

where ~ , h and X: are the holomorphic and antiholomorphic parts of the basis X , (2.15).

Substituting the mode expansion (2.3) and (2.151, we get the conserved charges

The shift by Xn also generates conserved charges

The state annihilated by these charges are again given by (2.7). Note that these charges are precisely the combinations (2.17) that appeared in the definitions of mode operators for strings quantized on this surface. Since we must use the same basis function, it is clear that these methods are just slightly different views of the disk state.

The extension to the ghost sector is straightforward. This time one just uses the bases (2.19) and (2.20) and gets the conserved charges by the Noether method precisely in the combinations (2.21) or (2.221, and, hence, the state (2.11 1.

D. Neumann function method

Using the mode expansion (2.31, we can compute the correlation function for X(z) in the presence of the boundary operator:

As usual in conformal field theory, we assume the in-

equality ) z ) > ! z ' / in (2.271, which ensures the conver- gence of the series that appear in the evaluation of (2.27). This is the Neumann function on the disk of radius r obtained from that on the sphere by adding the contribution of the "image charges" z - r 2 / Z Our final method is to use (2.27) as the definition of the boundary operator for the disk.

Let us then define the "Neumann coefficients" by ex- panding the Neumann function as

where

If we define

we easily see that this boundary operator, when evaluated with the mode expansion (2.3), reproduces (2.28) which is by definition (2.27). In the present case, Nnm is given by (2.29) and (2.30) is precisely the boundary operator we defined before. This definition is quite general and allows general definitions of such "boundary operators" once the appropriate Neumann function is given. This method is similar to that discussed in string-field theory in Ref. 3 1 and originally by and el st am.^^

One can get yet another view of this definition as follows. If one differentiates (2.27) n times with respect to z' and sets z l=O, one finds (neglecting ln j z' ) term)

which is precisely the basis function (2.15). This is to be expected since the Neumann function is constructed so that it satisfies the boundary condition. Thus, the Neu- mann function may be viewed as a generating function for the basis that satisfies the boundary condition. This is a very useful observation when one wishes to develop the operator formalism on a given surface. Once we know the Neumann function on a given surface and if we want to develop the operator formalism on a surface topologi- cally equivalent to the first surface, we only have to find a conformal map from the first surface to the second and transform the Neumann function, from which the basis functions are generated just by differentiation. We will use this method later on to construct the vertex in string-field theory for an arbitrary conformal mapping.

The same method also applies to the ghost sector. Since it is straightforward, it would be sufficient to give the Neumann function which we now regard as the definition of the boundary operator:

QUANTIZED STRINGS ON RIEMANN SURFACES AND THE . .

1 z 2 ( R 1 C ( Z ) B ( W ) ~ g ~ ) = - - ,

z -W w2

r 2 I ( R I c ( r ) B ( ~ ) / B g h ) =-- ,

IW - r 2 w 2

1 z 2 ( n I c(z)B(E) I ~ g ~ ) = - - , ZE-r2 E

1 r ( n 1 C(r)B(m) 1 , z-W W

where we have used the ghost vacuum defined

FIG. 3 . The disk state corresponding to the string interaction in the conformal plane.

by

vertex ' three-

- (a 1 =(O 1 F - l ~ - l ~ o (2.33) then, it is more natural to first consider the disk on the

from the conjugate SL(2binvariant vacuum ( 0 1 . It is not difficult to see that definition (2.32) uniquely defines the boundary operator (2.11) as for X.

To apply these methods to the case of cross cap is quite straightforward. The boundary conditions for the cross cap of radius r r e T are

We then find that these conditions are equivalent to (2.6) with r replaced by ir. So all the discussions proceed in the same way and the cross cap operator is given by

and the Neumann function by

For the ghost, the boundary conditions turn out to be

which are again derived from the BRST invariance. Again the discussions are completely the same as the disk with r replaced by the imaginary radius ir, and the operators and the Neumann functions are obtained from (2.1 1) and (2.32), respectively, by the replacement r+i r . This completes our exposition of the four methods to define the "boundary operator."

111. VERTEX IN STRING FIELD THEORY

In this section we show that the techniques described in the preceding section can be used to define the three- vertex in open-string-field t h e ~ r ~ . ~ - ' ~ More generally the N-string vertex which represents scattering of N strings can be derived.

In conventional approach, interacting string-field theory is defined by specifying the sewing of three strings and then conformally mapping the sewed state onto the conformal plane. We note that whatever the sewing prescription is, the mapped image must be a single sheet of the disk (or equivalently the upper-half plane) with three-string Hilbert spaces distributed on the boundary (Fig. 3). Instead of starting from the sewing prescription,

conformal plane and define the interaction of the strings on the world sheet by giving a conformal mapping. It is clear that whatever the resulting shape of the world sheet, this defines a meaningful field theory on the conformal plane on which any calculation may be performed. Thus in this approach there exists one string-field theory for a given conformal mapping. In particular, interactions nonlocal on the world sheet such as the Caneschi- Schwimmer-Veneziano vertex may be considered. Simi- lar discussions have been given in Ref. 28. We note that whether only the three-string interaction is enough to reproduce all scattering amplitudes is another question to be addressed separately. T o resolve this problem, one must show the full nonlinear BRST invariance of the theory such as Witten's (where associativity of the vertex is the crucial property).5,6328,34 If this is not satisfied with the three-vertex alone, then additional interactions are necessary as in Refs. 2-4.

Let us discuss the N-string vertex. Any method in the preceding section may be exploited; here let us use the conserved charge method. Suppose we have a disk state which is actually a superposition of N-string Hilbert spaces attached on the punctures u, ( r = 1, . . . , N ) distributed on its edge. We suppose that each string has its own conformal mapping u = f , ( z ) to the world sheet whose coordinates are given by r, + i u , =Inz =In f; ' (u for u in the rth string region (see Fig. 4). We take u, = f,(O). The overlapping conditions are written as

for u in the overlapping region. Our task is to determine the state represented by the disk in terms of the N-string Hilbert spaces.

Following our method, we consider the basis functions

FIG. 4. N-string vertex in the conformal u plane and in the z plane of each string mapped by the conformal map u = f,(z).

1840 NOBUYOSHI OHTA

satisfying the Neumann boundary condition on the disk. N(z ,z l )=- ln[f r (z) - f S ( z f ) ] On the u plane, they are -

- ln[ l - f r ( z ) fS (z1) ]+H.c . , (3.4)

xn =u -"+ .

Note that this is given by

(3.2) where we assumed that z and z ' belong to rth and sth string regions, respectively. From this, the basis functions are generated by

tions from the "image charges" and have only the effect (3'3) of making the contour integration originally defined over

semicircular boundary the closed one (known as the Actually, in order to express the disk state in terms of the "doubling" of the open string) and they will be neglected N-string Hilbert spaces, it is more convenient to consider in the following. the basis functions on the world sheet where the modes The Neumann function in (3.5) has a singularity are defined. They can be obtained by transforming the ln(z - z l ) when z and z ' are close. Let us extract the Neumann function to the N-fold z plane: singularity

1 an

We then find the conserved charges

1 an Xn=--- [-ln(u -u l ) - ln(1-Eu ' ) ]

( n - I ) ! au'.

where

~ h ~ , ~ i ( z ) = - -N(z ,z l ) ( n - I ) ! az'"

(3.5) . r ' = O

u ' = O

dz dz' N:, =$o-$o-z-m-l z ' -"- ' [ ln[ f r ( z ) - f s ( z ' ) ] -~ r s ln (z -z1)] . 2 ~ i 2 ~ 1

(3.8)

The second and rest terms in N ( z , z l ) are the contribu-

Here we have rewritten the derivative with respect to z ' by the contour integration. The contours are taken around the origins of rth (for z ) and sth (for z ' ) string z planes (corresponding to the punctures ur in the u plane).

The conformal mapping should be taken such that the logarithmic term in (3.8) is analytic at the origin z =z f=O and may be expanded in z and z'. (This is true for all known ) Then N:, is nonzero only for m, n 2 0. For m, n > 0, partial integrations give

1 dz dz' -, ,-, f:(z)fS'(z1) N = - $ ~ - $ ~ - Z z

mn 2 ~ i 2 ~ i [ f r (z) - fS (z ' ) l2

(the second logarithmic term vanishes), which is cast into

1 N" =- du d u ' 1 mn mn $ u , ~ $ u s ~ ~ f ~ l ~ ~ ~ l ~ m ~ f ~ l ~ ~ ' ~ l ~ n ( U - u ~ ) ~ '

This is a well-known formula in light-cone string-field I m

theory.233 The other coefficients are evaluated as (r i (si ( v = ( o / exp -f 2 a . (3.12) r,s m,n = O

In[ f r ( 0 ) - f s (0 ) ] , r f s , finds

1 Recalling the definition (3.5) of the basis function, one

lnf:(O), r =s ,

The disk state or the N-string vertex is the one annihilated by the charges (3.7):

3 8 - QUANTIZED STRINGS ON RIEMANN SURFACES AND THE. . . 1841

This implies

( V I X (z )X(z l ) 10) =N(z ,z l ) ,

We can remedy this by introducing additional factors into the Neumann function3'

(3.14) 7

as can be easily checked from (3.12). fi [fr(z)-fi(0)] Finally we note that the Neumann function (3.4) is 1 i = I Ngh(z,z'-) = f r ( z ) - fS (z1) 3

essentially invariant under the SL(2) transformation in the following sense. Namely, under the SL(2) transfor-

n [fS(z1)-f i (o)] i = 1

mation,

it transforms like

The last two terms depend only on the single variable z or z' and such terms are irrelevant in computing scattering processes since they automatically drop out from the amplitudes owing to the momentum conservation. This is reflected in the vertex. In fact, one can see that the vertex is SL(2) invariant if one also uses the momentum conservation.

Let us now turn to the ghost vertex. As before, it is convenient to consider the Neumann function for the ghosts which generates the basis functions. Using the transformation properties (2.4) of the ghost fields, this would be, on the z planes,

where three arbitrary punctures labeled by i = 1,2,3 are picked up. Because of SL(2) invariance, the resulting amplitudes will not depend on the choice of the punctures. This modification corresponds to making picture changing three times in the Hilbert spaces labeled by i.'9331 The necessity of introducing these factors here is thus a consequence of SL(2) invariance and actually is related to the existence of three zero modes in the ghost field C on the sphere.

The basis is then generated from (3.19)

The resulting conserved charges are then derived

However, the disk state is expected to have an invariance under the SL(2) transformation whereas (3.17) transforms dz'

Q ? ' C ' = 2 $,-B~"(z')c'"(z') under (3.15) like 27ri

(3.18) where the ghost Neumann coefficients are given by

The N-string vertex which is annihilated by (3.21) is then where , ( a 1 is the rth string adjoint vacuum of the found to be SL(2binvariant vacuum 1 0 ), defined by

N

( vgh 1 = n ,<a 1 eXP , r ( G l o ) r = l . (3.24) r = l

The case N = 3 corresponds to the string field-theory ver- (3.23) tex.

1842 NOBUYOSHI OHTA

Here a comment on the ghost number is in order. With the usual definition of ghost-number operator

which is anti-Hermitian, both ( 0 ) and (6 have ghost number - +. (The fact that anti-Hermitian operators have real eigenvalues is due to the indefinite metric of the Hilbert space.) The inner product ( n , n 2 ) of the states with ghost numbers n and n is nonzero only if n , = n,, as can be seen as follows:

hence (n , -n2 ) ( n I I n 2 ) =O. Note further that conjugate vacuum ( 0 1 has ghost number ++ due to the anti- Hermiticity of Q, and (8 = ( 0 ( c - ~ c ~ c , has ghost number -+ but not + ++ 3, and hence nonvanishing inner product (a / 0 ) = 1 . Our vertex (3.23) thus has ghost number - 3 N / 2 .

By our construction, our vertex is manifestly SL(2) invariant. Now we wish to show that it is also BRST invariant. From all our discussions, it would be clear that string fields, when placed next to the vertex, produces the Neumann functions. If these do not produce singularities which prevent deformations of the contours of the BRST charge, the contours originally defined around punctures may be deformed and joined and then shrunk to a point, showing the invariance of the vertex. There are poten- tially dangerous points which prevent the deformation in string-field theory, namely, the "interaction points" defined by a ln f y l ( u )/au =o. Following the Kyoto

we may regularize the BRST charge on the world sheet and compute the contributions of the singularities to the BRST charge on the vertex by using the u- plane correlation function

The result is then found to be3'

where a is a constant and u , is the interaction point considered. Thus the &--string vertex is BRST invariant for the critical dimension d = 26.

The above formulation may be rewritten as follows. Using the simple equality

one finds

2 dz dz' h-;g;ngnh)n= $ O p $ O p Z j m - Z Z ' - n t I 1 2 ~ i 2 ~ i

- 1

+ C I ~ $oPz 2n-i 2n-i

where constants C, (i = 1,2,3 are given by

which are nonzero for n 2 1. We then expand the exponent in (3.23) for the last three terms in (3.31). With the SL(2)- invariant vacuum to be put on the right-hand side, only the zero-mode terms C A I ' with n = 1 remain (others with n 2 2

38 - QUANTIZED STRINGS ON RIEMANN SURFACES AND THE. . . 1843

vanish on the vacuum). (If the three points i = 1,2,3 are chosen on the same string, other zero modes c p and cyl1 may survive.) If we treat these zero modes as operators to be inserted, the remaining vertex is

dv ( F eh = n , ( G 1 exp fi ' , e ~ ) r s b ~ ) ~ ~ s ' n z M;, b:nrl ,

r = l r . s m 2 - l 1 r ,m

where fi ;,hirs is the first term in (3.3 1) and

The total ghost number of this modified vertex is - t N + 3. For N = 3, the modification has the effect of changing all three SL(2)-invariant vacua to the down vacua / l ) - c l / O), on which the string fields are built in the formulation used in most papers on open-string field theory."I0 Remember that this modification is due to the additional factor we introduced in order to have SL(2) invariance in our vertex. In contrast, the string fields with the first vertex are built on the SL(2)-invariant vacuum and the kinetic term needs insertions of zero modes compared with the usual formulation. These two formulations are of course equivalent. One could also try to construct the theory on other "picture changed" vacua, as discussed in Ref. 31. The formulas (3.22), (3.231, (3.351, and (3.36) have been also obtained in Ref. 28 by a different technique. There it is also checked that these give the standard results for witten's5 and other vertices (see also Refs. 6 and 36). The BRST invariance of the vertex (3.35) may also be proved similarly to the case of the vertex (3.23).

What is remarkable in this formalism is that it is not necessary to introduce the cumbersome "prefactors" to our N vertices. In order to have the BRST invariance

which is the most important property of the vertices to ensure the decoupling of the ghosts, it is in general necessary to introduce prefactors in addition to the naive over- lap ve r t i ce~ .~ , 5 t 6 9 3 1 These prefactors actually have the effect of recovering the SL(2) invariance of the vertices that has not been incorporated in the construction of the naive vertices. This is why our manifestly SL(2)-invariant N vertices are automatically BRST invariant in the critical dimension.

IV. TORUS AND HIGHER GENUS

In order to extend the formalism to higher-genus Riemann surfaces, it is easiest to exploit the informations contained in the Neumann functions, as we have seen for the disk state in Sec. 111. In this section, we discuss the simple example of the torus explicitly and then indicate the generalization to higher genus. A general discussion has been already given in Ref. 27 on the basis of the mathematical notions of meromorphic sections, etc.; for completeness we reformulate the formalism by using the Neumann functions which contain all the necessary information.

A torus is represented as in Fig. 5 with the inner boundary / u = q I < 1 being identified with the outer boundary 1 u = 1 by u e q u . This is mapped into the parallelogram in Fig. 6 by u =e2"". The Neumann function in the z plane is

where r=lnq /27~i. As discussed in Sec. I1 D one gets the basis functions by differentiation of (4.1):

.r 7T We see that this is indeed invariant under z -z + 1 and en =apn - 2 nNnm am - -(al -a1 )6,, , .

I m r (4.4) z --tz + T and is a solution of X field equation. The first m =I

term in the sum is divergent for z =O. This is because the B~ the same token, using 2, for the basis, one finds original Neumann function (4.1) has a singularity l n ( i -z ' ) when z is close to z', and so X, has a singularity - X

7T Q n = G - , - 2 n N n m a m -----(GI-a1)6,,, . like z -". It is not difficult to see that this is the only Imr

(4.5) m = l

singularity in XI , at the origin and hence it can be expanded as These uniquely characterize the "torus vacuum25"

The conserved charges are then

NOBUYOSHI OHTA

FIG. 5. Torus defined by the identification u-qu with 9 I < l .

which is a state of a torus with puncture at z = O (Fig. 7). Recalling the definition of N,, , one has

1 Z' N ; Z , Z ~ = 2 - I T I n + I N , ~ Z ~ Z ~ ~ + C . C .

, = I n,rn=l

- n ( z -z'-T+T'j2 , (4.7) 2 I m r

and

( T I X(z)X(z1 j 10) =N(z ,z1 ) . (4.8)

Similar construction may be done also on the u plane. It is now straightforward to extend the above construc-

tion to higher genus and define the "g vacuum," and the result may be summarized by24327

( g 1 X ( z ) X ( z t ) 1 0 ) = -In 1 E ( z , z l )

where E is the prime form, r is the period matrix, and w is the Abelian differential. This defines the "g vacuum"

a " 1 - 1 +y 2 -i Anal , - A,B, )-

n . r n = l I m r

where N,, and A, are defined by

FIG. 6. Torus represented by a parallelogram with opposite sides identified.

FIG. 7. Torus vacuum with a puncture at z =O. Generaliza- tion to the "g vacuum" is obvious.

respectively. Here we have suppressed an index running from 1 to g for w and A , .

For extending the construction to the ghost sector, it is only necessary to know the Neumann function. How to obtain it has been discussed by ~ a r t i n e c , ' ~ and it is given, for example, by the Riemann 0 series (here we discuss only holomorphic part):

where TCSL(2,C) is the covering transformations which corresponds to the Riemann surface considered. Surfaces of higher genus g 2 1 are obtained by considering the group T generated by g elements y iESL(2 ,C) , i = 1 , 2 , . . . , g .

For the torus, y may chosen to be a dilation y ( u ) =qu and the elements of are gnu i- rn < n < + co ). It is still necessary to regularize the sum in (4.13) and one then obtains

In order to make this single valued on the surface, however, we must add the zero-mode term12

In general, the Neumann function G on the Riemann surface of genus g has the structure12

where p'", i = 1, . . . ,3 ig - 1 ) (for g 2 2 ) is a basis for holomorphic quadratic differentials [zero modes for B (z)], M is the generalization of the period matrix, and F"' is a basis for the automorphic integrals whose 8 derivatives span the space of harmonic Beltrami differentials. The last term in (4.16) corresponds to (4.15)

3 8 - QUANTIZED STRINGS ON RIEMANN SURFACES AND THE. . .

in the torus case. Just as in the case of X, the "torus vacuum" or "g vac-

uum" is defined to reproduce the Neumann function (4.16) as the correlation of B ( z ) and C ( w ) , with M,;'F'~'(w) part being treated as insertions of B fields modified as

where the contour is taken around the puncture at z = O (see Fig. 7). This gives the correct measure for the moduli This is analogous to the modification of the N-string vertex for g = O to treat zero modes of the C fields as insertions in Sec. 111. One then generates the basis functions for B and C from G by differentiation and has also (3g -3 )cp"' as additional basis for B field. By the Noether method, one gets the conserved charges using these basis functions. The p"' gives 3g - 3 conserved charges which consist of solely annihilation operators of C field

where the contour is taken on the puncture. It would be now clear that after all the procedure we have described, the g vacuum is given by

where the Neumann coefficients NLg,!' are defined by

or, similarly to (3.21),

The ghost number of ( g g h is - 3g + 3 - +. This means that in order to have nonzero expectation values, we must insert 3g -3 more B's than C s . This is the number of moduli and the (3g - 3 ) B insertions (4.17) precisely reproduce the correct measure for moduli integra- t ions~ l , 12 '27 derived in the path-integral approach.'4-17

These results are related to the Riemann-Roch theorem. The above general discussion is actually valid only for

g 2 2, and the torus case g = 1 is a little special, since we have already seen that (4.15) is necessary to make the Neumann function single valued. This is the reflection of the special situation in the torus where there are one zero mode for both C and B fields. Namely, (4.15) indicates the existence of one zero mode in B (and one moduli), and the factor u ' / u in Eq. (4.14) means the existence of a zero mode in C. This implies that we must have each inser- tion of both B and C charges. The zero modes on the torus are just the constants for both B and C and hence the vacuum is given by

where b is a linear combination of the annihilation operators 6, (n 2 - 1 ) of antighost B derived similarly to (3.35) and (3.36). The ghost number of this "torus vacuum" is therefore given by the same formula below (4.211, in agreement with the Riemann-Roch theorem. (We have seen in Sec. I11 that this is true also for g =O.)

Finally we end this section with a brief comment on the BRST invariance of the "g vacuum." I t is straightforward to introduce punctures on the Riemann surfaces and construct the vertices as for the disk state in Sec. I11 (Ref. 271, and we would like to argue the BRST invariance of the vertices of which a special case is the "g vacuum." The argument is actually almost the same for the disk case in Sec. 111. The BRST charges are defined by contour integrations around each punctures as in Fig. 8. If the deformation of the contours is allowed, they can be joined and then be pushed to the other side to shrink to a point. That such a deformation is allowed is ensured by the fact that the Neumann function which is used to define the "g vacuum" and hence appears in the evaluation of the BRST charge as in Sec. I11 is defined to transform properly under the conformal transformation. The resulting integral vanishes because the Neumann function satisfies the appropriate boundary conditions. However, if one had included the integral over moduli, there would appear the BRST anomaly from the boundary of the moduli space (in particular, from the dilaton tadpoles) which may be removed by the renormalization of the slope parameter and mass or other cancellation^.^' This is an interesting subject and deserves further investiga- tion.

V. CONCLUSIONS

We have thus reformulated the operator formalism on the Riemann surfaces which may be defined by four methods. These methods have their own advantages. For instance, the first method allows very intuitive understanding of the formalism, and the Bogoliubov transformation is the standard first quantization of the string on the Riemann surfaces yet, we believe, most clearly eluci- dates the meaning of the conserved charges which are actually the mode operators. We have seen that all the informations necessary to set up the formalism are contained in the Neumann functions for the surfaces.

The application of these methods enabled us to define the string field-theory vertex for a very general conformal mapping. Also the N-string vertex which may be used for computing N-string scattering amplitudes directly can

+ other contours FIG. 8. Contour deformation proving the BRST invariance

of the generalized vertex.

1846 NOBUYOSHI OHTA - 3 8

be constructed. T h e BRST invariance of these vertices may be proved easily in this formalism and will lead t o amplitudes free from ghosts. W h a t remains for having a complete string-field theory is t o make a choice of the conformal mapping which gives cyclically symmetric

t four vertex ( VI234 / -- ( V125V534 1 , which ensures the full gauge invariance of the a c t i ~ n . ~ ~ , ~ ~ T h e higher genus "vertices" a re also useful for investigating the higher- o rder effects in string theory. I t would be interesting to examine the loop corrections t o the field equations and mass renormalization in string theory.25s29v37 I n the present formalism, these can be all understood a s t h e cancellation of the BRST anomaly. M o r e important is t o ex-

amine if any insight into string theory beyond loop expansions may be gained with the help of t h e formalism. We believe tha t these a r e feasible problems in this formalism. Another urgent problem is t o extend this t o super- strings and examine similar problems in these theories. W e hope that our rather elementary exposition would be useful for such extension, which is currently under study.

ACKNOWLEDGMENTS

I a m grateful for useful discussions with K. Arakawa, M . Horibe, A. Hosoya, and W . Ogura. Thanks a r e also due to A. Hosoya for careful reading of t h e manuscript.

'For a comprehensive review and references, see M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory (Cambridge University Press, Cambridge, England, 1987), Vols. 1 and 2.

2M. Kaku and K. Kikkawa, Phys. Rev. D 10, 1110 (1974); 10, 1823 (1974).

3 ~ . Cremmer and J. L. Gervais, Nucl. Phys. B76, 209 (1974); B90, 410 (1975).

4 ~ . Hata, K. Itoh, T. Kugo, H. Kunitomo, and K. Ogawa, Phys. Rev. D 34, 2360 (1986); 35, 1356 (1987), and references therein.

5E. Witten, Nucl. Phys. B268, 253 (1986). 6D. J . Gross and A. Jevicki, Nucl. Phys. B283, 1 (1987); B287,

225 (1987). 7E. Cremmer, A. Schwimmer, and C. B. Thorn, Phys. Lett. B

179, 57 (1986). 8S. Samuel, Phys. Lett. B 181, 255 (1986). 9N. Ohta, Phys. Rev. D 34, 3785 (1986). IOA. Eastaugh and J. G . McCarthy, Nucl. Phys. B294, 845

(1987). "S. Giddings and E. Martinec, Nucl. Phys. B278, 91 (1986). 1 2 ~ . Martinec, Nucl. Phys. B281, 157 (1987). I3A. M. Polyakov, Phys. Lett. 103B, 207 (1981); 103B, 211

(1981). 1 4 ~ . Friedan, in Recent Advances in Field Theory and Statistical

Mechanics, edited by J . B. Zuber and R. Stora (Elsevier Sci- ence, Amsterdam, 1984); 0 . Alvarez, Nucl. Phys. B216, 125 (1983).

I5J. Polchinski, Commun. Math. Phys. 104, 37 (1986); G. Moore and P. Nelson, Nucl. Phys. B266, 58 (1986).

1 6 ~ . D'Hoker and D. Phong, Nucl. Phys. B269,205 (1986). 17A. Belavin and V. Knizhnik, Phys. Lett. 168B, 201 (1986); G.

Moore, Phys. Lett. B 176, 369 (1986). ''A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl.

Phys. B241, 333 (1984). 19D. Friedan, E. Martinec, and S. Shenker, Nucl. Phys. B271,

93 (1986). 20A beautiful account may be found in M. Peskin, in From the

Planck Scale to the Weak Scale: Towards a Theory of the Universe, proceedings of the Theoretical Advanced Study In- stitute in Particle Physics, TASI 86, Santa Cruz, California, 1986, edited by H. E. Haber (World Scientific, Singapore, 1987).

2 ' ~ . Ishibashi, Y. Matsuo, and H. Ooguri, Mod. Phys. Lett. 2, 119 (1987).

22S. Saito, Phys. Rev. D 36, 1819 (1987); Phys. Rev. Lett. 59, 1798 (1987).

2 3 ~ . Alvarez-Gaume, C. Gomez, and C. Reina, Phys. Lett. B 190, 55 (1986); N. Kawamoto, Y. Namikawa, A. Tsuchiya, and Y. Yamada, Commun. Math. Phys. 116,247 (1988).

2 4 ~ . Vafa, Phys. Lett. B 190, 47 (1986); 199, 195 (1987); see also P. Di Vecchia, K. Hornfeck, and M. Yu, ibid. 195, 557 (1987).

2 5 ~ . Ooguri and N. Sakai, Phys. Lett. B 197, 109 (1987). 26E. Witten, Commun. Math. Phys. 113, 529 (1988). 27L. Alvarez-Gaume, C. Gomez, G. Moore and C. Vafa, Nucl.

Phys. B303,455 (1988). 2 8 ~ . LeClair, M. Peskin, and C. R. Preitshopf, SLAC Report

Nos. SLAC-PUB-4306 and SLAC-PUB-4307, 1988 (unpublished); A. LeClair, Nucl. Phys. B297, 603 (1988).

2 9 ~ . G. Callan, C. Lovelace, C. R . Nappi, and S. A. Yost, Nucl. Phys. B293, 83 (1987); see also M. B. Green, Phys. Lett. B 201, 42 (1988).

3 0 ~ . Polchinski and Y. Cai, Nucl. Phys. B296, 91 (1987). 3 ' ~ . Kunitomo and K. Suehiro, Nucl. Phys. B289, 157 (1987). 3 2 ~ . Hosoya and H. Itoyama, Fermilab Report No.

FERMILAB-PUB-87/111-T, 1988 (unpublished); M. Horibe, A. Hosoya, H. Itoyama, and N. Ohta (in preparation).

3 3 ~ . Mandelstam, Nucl. Phys. B64, 205 (1973). 34S. P. de Alwis and N. Ohta, Phys. Lett. B 188, 425 (1987). 3 5 ~ . Itoh, K. Ogawa, and K. Suehiro, Nucl. Phys. B289, 127

(1987). 36G. T. Horowitz and S. P. Martin, Nucl. Phys. B296, 220

(1987). 3 7 ~ . Yoneya (private communication). See also P. Mansfield,

Nucl. Phys. B283, 551 (1987); A. Sen, ibid. B304, 403 (1988).

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Quantized strings on Riemann surfaces and the string-field-theory vertex