Covariant string field theory. II

PHYSICAL REVIEW D VOLUME 35, NUMBER 4 15 FEBRUARY 1987

Covariant string field theory. I1

Hiroyuki Hata* Research Institute for Fundametltal Physics, Kyoto Uniuersity. Kyoto 606, Japan

Katsumi Itoh, Taichiro Kugo, Hiroshi Kunitomo, and K a k u Ogawa Department of Physics, Kyoto Uni~lersity, Kyoto 606. Japan

(Received 29 September 1986)

The covariant string field theory is presented in its full detail for the closed-bosonic-string case. The properties of the interaction vertex of three closed strings are fully clarified. We thus establish the nilpotency of the full nonlinear Becchi-Rouet-Stora (BRS) transformation as well as the BRS invariance of our gauged-fixed action. This, on the other hand, also establishes the gauge invariance of our gauge-unfixed action and the group law of the gauge transformations, which were reported previously. Thegeneral A'-po~nt amplitudes for arbitrary external closed-string states are computed explicitly at the tree level. In particular, when the external states are on shell and physical, we prove that the amplitudes become independent of the choice of the external string-length parameters and correctly reproduce the usual dual amplitudes.

I. INTRODUCTION

At the end of last year we proposed a manifestly covariant field theory of interacting bosonic strings in two short papers',2 (hereafter referred to as I and 11) for the open- and closed-bosonic-string cases, respectively. There we constructed a gauge-fixed action as an extension of the preceding Siege1 free action3 to the interacting case, and proved its invariance under a nonlinear nilpotent Becchi- Rouet-Stora (BRS) transformation. Subsequently, in Ref. 4 (referred to as III), we presented a gauge-invariant actions with local gauge symmetry for both the open- and closed-string cases on the basis of the string vertices constructed in I and 11. The group structure of the gauge transformations was also clarified there.

Since the proofs for various statements given in I and I1 had to be very short by the nature of Letter articles, we fully spelled out their details in our recent paper"referred to as IV), restricting to the open-string case. There the quartic open-string interaction term was determined explicitly for the first time. We gave also the explicit computation of general N-string amplitudes at the tree level and the complete calculation of the zero-slope limit of the gauge-fixed action and the BRS transformation.

T h e purpose of the present paper is to give such full details for the closed-string case as a continuation of paper IV. (For other approaches to string field theory we refer the reader to many papers cited in IV.) In principle there is no difficulty in extending the various calculations in open-string theory performed in IV to the closed-string case since the closed string is more or less a "product" of two open strings consisting of right-moving modes and of left-moving modes. It should, however, be noted that there has appeared no literature which gives explicit com- putations of closed-string scattering amplitudes in the framework of string field theory. T o the extent the present authors know, this is true even in the light-cone gauge string field theory7-"nd even at the tree level.

Owing to this unfortunate (probably accidental) situation for the closed string, we cannot use the knowledge in the light-cone gauge string field theory which was actually very useful in the open-string case. For instance, the Cremmer-Gervais identity9 for the determinant factor of the Neumann function was essential in proving the 0 ( g 2 ) nilpotency of the BRS transformat~on in the open-string case, but its generalization to the closed-string case (as well as its proof of course) has not been given in the literature.

This paper is organized as follows. In Sec. I1 we explain some basic properties of our closed-string field and BRS operator QB and fix the notations and conventions used in this paper in conformity with those adopted in the prevlous open-string paper IV. In Sec. I11 we define the * product, which gives a closed-string field @* \V from arbitrary two closed-string fields Q, and Y, referring to a three-string vertex functional V. We show that one can easily construct V based on the knowledge of our previous open-string vertex Vope" in such a way that the distribution law

holds (at space-time dimension d = 2 6 ) . In Sec. IV we prove that the * product thus constructed satisfies a Jaco- bi identity:

again at d =26. It will be explained in detail how this identity is guaranteed to hold by the duality property of the theory, as was first pointed out in I and 11.

We present the gauge-invariant action with a local gauge symmetry obeying a simple group law and also the gauge-fixed BRS-invariant action in Sec. V. Although they are identical with those reported already in I11 and 11, respectively, we cite them here again in the present no-

1318 1987 The American Physical Society

3 5 COVARIANT STRING FIELD THEORY. I1 -

tations for completeness and give also a somewhat simpli- fied proof of the BRS invariance of the gauge-fixed action. The discussion of gauge-fixing problem is omitted which can be found in 111.

In Sec.VI we show that our covariant theorv with gauge-fixed action indeed reproduces the usual dual amplitudes for general N-closed-string scattering at the tree level if the external states are on shell and physical. In particular the explicit computation of the four-string tree amplitudes is presented in detail. Furthermore, based on the general N-string amplitudes obtained there, we prove that the on-shell physical tree amplitudes are completely independent of the choice of the length parameters a, of the external strings aside from the overall conservation factor 6( x:=, a, ).

Section VII is devoted to the summary and discussions. In particular we discuss how the problem of unphysical string-length parameter a characteristic to our covariant formulation can be solved in a satisfactory manner, referring to our recent proof for the a independence of the on- shell physical amplitudes at any loop-order level." We also make a comment that there is a severe difficulty in extending the Witten-type vertex to the closed-string field theory.

In Appendix A we give some basic definitions and properties of the Neumann functions associated with closed-string diagrams, which will probably be very useful since there seems to exist no literature available for the closed string as mentioned above. Appendix B is devoted to a path-integral evaluation of the 1Y-string effective vertices which appear in the computation of general IV-string amplitudes. In Appendix C we present the proof of the Cremmer-Gervais identity generalized to the closed-string case, which plays a key role in proving the Jacobi identity (1.2) in Sec. IV and also in obtaining the dual amplitudes in our theory in Sec. VI.

11. STRING FIELD

The closed-string field Q is a functional of string coordinate X p ( u ) ( p = 1, . . . , d ) and Faddeev-Popov (FP) ghost and antighost (Hermitian Grassmann) coordinates c ( a ) and F ( u ) :

Basically we follow the notational conventions adopted in our paper IV for the open string. [Therefore, in particular, c ( u ) and ?(a) are exchanged with in-,(a) and in-,(a), respectively, compared with the notations in I-111.1 However, since the zero mode co of c (a) (which is absent in open string) requires a special treatment here, we adopt the momentum (77;) re resentation only for that mode k' and hence co = -id/&,. We have separated n-: explicitly in (2.1) and therefore c ( a ) should be understood to contain no zero mode there and henceforth, when it appears as an argument of @. The variable a denotes "string length" which plays a role similar to p + in the light-cone gauge formulation. '-"

The coordinates Xp, c, and F are periodic functions of u with period 277, and are expanded into the oscillator modes as follows:"

with a prime denoting a / a u , where P p ( u ) , n-,(u), and v , (u) are the momentum variables - iS /6xp(u) , - i6/6c ( a ) , and - i6/6?(u), respectively, and c (0) in ( 2 . 2 ~ ) is the original one with the zero mode co not omitted. The oscillator modes a i" ' ,c~" , T :' ( i = i ) satisfy the properties

The ground state of the oscillators is denoted by ( 0 ) : of the field functional @[Z] (2.1) equivalently by a ket

( a i ~ k ~ ( ? I - I ? vector. Keeping the "coordinate representations" for the ,cn , c n )0)=0 ( n 2 l ) . (2.4) "zero-mode variables" ( x p , ~ ~ , ~ ; , a ) and separating out

Then we can represent the nonzero-mode-dependent part the ghost zero modes To and n-i explicitly, we can write

1320 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA - 3 5

the string field @ in ket representation as

The physical modes of the string are contained in the bosonic 4 component, and so we assign to the component fields d, 4, X, and 77 the (net) FP ghost number NFP =0, - 1, + 1 ,O, respectively, and also the corresponding Grassman-even/odd properties. Thus our string field

@ carries the ghost number iVFp = - 1 and hence is Grassmann odd.

The string field @ satisfies a reality condition:I2

Similarly to the open-string case of IV, this condition can equivalently be expressed in terms of bra-ket notation as

Here r and d r denote a set of zero-mode variables ( x , , ~ - [ ~ , r : ' ~ ' , a ~ of rth string and its integration measure dx,&Fid%-:"'da,/2~, respectively,13 and ,(O is the bra vacuum of the oscillators of the rth string. Note that the "two-string vertex" ( E ( 1,2) converts the coordinates z"' of string 1 into 2 1 2 ' of string 2; i.e.,I4

( k ( 1,2) 1 (~~~~'(a),c(~'(a),~"'(a),~:(~',a~)= ( E ( 1 , 2 ) 1 (xPf2'( --a), -cI2'( - U ) , F ( ~ ' ( -a),%-:(2 , - a 2 ) . (2.9)

The closed-string field @ is subject to another impor- zero mode c o = -ia/a%-:.I It is important to note that tant ~ o n s t r a i n t : ~ this BRS charge of closed string is just the sum of two

"open-string BRS charges" consisting of 2 modes. P@=@ or ( L + - L _ ) @ = O , (2.10) Indeed, for the open string, the variables

expressing the invariance under an arbitrary shift of the origin of the a coordinate, where . d ' ~ ( a ) - ( a - 1 ~ $ ( a ) , a ~ k ( a ) , a - 2 c k ( a ) ) (2.13)

[If we try to retain the %-: variable, however, it will turn out not necessary to impose the constraint (2.10) a priori. See Sec. V.]

The BRS operator QB in the first-quantized theory plays the role of a kinetic operator in the string-field theory. It takes the same form as for the open string except for the a-integration region (and an overall factor of 2 which is introduced for convenience):

were defined only over a€ [O,%-] but one could introduce new (single set of) variables d;,, = (a- ' A P , a c , a p 2 C )

w defined over a € [ -r,n-] by .d:,,(a)=~(a).d;(a) + B ( -a),d.!( - 0 ) . Then the BRS charge of the open string can be written in exactly the same form as above except for the factor 2 [see Eq. (2.19) in IV]:

QB=2& J P T r d a { i % - , [ f ( 7 ) " P , ~ , + n p ~ ~ i ~ v t ) I J

Therefore, in comparison with (2.12), we immediately see +i(c'T-r;7rc)] that the present closed-string BRS charge has the form

- I

as was stated above. (2.12) Thus QB+ of the t modes have the same expression in

[Here c in (2.12) is again the original c (a) containing the terms of the oscillators as for the open string:

35 - COVARIANT STRING FIELD THEORY. I1 1321

m

L ~ ~ ~ ( ' ' = 2 ( n +m):c;5;cz1: , m = - m

where L: and LZP are the Virasoro generators of the usual string modes and FP ghost modes, respectively. Since { QB + , QB - ] = 0, the nilpotency of QB ,

Q B ~ = O , (2.17)

follows from ( eBi 12= ( 2 ~ : ~ ~ " ) ~ = 0 which hold only when d =26 and a ( 0 ) = 1 as first found by Kato and ~ ~ a w a . ' ~ . ' ~ It is convenient to make the dependence on the FP ghost zero modes T o and n-: explicit to rewrite QB as

Here eBi stands for the part of QBi which contain neither ck" nor ? b*'.

111. THREE-STRING VERTEX

It is not necessary to introduce a four-string interaction term in the case of a closed string, as was first pointed out in 11. Thus we have only to consider a very simple form of BRS transformation.

or a gauge transformation

the details of which we shall discuss in Sec. V. Here the * product of two arbitrary closed-string fields Q, and \V is defined in the same form as for the open string case [(5.68a) in IV] as

by using a three-string vertex functional V. This may equivalently be rewritten into the bra-ket notation"

if Q, and \V are Hermitian ( E = + 1 ) or anti-Hermitian t ( E = - 1); i.e., Q, [z]=E~Q,[Z] .

In this section we construct the three-string vertex V suitable for the BRS transformation (3.1) and the gauge transformation (3.2) of the closed string. The nilpotency of BRS transformation or the closure of the gauge transformation algebra demands an equation

as is clear from the corresponding equation of open string in IV. Further, the ghost number of Q,*@ must coincide with that of QB@ and hence the vertex V should carry N F p = -2. [Note that Q B , Q,, and the measure d Z r (or d r oc d x r d ~ [ ' d r ~ ' ' ' d a r ) carry NFP = + 1, - 1, and + 2, respectively .]

This vertex V for the closed string can be constructed essentially as a product of two "open-string three- vertices" VoPen for the ? modes, as was already done in 11. Here we explain this procedure in the present notation in somewhat more detail.

Similarly to the open-string case, we look for a solution V of (3.5) proportional to the following overlapping S functional:


Here we are considering the case a l , a 2 > 0 , a3 < 0 (al+az= ( a3 I ). This 6 functional expresses the splitting of string 3 into strings 1 and 2 (or the joining of 1 and 2 into 3 ) at the point u3= fn-a>/ 1 a3 as is depicted in A of Fig. 1 . The extension to all the cases of a , is done straightforwardly and the corresponding string configurations are given in Fig. 1 . Just as in the open-string case, the 6 functional (3 .6) also implies the connection conditions

for the conjugate variables

and hence also for

i.e., for the variables d F = ( a - ] ~ $ , a ~ ~ , a - ~ C + ) introduced in (2.13). Conversely, the 6 functional (3.6) can be equivalently characterized by the connection conditions

The solution to this equation is unique1 given by (3.6) u i 6 ! 3 7 to factors commutative with eldi +82dy'-dk .

Note here that the connection conditions have split into two independent sets of equations for the + and - modes in (3.9).

Now recall that the open-string vertex VnPen had the form

- with a ghost factor G ( a I ) = ~ n - a , i n - ~ ' ( u ; ( r = 1 , 2 , or 3 ) at the splitting/joining point u1 and that VgPen( l , 2 ,3 was the 6 functional which satisfied the same sets of connection conditions as (3.9) for the variables ,d:(u) defined over 0 5 u 5 rr. In terms of the previously introduced variables d?a,,,(u) = O(a)d","(u) + O( -u)dl!( - a ) defined over -T j a j rr, those connection conditions can be rewritten into a single set of equations

with the same el,> and as given in (3.6). Therefore the open-string 6 functionals VgPen( 1 &,2&,3* ) with the f mode variables r i of closed string substituted satisfy the desired connection conditions (3.9) for the k modes, respectively, and hence the present closed-string 6 functional V o ( 1 ,2 ,3) (3.6) must coincide with the product V~Pen(1+,2+,3+)V;Pen(1-,2-,3-) (up to a commutative factor).

Furthermore, since the open-string vertex VOPen (3 .10) was constructed in I and IV so as to satisfv (z:=, (in d = 2 6 ) , the fact Q B I =2Qjpen [ d g e n ( u ) - d y ( k o ) ] noted in Sec. I1 immediately leads to''

Therefore the product Vopen( 1 + , 2 + , 3 + ) Vopen( 1 _ , 2 - , 3 - 1 gives a solution of the closed-string vertex V which satisfies Eq. (3.51, ( z : = , Q[ ' )v =o, since Q B = Q B + +QB- and is proportional to the 6 functional (3.6). Thus, from

1 VOpen) given previously by (3.54) in IV, we find

FIG. 1. The string configurations of the overlapping 6 functional in the three-string vertex V, for the cases A a j = l a l / + a 2 1 , B lal = a 2 + a j , a n d C a ~ = ' a i l + a , .

35 - COVARIANT STRING FIELD THEORY. I1

! 3

/ ~ ( 1 , 2 , 3 ) ) ap2(al,a2,a3) n n i l - T b " " ' w ~ ' ) ( 0 ) 6 ( 1 , 2 , 3 ) , i r=l

where

I

Here the Fourier coefficients of the Neumann function, N rm and 8 ', , are given explicitly in Eq. (3.11) of IV, which are the same as those defined in Refs. 8 and 9 in the light-cone gauge formulation if our string-length parameter a is identi- fied with the light-cone momentum a =2p+.

The right-hand side (RHS) of (3.13), however, does not give the final answer, since it carries ghost number N F P =O contrary to the desired value N F P = 2 . We still need a factor with N F P = - 2 by which to multiply the vertex (3.13). As a preparation for looking for this factor we first note that the expression (3.13) itself has the same form if it is rewritten in terms of the I and I1 modes defined below instead of the k modes used in (3.13):

Since this change from the i modes to the 1,II modes is an orthogonal transformation, Eq. (3.13) clearly can be written as

where wi[il and F I , I I are the same functions as wg' and F* defined by (3.14) with the + modes replaced by the 1,II modes and ~ b " " ' are, from (3.15) and i2.3),

That is, this vertex is still given by the product of two "open-string vertices" VOPe"( ll,21,31)Vopen( 111,211,311) even if written in terms of the 1,II mode variables. It is convenient to recall here the following identity for the open-string vertex VoPen [Eq. (3.52) in IV]:

1324 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA 35 -

where s may be any of 1 , 2 , and 3 (mod 3) . Note that the parts enclosed by the square brackets in the RHS's of (3.18) are the 6 functional VgPen aside from the a , p conservation factor (3 .14d) .

Now, to make the vertex V carry the correct ghost number XFp = - 2 , we multiply the vertex (3.16) [=(3.13)] by the following factor with .VFp = - 2 (Ref. 2 ) :

This is because this choice of the multiplication factor yields a very simple expression for the I1 mode part of the vertex as is seen from the identity (3.183:

where again the part enclosed by the square bracket may be regarded as the 6 functional VnP"'( 111, 211, 3 of the I1 modes since the missing factor ( 1 + i 2 4' 3"!1d1r'r 1 1 ' ) is

1 by the presence of the front factor T ' " ' = 6 ( r 1 " ' ) . [A more important reason for the choice of the multiplication factor (3.20) will become clear in the next section.]

As a matter of fact, however, the multiplication of the factor (3.20) violates the desired property 2, Q ~ ' V = O which (3.16) already satisfied. Fortunately this is easily remedied by multiplying the vertex further by the projection operators Y r ' of strings r = 1-3 defined in (2.1 1):

0 Indeed, since {QB,TOI1 / a c [ Q B , ~ 7 i c ] = - ( L + - L - ) and 9 ( L + - L - ,=0 by (2.18) and (2 .11) , the factor (3 .20) an- ticommutes with 2,~:' In the presence of the projection operator 9 123, and further slnce [ Y , Q B ] =0 the property 2, Q ~ ' V = O is maintained. [This expla~ns the reason why we have used T;" rather than T h l ' to construct the factor (3.20) with ,VFp= -2.1 Thus from i3 .16) , (3 .21) , and (3.17), we finally obtain the desired closed-string ver- t e ~ . ~

Here the oscillator-independent normalization factor ( a l a 2 / a 3 ) p 2 ( a l , a 2 , q ) is chosen in such a way that the next 0 ( ~ ' 3 requirement of BRS nilpotency or the gauge invariance will be satisfied. This expression (3 .23) is now valid for all regions of ar ' s .

Another convenient expression of the vertex (3.23) for the purpose of the next section is obtained by applying the identity (3.18) to the I mode part of (3.23):

X G ( U ~ ) ~ , 1 V 0 ( 1 , 2 , 3 ) ) . (3 .24)

Here 1 Vo( 1,2,3 1 ) stands for the 6 functional (3 .6):

x a [ ~ a r - l T ~ ( r j s 1 ,

and G (ul is the following ghost factor at the interaction point

G ( u 1 3 = y%ar+i[7ir1(u?')+~r'( -uy ) ]

= d % a r i 7 i r 1 ( u y ' ) (3 .26)

with r being any of 1 , 2, and 3. (For instance, the interaction points a?' are rr,0,7ia2/ a3 for r = 1,2,3, respectively, in the case a l , a 2 > 0 , a3 <O of A in Fig. 1 . ) In deriving expression (3.261, we have used the fact that the open-string 7iF(ul) factor appearing in the formula (3.18) should read, for the present I mode of closed string,


by (2.3) in IV and (2.2c), (2.3), and (3.15). Finally, in this section we note some symmetry properties of our vertex. First is the following cyclic symmetry:

7 ~ : ' ~ ) / ~ ( 1 , 2 , 3 ) ) =v;(" I ~ ( 2 , 3 , 1 ) ) = 7 ~ : ( ~ ) 1 V(3,1,2)) . (3.28)

This is clear from the expression (3.23) since it takes the same form as that of open string other than the factor

( a l a 2 / a 3 ) n c 6 x a,-'.rr:"' i which possesses the property

Next is an antisymmetry property under the exchange of 1 and 2:

T o prove this we recall the following equation for the overlapping. 6 functional VgPen( 1,2,3) ) of the open string: l9

where R was the twist operator by which the open-string oscillators a, =(a:,c,,q ) were transformed as Ran R- = ( - )"a,, . If we note that the translation opera-

rff(L+ -L-) tor e of the a coordinate by an amount 7~ plays the same role in this closed-string case as R did in the open-string case, we can translate (3.30) into an equation for the present 6 functional Vo ) of the closed string:

Here we have used 9 ( L + - L - ) = O in the last equality. From the expression (3.24) and the fact that the ghost factor IIc is odd under the exchange of 1 and 2, we see that (3.31) gives the above equation (3.29). Third is the following Hermiticity property:

or equivalently, in the functional notation,

Equation (3.32a) follows immediately from the expression (3.23) and

The final property we note here is

v [ Z i , Z 2 , ~ , 1 = V [ Z I , Z ~ , Z ~ I . (3.34)

This is easily understood if we consider a "twist operator" R of closed string defined as an exchange of I modes, I , - I = ( ( T I c ( T ) - ( T I a, , , , c , , n#O (3.35)

under which the vertex (3.23) remains invariant while the variable Z is transformed into 2.

IV. 0 ( g 2 ) PROPERTIES OF THE VERTEX

A. Jacobi identity

In the previous section we have shown that the vertex (3.23) or (3.24) satisfies ( x i = , Q:')v=o, (3.51, which is

equivalently expressed as a distribution law of the BRS operator QE on the * product defined in (3.3) as4

with / @ I being 1 (0) when @ is Grassmann odd (even). This is exactly the same property as in the previous open- string case. [Note that (4.1) holds only when d = 26 and a ( 0 ) = 1 as in the latter case.]

A property characteristic to the present closed-string case is that the * product is essentially symmetric:

This property of course did not hold for open string in which case Q, and Y are matrix-valued first of all and in addition the overlapping 6 functional VgPen ( 1,2,3 ) ) be- came the twisted one when 1 and 2 are exchanged as not-


ed in (3.30). In the present closed-string case, however, the string fields carry no matrix indices and further the vertex ) Vi 1,2,3) ) possesses the symmetry (3.29) thanks to the presence of the projection operator .P 123. Equation (4.2) follows from (3.29) and the definition (3.4) of the * product. The Hermiticity property (3.32) of the vertex can also be rewritten into the *-product notation. For string fields satisfying (anti-)Hermiticity Q T [ z ] =E@@[Z], Y ~ [ z ] = E ~ Y [ Z ] , we have from (3.32b) and (3.3)

i @ * Y ) + [ z ] = € @ c p i ~ * @ ) [ 2 ] . (4.3)

Note that the same equation holds also in the open-string case.

Now let us turn to the main subject of this section. The nilpotency of the BRS transformation (3.1) requires the

equation

to hold. The identities QB2=0 and (4.1) guarantee this at 0ig0) and O i g i ) , respectively (at d =261. The 0ig2) term of (4.4) demands an equation

by using the symmetry property (4.2). As was first recog- nized clearly in 111, Eq. (4.5) is actually satisfied by the following rather interesting identity, which we call the Jacobi identity: For arbitrary three closed-string fields

( r = 1,2,3),

or equivalently, by (4.21,

@ " ) * ( @ ' 2 J * @ l 3 ' ) + ( - ) I ' 2 +I3 1 @ ( 2 1 * ( ~ ( 3 1 * @ ( 1 ! ) + ( - ) 3 ~ 1 + 2 ) @ ( 3 i * ( @ ( 1 ' * @ ( 2 ) ) = 0

where / r denotes @"' ; . A brief proof of this identity was given already in I1 by taking the example of special diagrams of strings' configuration. To give the full details of this proof is the purpose of this section.

In terms of the bra-ket notation to perform unambiguous calculations. Eqs. (3.31, First let us express (@'I ' * d2') * @"' '

(3.341, and (4.3) lead to

and hence equivalently,

1 ((@"'*@'2')*@"')(4)) = E ~ E ~ E ~ J d 5 d 3 ( ( @ ' > *@I1 ) ( 5 ) (@"'(3) I ~ ( 5 , 3 , 4 ) )

Here 6,'s denote the Hermiticity sign factors, @""[z] = E , @ ' ~ ' [ Z ] ( r = 1,2,3) . We can assume such definite Hermiticity properties for @""s without losing any generality in proving (4.6) since any string field can be decomposed into the Her- mitian and anti-Hermitian parts. [Note that the order of @"' and @I2' is reversed in the RHS of i4.9).] We need the bra-state representation of @"'a@"', which is obtained by taking the Hermitian conjugate of (3.4) and by using (3.291, (2.7), (2.91, and (3.24) as follows:

w ~ t h 9 .9c1' .Y'2 . Subst~tutlng this into (4.9) and writing

a"= 1" -77 ~ d B / 2 r ) e x ~ [ i B ( ~ ~ ' ~ ~ ' ) ] ,

we obtain

with an effective four-string vertex

35 COVARIANT STRING FIELD THEORY. I1

Here we have displaced 8 into 8 + n in order to make the expression at 0 = 0 have better correspondence to the open- string one.

As is clear from the b-functional meaning of ( Vo( l ' , 2', 5 ) 1 and 1 VO( 5,3,4) ) and from the similar expression of the open string, this effective vertex I A ) is simply proportional to the four-string overlapping the 6 functional

/ vk4'( 1,2;3,4;8)) having the same form as the three-string one (3.25):

By noting that the operator e x p [ i 8 ( L ~ ' -L ? ' ) I in (4.12) twists the intermediate string by an angle 8, we can draw the figures of the four-string configurations represented by the 6 functional / v:'( 1,2;3,4;0) ). They are given explicitly in Fig. 2 for the case a l , a 2 , a 3 ) 0 , a 4 < 0 . As was explained in detail in 1V for the open-string case, N L y ' s appearing in (4.13) are Fourier coefficients of the Neumann function N ( p , @ defined over the closed-string diagram ( p plane) corresponding to such four-string configuration (see Appendix A).

By the same procedure as was used in IV, the proportionality constant of I A ) to 1 vb4' ) is found by calculating the coefficient of the vacuum term o)%( 1,2,3,4) in (4.12). Thus we easily obtain

The two factors of l / a 5 come from the &b5' and d r y 5 ' integrations, and the contractions of the + and - oscillator modes of the intermediate string 5 yielded the determinant [det( 1 -iT 66k 55)]-'d-2"2 and its complex conjugate, respectively. (The bosonic oscillators contributed to the exponent d and the ghost oscillators c ~ " ' " , ~ ~ ' " ~ ' to -2.) Substituting (4.14) into (4.1 1) and using

FIG. 2. The four-string configurations represented by the overlapping 6 functional v ~ ; " ( o ) ) appearing in the first term of (4.19) in case (i) a,,az,a3 > 0. The dots stand for the origins of strings 1-3 and the values of the a coordinate of string 4 are indicated at some characteristic points Di =a, / a4 , .

1328 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA

i 1 3 =a5 2 E ~ s f a r 2 r r ~ ( s ) r r ~ ~ ) 123) ,

r , s , t = l

we finally obtain the desired expression:

I 1 ( ( @ ( 1 ' * @ ! 2 ' ) * @ ( 3 1 ) ( 4 ) ) = ( ~ ~ ~ ~ 6 ~ / f f ~ ) d 1 d 2 d 3 ( @ ' l 1 ( l ) 1 ( ~ ' ~ ' ( 2 ) 1 ( @ i 3 ) ( 3 ) 1

where D is the following positive numerical factor defined by

We apply this formula (4.17) also to the second and third terms of (4.6), and then eliminate the front sign factors there such as ( - ) " + ) by reordering the "external" strings (@'" / into the natural order (@"I ; ( Q ' ~ ' / (@I3' / ; for instance, for the second term,

( - ) l l l i 1 1 + 1 3 1 ( a ( 2 1 ( ~ ) l ( a i 3 ] ( 2 ) l (@!11(3)1 = ( @ ( 1 ) ( 3 ) 1 ( ~ ( ~ ~ ( 1 1 1 ( @ ( 3 ) ( 2 ) 1 .

Renaming the integration variables 1,2,3 cyclically, we can write the three terms on the LHS of (4.6) as

/ (LHS of (4 .6 ) ) (4 ) )

= ( E 1 E 2 E 3 / a 4 ) J d l d 2 d 3 ( @ ( 1 1 ( 1 ) ( ( ~ ( ~ ' ( 2 ) ( ~ ( ~ ' ( 3 ) / w

Here we have used the fact that defined in (4.16) is cyclically symmetric and hence common to the three terms in (4.19). This property is the most important reason why we have chosen the multiplication factor (3.20). We suspect that the choice (3.20) may be the unique factor possessing such a property. For a clear distribution we have denoted the names of the intermediate strings and their twisting angles by (5,0p ), (7,0Q 1, (9,0R for the first to third terms, respectively.

B. Proof of Jacobi identity

We now start to prove that the three terms in (4.19) cancel among themselves possessing the same set of values

of the zero-mode variables ( p r , ~ ~ ' , r r ~ ' r ' , a r ) ( r = 1,2,3); that is, the cancellation occurs in the integrand of d 1 d 2 d 3 integration which is enclosed by the large parentheses. We consider an arbitrary fixed set of the zero-mode values, in particular, a fixed set of (al,a2,a3), henceforth. Without losing generality we can assume a4= - (al +a2 +a3) < 0, since the Neumann functions and the overlapping 6 functionals vb4' ) depend only on the ratios of a,'s and hence the cancellations among three terms of (4.19) occur in the same way even if we change all the signs of al-a4 simultaneously. There are seven cases of the sign combinations of al , a2, and a3 (for a4 < 0):

( i ) al,a2,a3 > 0 ,

( i i) a l , a z > 0 and a3 (0 , plus two cases obtained by cyclic permutation ,

(iii) a l > 0 and a2,a3 < 0, plus two cases obtained by cyclic permutation .

COVARIANT STRING FIELD THEORY. I1

FIG. 3. The overlapping structures of four strings 1-4 for their half portions with O < u < T, implied by the 6 functionals P, ~6~ ' (1 ,2 ;3 ,4 ;0~) , Q, ~ :~ ' (1 ,2 ;3 ,4 ;6~) , and R , ~ :~ ' (1 ,2 ;3 ,4 ;8~ ) in (4.19). These configurations are for the case ( i) a,,a2,a3>0 and at the twisting angle HP,Q,R = O . The configurations at O P , ~ , ~ + O are obtained by twisting the intermediate strings (indicated by solid-dotted double lines) by angles 8 p , ~ , ~ .

It is, however, not necessary to consider the case obtained by cyclic permutations of 1,2,3 since the quantity enclosed by large parentheses in (4.19) is cyclically symmetric. Thus we need consider the three cases explicitly written in (i)-(iii) of (4.20).

The four-string configurations represented by the overlapping 6 functionals 1 vi4' ) appearing in (4.19) can be drawn for all the cases and Fig. 2 is an example for the first term of (4.19) in the case (i) al,a2,a3 > 0. For ease, however, it is more convenient to draw figures as shown in Figs. 3-5. The figures there represent only the overlapping structures of four strings for their half-portions with 0 j a .rr and only at the twisting angles eP,Q,R =O. But, from those figures one can immediately recover the other half-portions and can easily imagine the configurations at general angles OP,Q,R by twisting the intermediate

strings. The diagrams P, Q, and R in Figs. 3-5 correspond to the first, second, and third terms of (4.19), which we call P, Q, and R terms henceforth. Figures 3, 4, and 5 are for the cases (i) al,a2,a3 > 0, (ii) a1,a2 > 0, a3 < 0 , (iii) a , > 0, a2,a3 < 0, respectively. We consider these three cases separately.

Case ( i ) : al , a2,a3 > 0

Let us start with case (i) al,a2,a3>0, which was discussed explicitly in 11. From the diagrams in Fig. 3 corresponding to this case (i) we immediately notice, for instance, that the four-string overlapping structure of the diagram P at O p =0 for the other undrawn half-portions ( -.rr j a j 0) is identical with that of diagram Q if the intermediate string 7 is rotated by an angle OQ =.rr. Since the overlapping 6 functional is unique if it satisfies the same set of connection conditions and a normalization condition as in (4.13) [which contains the vacuum term I 0)%( 1,2,3,4) with weight 11, it thus follows that

This type of equality turns out in fact to be a special case of more general equalities which exist for arbitrary twisting angles O P , P , R . They can be found by examining the four-string overlapping structure by drawing the diagrams of the form of Fig. 2, or by examining the connection conditions for the coordinates . d ~ = ( a ' ~ ~ $ , a ~ ~ , a - ~ C ~ ) by oing back to the original defining equation (4.12) of

7 4 , 1 V o ) from 2 three-string vertices Vo ) . Let us explain the latter method in some detail: From Eq. (4.12), we have

FIG. 4. The four-string configurations for case (ii) a l ,az > 0, P Q I?

a, <O. In this case, the Q and R configurations each separate into two distinct structures depending on the relations of the FIG. 5. The four-string configurations for case (iii) a , , >0, string lengths as indicated. a2,ai < 0.


X ~ 0 ( 5 , 3 , 4 ) ) 1 E ( l 1 , l ) ) I E(2 ' ,2 ) ) , it finally becomes

d:'(n-+aS/ 1 a 4 1 [ [ ~ p - ( u ~ / u ~ ) a ~ ] ] ) On this expression, we operate .d$'(ol ), for instance. Then it becomes d l : " ( - u l ) on 1 E ( 11,1)) and changes in front of 1 Vo(5,3,4)), with the understanding that into d ' l ' ( n - + u l u l / a 5 ) in front of ( Vo( l t ,2 ' ,5) 1 . By us- [ [ a ] ] =u(mod2n-1 denotes the value in [ -n-,a]. In this ing way we obtain (for -n- 5 u ~ , ~ , ~ <n-)

These are the connection conditions for the P term. Those for the Q and R terms are obtained from this result by re- placing (1,2,3;as=al + u 2 , e p ) with (2,3, 1 ; a 7 = a 2 + a 3 , Q ) and (3, 1 ,2 ;a9=a3+a l ,BR ), respectively. By comparing the connection conditions of the P and Q terms, for instance, we easily find the equality

for V, in the region ( q, 1 < a2r, since both sides can be seen from (4.22) to satisfy the connection conditions

(In order to avoid complications, we have written here the connection conditions which are valid in fact only for I O , ( L I ; ~ - L " ~ )

0 < a,u2 i q, +a2n- and 0 < u3 2 n-.) In the present context, the cumbersome a-translation operators e appearing in (4.23) can be dropped owing to the presence of the projection operator ?1234= 9(1)9'2)9i3i9'4' in (4.19).

In this way we find the following equalities between the P, Q, and R 6 functionals in (4.19):


Note that these equalities already cover all the possible values of 6p,Q,R. Therefore the three terms in (4.19) split into three pairs, the two terms in each of which contains an equal 6-functional part 9 1'34 vL4' ). We call these three pairs PQ, QR, and R P terms, respectively. Further- more the two ghost factors G (U:'~)G (u:~", etc., in (4.19) are also common but with opposite order between the two terms in each pair. Indeed, since the four-string configurations are common between the two terms in each pair, the ghost coordinates G ( u I ) at the corresponding interaction points between them also coincide of course. So comparing the interaction points in the P, Q, R configurations in Fig. 3 (or in Fig. 7 below) we clearly have the following identifications of ghost factors in each pair:

showing actually that the ghost factors are contained in opposite order. Thus we have that the two terms in each pair have the same ghost factors and the same 6 functionals but they have relatively opposite signs. Therefore, in order to prove the desired cancellation of (4.191, we have only to show the equalities of the coefficients D times the 6-integration measure d e p , ~ , ~ , since sgn(a5 ) = sgn(a,) =sgn(cr9)= + 1 in case (i); for instance,

in the PQ term. These equalities in fact result from the duality realizing

mechanism just as in the open-string case discussed in IV. In order to see this let us consider the string diagrams drawn in Fig. 6 which correspond to the overlapping 6 functionals P,Q,R in Fig. 3 at 6P,P,R = O if the interval T of the two interaction times is taken to be zero. These string diagrams in fact represent the half-portions (corresponding to the 0 5 a j 7r part) of the surfaces swept by the closed strings 1-4 depicted in Fig. 7. The general closed-string diagrams corresponding to the configurations with arbitrary twisting angles 8p,Q,R are obtained by rotating the intermediate strings by angles 8P,Q,R as indicated in Fig. 7 and therefore they are characterized by two parameters: the time interval T and the twisting angle 8.

FIG. 6 . The string diagams which correspond to the configurations P,Q,R with O P , P , ~ = O in Fig. 3 when the time interval T goes to zero.

FIG. 7. The closed-string diagrams corresponding to the configurations P , Q, R in Fig. 3.

In the same way as in the open-string case the surfaces, called p planes, of closed-string diagrams are parametrized by the coordinate p = r+ i a =a,(&, + ia , ) + r;' + i/3, as explained in Appendix A.

The p plane of any closed-string diagram is mapped into the (entire) complex z plane by the well-known Man- delstam mapping:

Here we have already fixed the arbitrariness of choosing the Koba-Nielsen variables Z 1-4 owing to the projective invariance by setting

Once this "gauge" of projective invariance is fixed, any closed-string diagrams, which are characterized by two parameters ($,TI, correspond one to one to a complex variable Z3 through the Mandelstam mapping (4.27); that is, the correspondence (6 , T ) + + Z 3 is one to one.

To understand this correspondence more clearly, it is instructive to consider first the case where Z j is restricted to be real. Then the mapping (4.27) is identical with that in the open-string case, for which we know already the correspondence as shown in Fig. 8 (Ref. 6). The XPQ, XQR, and xRp are certain values of Z3=x(real) determined by and stand for the points at which the time interval T of the corresponding string diagram becomes zero. The string diagrams drawn in Fig. 8 may be regarded as the open-string ones or as the half-portions of closed-string ones. [Indeed, at 6P,Q,R = O and 7r (mod27r), the half-portions of 0 j u, j 7r of closed strings are mapped into the upper half z plane and the other halves of - 7 r j a j 0 into the lower half-plane.] The attached symbols P ( O p = a ) , etc., denote the corresponding configurations of closed strings with indicated twisting angles. Now generalizing the analysis of the mapping (4.27) to complex values of Z 3 , we find the correspondence schematically drawn in Fig. 9. The complex Z j plane spilts into three regions P, Q, and R which correspond to the closed-string diagrams of P, Q, and R configurations in Fig. 7, respectively. As the twisting angle Op of the closed-string diagram P of Fig. 7, for instance, is varied from 0 to 27r with the time interval T = To > 0 kept fixed, then the corresponding point Z3 generally moves along a closed path as indicated by the dotted line in Fig. 9. The solid lines along the boundaries of the three regions are such closed paths corresponding to the T = O closed-string diagrams P, Q, and R in Fig. 7. We have indicated the values of twisting angles 8 P , P , ~ at some characteristic

HATA, ITOH, KUGO, KUNITOMO, AND OGAWA

FIG. 8. The string diagrams corresponding to the Koba-Nielsen variable Z3 = X (real) in various intervals, for case ( i ) al,az,a3 > 0.

points. The triple points, denoted by X and Y in Fig. 9, correspond to the particular string configurations in which the two interaction points coincide. It should be noted here that the three boundary segments from X to Y separating the regions P and Q, Q and R , and R and P just correspond to the above encountered three regions of twisting angles, PQ, QR, and RP, for which the equalities (4.24) of overlapping 6 functionals were found to hold. This coincidence is, of course, not accidental since the point on the boundary, e.g., of P and Q domains, corre-

sponds to a single four-string configuration which can be realized as the T+O limits of both P-type and Q-type string diagrams and hence the corresponding P-type and Q-type 6 functionals necessarily coincide.

From Fig. 9 we now understand that the Z 3 variable corresponding to each type of closed-string diagrams P,Q,R in Fig. 7 spans only a part of the complex plane even if T and 0 are varied over [0 , cc ] and [0,2rr], respectively, but the sum of the three just covers the entire corn- plex Z , plane. On the other hand, we know that the corn-

FIG. 9. Three regions P, Q, and R of the Koba-Nielsen variable Z3, which correspond to the closed-strlng diagrams P, Q, and R in Fig. 7 [case (i) al,a2,a3 > 01. As the twisting angles OP.Q,R of the T =O diagrams P, Q, and R in Fig. 7 vary from 0 to 27r, the corresponding point Z j turns around once along the boundary of each region.


plete dual amplitude of four- (closed-) string is given by integrating the Z 3 variable over the whole complex plane. Therefore the contribution of each of the diagrams P, Q, R in Fig. 7 must yield the part of the Z3 integration restricted in each domain of the full dual amplitude. But the fact that the dual amplitude is a d Z 3 d Z f integral of a smooth function of Z3 means, in particular, that the two distinct amplitudes of the P-type and Q-type diagrams, for instance, have to coincide with each other at the PQ boundary when expressed in terms of the Z 3 variable. This must occur in order for the theory to reproduce the dual amplitudes and is indeed the case in the light-cone

gauge string field theory and also in our theory. This fact guarantees the desired equalities (4.26) of the coefficients D.

Indeed D's are just the coefficients that appear in the amplitudes of the closed-string diagrams P,Q,R in Fig. 7. Although there seems to exist no literature that gave direct calculations of four-point amplitude for the closed- string case in the light-cone gauge string field theory as was done by Cremmer and ~ e r v a i s ~ for the open-string case, it is straightforward to extend the Cremmer-Gervais calculation to the closed-string case. Then the amplitude of the diagram P in Fig. 7 is found to be given by

where a, and N 66 are the same as in (4.15) and

The (ext( 1-41 I denotes the external string states and the effective four-string vertex I VY'( 1,2;3,4;Qp)) is given by the same equation (4.13) as the previous four-string 6 functional i vL4'( 1,2;3,4;Op)) if the Neumann function 3 F t there is replaced by that of the string diagram P in Fig. 7. As in the open-string case, the same formula (4.29) is in fact valid both in the light-cone gauge string field theory and in the present covariant formulation (see Sec. VI). [Of course, the longitudinal and scalar modes a~:' ( n > 1) as well as the ghost modes y _, and 7-, in v i4 ' ) are set to zero in the light-cone gauge case.] The amplitudes for the diagrams Q and R in Fig. 7 are given by the formula (4.29) if the indices (1,2,3;5,6;P) are replaced by (2,3,1;7,8;Q) and (3,1,2;9,10;R 1, respectively.

A crucial equation is the following identity which is derived in Appendix C by generalizing the Cremmer-Gervais direct proof of the corresponding one in the open-string case:9

- +m ( T + i a 5 Q p ) ( N Fop ),, = exp

Here again R&"' is the Neumann function LF (with r =s =j , n = m = 0 ) for the string diagram P in Fig. 7. The same form of identities hold also for the diagrams Q and R, of course by performing the above replacements. Note here that the LHS of (4.31) is exactly the quantity

/ dBp 1 D ( a l , a2 ;a3 ,a4 ;Qp) at issue introduced in (4.18), if the space-time dimension d is 26 and the a-integration measure p is chosen as in (3 .14~) :

( - )"z/;;N:;(a5,a3,a6)fi( - I rn .

3

p (a l , a2 , a3 )= exp r o ( a l , a 2 , a 3 ) / a , . (4.32) I Therefore the generalized Cremmer-Gervais identity (4.31) implies the desired equalities (4.26) between the coefficients D. Indeed, at any points Z3 on the PQ boun-

dary ( T = O ) in Fig. 9, the string diagrams P and Q of Fig. 7 with Op=p/a5 and 8Q=??-p/a7 become the same and the corresponding Neumann functions R Ly coincide. Hence both dBp I D ( a I , a2 ;a3 ,a4 ;QP =p/a5) and doQ / D(a2,a3;al ,a4;OQ=r-p/a7) equal the common quantity given by the RHS of (4.31) and the desired equation (4.26) holds. Clearly the same equalities hold on the QR and R P boundaries and thus we have proved the complete cancellation of the three terms in (4.19) for case (i) a l , a 2 , a 3 >O.

[Incidentally the Cremmer-Gervais identity (4.3 1) says the equality not only at the boundary but at any points inside the domains. Thus, it says that the contribution (4.29) of the string diagram P to the four-point amplitude becomes (at d =26)

HATA, ITOH, KUGO, KUNITOMO, AND OGAWA

lT J Z ~ E T ~ ~ I O I ~ ( P J d v d v L abc o*bc -'.= 1 1 =

i.e., the dual amplitude with Z3 integration restricted in the region P in Fig. 9, and hence that the full dual amplitude is reproduced by adding the contributions of the diagrams Q and R of Fig. 7.1

Cases (ii) al,a2 > 0, a3 < 0 and (iii) al > 0, a2,a3 < 0

We still need to consider cases (ii) a l , a , > 0, a3 < 0 and (iii) a , > 0, a 2 , a 3 < 0. In case (iii), however, quite similar arguments to the above apply as is inferred from the comparison of Figs. 3 and 5. That is, the amplitude of the decay process 4- 1 + 2 + 3 was relevant in case (i) and similarly it is the decay process 1 -2+ 3 + 4 in case ( i i i ) . I t is, therefore, not necessary to repeat the proof for case (iii). The only different point in the proof is the following: The signs of a 5 , a 7 , and a9 become different in case iiii); a ,=aI +a2 1 0 , a7=a2+a3 < 0, and ag=a3 +a l > 0. However, this change of sign factors ~ g n ( a ~ , , , ~ ) in (4.19) turns out to be compensated by the change of the identification of the ghost factors G ( u I ) . Instead of the previous identification (4.251, we now have the following one as is easily understood from Fig. 5:

Thus, in this case also, the previously observed equalities and opposite signs of the ghost factors between the two terms in each pair PQ,QR,RP are realized as the products s g n ( a )G i u I )G (a; including the sign factors.

Now let us turn to the remaining case (id. In this case different overlapping structure of strings appear in Fig. 4 in the Q and R configurations depending on a2 1 5 a? 1 and 1 a l 1 >< a3 , . However, since the following arguments apply essentially to any cases, we assume

/ a , / , a2 > 1 a3 for definiteness, in particular, in drawing figures. [The case-dependent part appears only in the ghost and sign factors as in (4.34) and we will discuss it in the final place.]

The correspondence between the string diagrams and the Koba-Nielsen variable Z3 through the Mandelstam mapping (4.27) in case (ii) is given in Fig. 10 for real Z 3 , which is essentially the same as Fig. 22 in IV. For general complex Z 3 , we have the correspondence schematically drawn in Fig. 11. (Digression: A n interesting point is

that this Fig. 11 tells us why the quartic string interaction term need not be introduced in the closed-string field action.' In the open-string case, actually, the existence of the region [ x _ , x + ] in Fig. 10 necessitated the quartic interaction term. Indeed, without the quartic interaction term, the interval [s- , x + 1 could not be covered in the Z3 (real) integration and the duality would be violated. In the closed-string case, however, the interval [ x _ , x + ] at issue turns out to be simply a part of the boundary of the region P in Fig. 11. Notice that any point inside the region P corresponds to the string diagram which is constructed by using only the ordinary cubic interaction terms (twice) and by twisting the intermediate string. Therefore this implies that the four-string configuration corresponding to the open-string quartic interaction term can be realized by using only the cubic interaction terms and twisting as a limiting configuration of T-0.) From Fig. 11 and the Cremmer-Gervais identity (4.3 11, we find the following equalities in this case:

( P Q ) V q = a 5 0 p = a 7 i ~ - ~ ) < ~ 1 7 7 i ,


FIG. 10. The string diagrams corresponding to the Koba-Nielsen variable Z 3 =x (real) in various intervals, for case (ii) al,az, > 0, a, < 0.

In this case there is no QR boundary. Instead, the real line interval [ x - ,x+ ] gives a PP boundary as is seen in Fig. 11,

FIG. 11. Three regions P, Q, and R of the Koba-Nielsen variable Z , , which correspond to the closed-string diagrams P, Q, and R in Fig. 4 [case (ii) al,az > 0, a3 < 01.


on which we have

The last equality D (8,) = D ( - Qp ) is a result of the Cremmer-Gervais identity on this PP boundary, but generally holds since the absolute value of det(1-N h61qii) is an even function of Op. When 1 a l 1 , ( a 2 I > i a , 1 is not assumed, a7=a2+a3 and a9=a3+al may become negative. For such cases, the a7 and a9 appearing in (4.35) should be replaced by j a, ( and a9 1 .

Since we have seen the equalities of the overlapping 6 functionals 9 ~ ~ ~ 4 v:' ) and the coefficients D , we have only to examine the ghost factors multiplied by ~ g n ( a ~ , ~ , ~ ) to show the cancellation of (4.19). First let us see the cancellation between the P and Q terms on the PQ boundary. By examining the diagrams P and Q in Fig. 4, we find the following identifications of ghost factors depending on the two cases a2 ( 5 I a, 1 :

Thus the expected relation

holds in either case and the cancellation occurs on the PQ boundary. Similarly, for the R and P terms on the RP boundary, we have

and they also cancel. Finally, for the P terms with Op and -Op on the PP boundary, we have the identification

This interchange of two interaction points u j25 and u:'" between the two configurations with Op and -ep can be understood if one actually constructs those two configurations by twisting the intermediate string by angles Op and -8, [ / a7 ; .rr 5 I a50p 1 5 (aS- I a9 ' )TI starting from the O p = O configuration. This is done in Fi 12. Therefore, in this case also, the ghost factors G(o$')G(oi") of the two P terms with Op and - Bp coincide with opposite signs, and the cancellation occurs. We thus have completed the proof of the vanishingness of (4.191, i.e., the Jacobi identity (4.6).

V. ACTIONS

The (gauge-fixed) BRS-invariant action and the gauge- invariant action for the closed-string system were already given in I1 and 111. Let us explain them briefly here.

Note that we have put a "metric" .rr: in this de f in i t i~n .~ ' If Q is Hermitian ( E @ = + 1) or anti-Hermitian ( E @ = - 11, @ t [ ~ ] = ~ @ Q [ 2 ] , then it can be rewritten into a more familiar form of an inner product:

Since the metric n-: has a commutator 1 ~ ~ , . r r : ] = i ( L + -L - ) with the BRS operator QB, this bilinear form admits a partial integration formula for Q B ,

A. Inner product if Q or \V satisfies the constraint Y Q = Q . First of all we need to define a bilinear form for arbi- It is convenient further to introduce a trilinear form de-

trary two string fields Q and Y: fined by


FIG. 12. The changes of the four-string configurations when Bp is varied, starting from the Bp=O configuration of the diagram P in Fig. 4. The bond shown by the solid area stands for the interaction point corresponding to the ghost factor ~ ( ( r : ~ ~ ) and the bond connecting the solid dots for that corresponding to G (0:").

If we recall the cyclic symmetry property (3.28) of T~" 'V[Z , ,Z~ ,Z , ] , it is seen to satisfy

' I ' + I A '['PA@]

= ( - ) " ' @ ~ + l y l ' [ ~ @ ~ ] , (5.5)

When all of the @, \V, and A are Hermitian or anti- Hermitian, the definition (5.4) can be equivalently rewritten as

For convenience we collect here the properties of the * product which was proved in the previous sections: distribution law,

Jacobi identity,

@ * ( \ V * A ) + ( - ) ~ Q ~ ~ ~ ' ~ + A 'T* (A*@)

+ ( - ) l A 1 " Q + I ' 'A* (@*\v )=o , (5.7b)

commutativity,

B. Gauge-invariant action

The gauge-invariant action S of the closed string is constructed in such a way that 6B@=0 is proportional to the equation of motion, just as in the open-string case:

where 6B@ is the BRS transformation given in (3.1). Thus the action is given by

s = @.QB@ + Sg@3 , (5.9)

with notations introduced above:

@ . Q ~ @ = J ~ I T : ( @ ( I ) ~ Q ~ l @ ( l ) ) , (5.10a)

@3 Ez [@@@I = J d l d 2 d 3 7 ~ ~ ' ~ ' ( @ ( 1 ) (@(2) ( W 3 ) /

Actually this action (5.9) is invariant under the following (infinitesimal) local gauge transformation:


6 @ = Q B l i + 2 g @ * i i , (5.1 1) carries F P ghost number N F P = - 1 , the parameter A must be Grassmann even, anti-Hermitian2' (A'[z] -

where A is a transformation parameter which itself is a = - A [ Z ] ) and carry N F P = -2. The invariance of (5.9) string functional, satisfying the constraint Y A = A . Since under (5.11) can easily be shown by using (5.11, (5.3), (5.4), our closed-string field @ is Grassman odd, Hermitian, and (5.7~1, and, in particular, the cyclic symmetry (5.5):

which indeed vanishes by the nilpotency of our BRS transformation, (6, j2=0, shown in (4.4). In this proof we have needed the constraint P A =A on the parameter A for the partial integration formula (5.3) to be applicable. But the constraint P@=@ on the field @ was not necessary. We will discuss this point later. It cannot be emphasized too much that this gauge invariance as well as the BRS nilpotency [or the properties (5.7a) and (5.7b)I exists only in d =26.

The group structure of gauge transformations is very simple in this closed-string case owing to the absence of the quartic interaction term. The commutator of two infinitesimal gauge transformations with parameters AI and A2 is calculated as

where we have used the commutativity (5.7~1, distribution law (5.7a), and Jacobi identity i5.7b). The last quantity is just a gauge transformation with parameter 2gA I * Az, and thus we obtain a very simple algebra

which closes off shell. Hence the structure constant is simply ( 2 g times) our three-string vertex functional V[Z1,Z2>Z31.

The above gauge invariance and closure of gauge algebra hold without imposing any constraint on the internal ghost number of Q. [The internal ghost number n ~ p is the ghost number n ~ p which is carried only by the coordinate c ( n ~ p = + 1) and ? ( n ~ p = - 1 ), and should be dis- tinguished from the usual (net) ghost number N F P which can be carried also by the coefficient fields in the "local" field expansion of @ (see IV for a more precise definition).] However, since it is natural that the gauge- invariant action consists only of component fields with zero ghost number when @ is expanded into an infinite

number of "local" component fields, we restrict @ to its internal ghost number n ~ p = - 1 sector a _ , . Then the d field in

which contains true physical modes, is restricted to the n ~ p =O sector do. The restriction on @ is consistent if the gauge transformation parameter A is also restricted to its n ~ p = - 2 sector AP2. We understand that @ and A are these restricted ones and A P 2 in the above action (5.9) and gauge transformation (5.11). Even then, the gauge invariance and the algebra (5.14) of gauge transformation remain valid as they stand.

Finally in this subsection we should note an important fact, that the antighost zero-mode variable rrf can actually be dispelled from the present formulation. In order to see the role of a f , let us make explicit only the rrf depen- dences of the field @, the transformation parameter A, the BRS operator QB, and the vertex V:

0 @ = @ ' + T ~ X ' ( @ I = -T0d+20, XI= -TOX+iq) , 0 A = A 1 + r r C A i ,

- Q ~ = Q ; + ~ ( ~ / ~ T : ) L - ~ ~ T : ~ Z ? (L-L+ - L , M r M + - M - j ,

1 2 O i l 1 0121 2 O ( 2 ) 0i31 2 0131 O i l ) V [ Z I , Z 2 , Z 3 ] = 7 ( a 3 T, T, +a1 rrc rrC +a2 T, rr, )V ' [Z; ,Z; ,Z; ] , a 3

3 5 - COVARIANT STRING FIELD THEORY. I1 1339

where the prime is generally understood to indicate that the sr: variable is not included. Then the gauge transformation (5.1 1) and the BRS transformation (3.1) are rewritten in terms of those component fields @', X', A', and A; as

6 @ ' = Q h A ' + i z A ~ + 2 g @ ' * ~ ~ , (5.16)

with the * product for the primed fields defined by

Further the action (5.9) becomes

Note, in particular, that the component field Y' appears only in the free term in the action (5.19) and plays simply a role of a Lagrange multiplier imposing the constraint

on @'. Furthermore, in this action (5.19), the term i@'zX1 by itself and hence the rest

are separately gauge invariant under (5.16) since E6@'= L ~ X ' = O follows from the constraint d A = A (i.e., LSA~=ZA;=O~ and the presence of the projection operator .YIz3 in V', (5.15). Therefore, if we impose the constraint

from the beginning to incorporate the "equation of motion" SS /SX '=~@'=O, (5.20), in advance, there had been no need for the X' component; indeed X' in fact dropped out not only from the action (5.19) but also from the BRS transformation of @':

GB@'=Qi@'+g@'*@' . (5.23)

Note also that the Xi component of A,Ai, does not appear in the gauge transformation of @':

Thus, the X' components, X' and A;, can be thrown away if we impose the constraint (5.22) on the field variable @. This is equivalent to dispelling the sr: variable completely from the formalism as announced above, and the action (5.21) as well as the gauge transformation (5.24) and the BRS transformation (5.23) takes exactly the same form as

in the open-string case in which the n-: mode was absent from the start. It is this formalism omitting n-: variable that was adopted in our previous papers 111 and Ref. 22. [It should be noted that the same formulas as (5.1)-(5.7) hold also in this formalism if rr: is omitted from the integration variables and the measure.] The necessity of the constraint (5.22) in this formalism is naturally under- standable as follows: The omission of the zero-mode variable sr: implies in the first quantization stage that one im- poses no gauge fixing for the uniform o-coordinate translation generated by L + - L - . Then L + - L - = O remains to a first-class constraint and hence necessitates the constraint ( L + -L j ) @ ) = O on the state. [If we wish to retain the n-: variable, it is not necessary to impose the constraint Y Q = @ on @ (except for @ A = A for the transformation parameter) as noted before.]

C. BRS-invariant gauge-fixed action

Since the action (5.21) and the BRS transformation (5.23) in the $-omitted formulation formally take the same forms as in the open-string case (with the quartic vertex v ! ~ ' set to be zero of course), it is obvious that the following action gives the BRS-invariant gauge-fixed action:

Here d is the "first component" of @ in the expansion (2.5)

carrying the F P ghost number NFp=O and hence is Grassmann even. As in the above sr:-omitted formulation of gauge-invariant theory, it is necessary to impose the constraint

but it is subject to no constraint as for its internal ghost number content. The products d .Ld and d 3 in (5.25) are given, from (5.151, (5.18), and (5.19) [or from (3.23) and (5.10) more directly], by

where now the zero-mode variables r = 1,2,3 and the vertex u no longer contain the ghost zero modes c, and 7::


A

The BRS transformation tjBd in this gauge-fixed system is given again by setting $ = X =T =O in the original BRS transformation (3 .1 ) of the d component:

4 s noticed before, if @ satisfies .9@=@, in particular, L X = L q = O , the setting X = v = 0 here as well as in SBIO is not necessary since X and 77 do not appear in S B d and 6 B ~ automatically. So we consider only such @ as an "extension" of d henceforth. Then, just as the open-string case in IV, (5.30) says a relation between 6 B 4 and the original BRS transformation 6 ~ d ,

which together with the (off-shell) nilpotency of 6 B leads t 0

This equation proves the on_-shell nilpotency of the present BRS transformation since the equation of motion of the present system 5 is just 6S^/6d = -2 (6B$)h=0=0. This is understood from the following equation valid for 6 5 under an arbitrary variation of @, 6@= -F06d, keeping $=X=v=O:

[The final dot denotes the inner product at the d-J , component level without T ~ , T : integrations as in (5.281.1

All these equations are the same as in the open-string case_and the invariance of 5 under the BRS transformation 6 B d also follows in the same way as in IV. Equation (5.23) and

lead to

but 6 ~ @ ' 6 ~ @ vanishes even before setting $=O. Indeed 1 ( d * d ) ( 3 ) ) = d l d 2 ( d ( 1 ) ( d ( 2 ) 1

+g2(@*@).(@*@1 (5.35) 1 3 ! n - I - - w;r ) - - \r2 - f i 2 2 X ~ ~ N S ~ + C L ~ - ' 2 N;-m,m

vanishes separately in each order in g: The O ( g O ) and s = l n > l m = i

o ( ~ ' ) parts result from e B 2 = 0 and the distribution law (5.7a), respectively. The O ( g 2 ) part is a consequence of X ( L n ( + I ( S I + Y - n I - I ( S I ) . the cyclic symmetry (5.5) and the Jacobi identity (5 .7b):

I Here Xrs is defined in (3.14b).

( @ + @ ) . ( @ + @ ) = @ . [ @ * ( @ * @ ) ] = O . (5.36)

We give here_the explicit expression of the new BRS V I . SCATTERING AMPLITUDES transformation 6 ~ d defined by (5 .30): From (2 .18) , (3.11, (3.41, (3.231, and (5.30) we obtain It is now an easy matter to calculate the closed-string

A

A

(5.37) scattering amplitudes based on the gauge-fixed action S,

8 B d = ~ B d + g d * d J (5.25). Since the action 2 has the same form as the open-

with the * product at the d-component level understood string one, quite a parallel argument to the previous one to be in IV again applies here.


A. Four-string scattering amplitude fact independent of the choice of a ' s of external states as we have recently shown at any loop order level.I0

We now calculate the four-string amplitude explicitly at Let us start with the closed-string diagram drawn in P the tree level in this closed-string theory. For definiteness of Fig. 13, which corresponds to the region P in Fig. 11. we take a ] , az > 0 and a 3 , a 4 < 0, corresponding to Figs. 10 By the Feynman rule readable from the action (5.251, its and 11, although the on-shell physical amplitudes are in contribution to the transition amplitude 7 is given by

= J d 5 d 6 ( R ( 5 , 6 ) 1 exp i u(,1,2,6)) v(5,3,4)) ,

where ( k ( 1,2) is given by (2.8) but with the To and rr: "")= I-., rr dep e i i ~ ~ + n ) i ~ ~ ~ - ~ ~ ~ ) modes omitted and (ext( 1-41 denotes the wave func-

(6.4)

tional of the external states: (Here the displacement of Op by rr was done simply for

( ext( -4) = ( p( ) ( p( 2 ) 1 (p( 3) ( q?(4) 1 , (6.2) making better the correspondence to the open-string case as in Sec. IV.)

Equation (6.1) is similar to (7.17) in IV. The front factor The effective four-string vertex ~ , (1 ,2 ;3 ,4 ;$~ ) ) intro- in (6. la) came from the propagator - ( 2 ~ ) - I duced in (6. lb) is an analogous quantity to the previous -

= - $(L + + L - )-I, for which we have used a trick 1 A( 1,2;3,4;03 ) defined in (4.12) in the proof of Jacobi identity in Sec. IV, and indeed reduces to the latter at

1 rn T(L++L-1 r = 0 (aside from the To- and rr:-mode factors, of course). - -=+lo d r e ,

2L (6.3) By the same arguments as given in Sec. VII and Appendix

H of paper IV, we obtain the following expression for and the 0 integration from the projection operator of the 1 A,) similar to Eq. (4.14) of IV for I A) (see Appendixes intermediate string 5: B and C ) :

where 1 uk4') is the normalized vertex functional given in terms of the Neumann function (defined in Appendix A) corresponding to the string diagram P in Fig. 13 as

- (4)rsi+)-Nl4)rs - (4)rsf- i - - i4)rs* Nnm - nm 9 Nnm -Nnm 9

with the time interval T i n diagram P of Fig. 13 related with 7 by

T =a57.

R Fe and k 66 are given in terms of the Neumann functions for the three-string vertices u(1,2,6) and u(5,3,4) by - [ ( n +m)/a5]lT+la5B) ( R Fe)nm = ( - I n & & :k(a5,a3,a4)2/Tz( - Ime ,

(6.5d) N6,6, = & N : 6 , ( a 1 , a 2 , a 6 ) f i , a 5 = - a 6 = a l + a 2 > 0 .

2T/a5 - d L + +LL), The factor e -eZ7 in (6.5a) originates from the constant part 2a (0 ) of L+ + L - in e

1342 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA - 35

Thus the amplitude of the diagram P in Fig. 13 is now found from (6.1), (6.5a), and ( 6 . 5 ~ ) to be given by

As announced already in Sec. IV, this result is the same as that in the light-cone gauge string field theory if we discard the "unphysical" modes at;*, Y - ~ , and 7-, in ")Pi), and is rewritten, when d =26, in terms of the Koba-Nielsen variables Z1-4 with the help of the Cremmer-Gervais identity (4.31) into

Here we are adopting the "gauge" (4.28) for the projective invariance and hence the integration variable is Z 3 , by 3 taking dVOb, =dV412. The amplitudes of the diagrams Q

2

and R in Fig. 13 are calculated in the same way and take the same form as (6.8) with the Z 3 integrations restricted in the regions Q and R in Fig. 11, respectively. Adding the contributions of the three diagrams P, Q, and R in

1

Fig. 13, we thus obtain the full dual amplitude with the Z 3 integration covering the whole complex plane.

P B. N-string amplitudes a t the tree level

The correspondence is now clear between the tree amplitudes in our covariant theory and in the light-cone gauge string field theory. As explained in IV, the translation rule from the amplitude in the light-cone gauge to ours is simply to put p - = O and to replace the O ( d -2) invariants p, .ps and a: "".a' ' "" by the OS (d/2) in-

( f I ( S ~ f [ + , sy Further variants p;pS and a: ,""'.an +2iym y .- comparing the previous open-string four-point amplitude in IV with the present closed-string - one (6.8) with (6 5 ) , we notice that the operator a?k .N :ma?m in the exponent of the effective vertex in the former is replaced by

in the latter. This must be a general rule since it is intuitively an almost trivial consequence of the fact that the oscillators of closed string separate into the + and - 10 ) 0. 9 modes (in the three-string vertex as well as in the propagator) which each correspond to the open-string oscillators. We indeed prove this rule " N ym -N :A*'" in Appendix B generally for the N-string effective vertex based on the path-integral method.

Therefore, generalizing the Mandelstam's result for the

(jjX/ R

N-point open-string amplitude in the light-cone gauge, we obtain the general iV-point closed-string amplitude in our FIG. 13. The closed-string diagrams for the four-string covariant theory at the tree level: scattering for the case a, ,a , > 0 a3,a4 < 0.


H~~~ &?;;+)-(NrsI-' - * - ,, =NFm are the Fourier components of the Neumann function for the corresponding N-string diagram, which are defined generically in Ap- pendix A .

For the purpose of discussing the on-shell amplitudes, however, it is more convenient to rewrite (6.9) further as follows. Writing

F1 \ I e / 0 ) = e ~ . c ( e ~ . ~ e ~ ( b ' 1 0 ) ) 3

2v (6.10) L.C= 2 C,L;'~; ,

r=l ir

and choosing the (complex) parameters <,f and cr- equal to N& and &?;;dC, respectively, we apply to the part

L.< F"?" e e / 0 ) the formula

Then, clearly, the coefficients % y;+'=Nrm in the exponent F'"' in (6.9b) turn out to be replaced by

N F ~ = N ;,,,exp( -nN&-mN&) (6.12)

and &? FA-' by their complex conjugates. Extracting the purely zero mode parts $pr.pS(N %+# g* ,*) from

F("' e 0 ) and taking care of the first factor ePL 'c in (6.10) and the last factor exp( 1 -pr2/8)<; in (6.1 lj, we find

with 2' indicating the summation excluding n =m =O. Noting further that

with L "'= 2( L 'j' + L ?' ) being our kinetic operator in (2.181, and using the expression

given in (A9) in Appendix A, we obtain the following desired expression for the N-string amplitude (6.9): 'Y n dZrdZ,?

.YX = ( 2 ~ ) ~ + '6 4 g ( g / 4 ~ ) ' - ~ P, .PI 12 n i Z r - Z s . & . v ,

dvabcdvo*bc r > s

with the exponent operator F'"" given by (6.13). physical, our tree amplitudes 7.11 become in fact independent of the choice of a r ' s and just coincide with those

C. a independence of on-shell physical amplitudes given in the light-cone gauge string field theory. The same form of proof as given for the open-string case in IV

Now with the expression (6.161, it is an easy matter to applies also here to the closed-string case. prove that, when then external states are on shell and As explained in IV, the physical one-string state I p ( r ) )


is a state in which only the transverse DiVecchia-Del Guidice-Fubini (DDF) modesz3 A:'" ( i = 1,2, . . . ,241 are excited:

Thus, for on-shell physical amplitude .YAV, the matrix ele- ment Jfs in (6.16) reduces simply to

Here v * generally for a vector v denote u = u u 0 / . The BRS operator Qf and the modes ak:: as well as the ghost modes yl- ' and 7 2 ) commute with A:'", and hence we have for the external state (ext( 1-N) = ny=, ( p ( r ) consisting of such on- shell physical one-string states ( p ( r ) I

Here we have dropped 9 l . . . by using the constraint 9"' / p ( r j ) = p ( r ) ) and have not yet used the property

( i l l r ) O = (ext / a _ , , + . The a dependence of "/.&, is contained in the coeffi-

cients N,':, of (6.19) aside from a trivial conservation factor 6 ( 2 a , ) . We now show that A.v in (6.19) is actually independent of a r ' s in the same way as in IV. Under an arbitrary variation of ar 's , ar+ar +Sar , keeping their total conservation xr 6 a r =0, AM changes by an amount

But we can prove the following identity for this change of coefficients, 6NFm =~N:~+'=(sN:~--')* as shown in Appen- dix A: .v 1 1 "-1 1 m - 1

6 N G = - 2 6ai fjrs-Nf+rn,~+- x ( n - k ) N ; - k , m ~ ~ o + - 2 ( m - ~ ) N : , - ~ N ~ , ~ i=l I a r k=l a s k = ]

By using this and

- a ( r ) - [ + l t r ' { Q s , c , ] = - x a ? ~ l $ ' m . a ~ ~ r ' + ( g h o s t t e r m s ) ,

m=-SC

we easily find that (6.20) can be rewritten into the form

[See the same calculation performed in (7.51) of IV.] This equation already proves the desired a independence of -K .v ,

6 . V N = 0 , (6.23)

since the most left operator in (6.22), either Q g' or T !?I(", annihilates the physical external state (ext( 1-N) I by (6.18). Now that we have proved the a independence o f . il\, we can take a part~cular choice for the a, 's of external strlngs

With this choice the "Neumann function" N,'s, in (6.19) becomes the same as that in the light-cone gauge string field theory in which the a parameter is 2p+ by definition. Furthermore the choice (6.24) enables one to rewrite -K lv (6.19) into

that is, all the a::; modes drop out in the D D F operator A,!"' and in the exponent F i . A key relation leading to this magic (6.25) is

shown in (A161 in Appendix A. Indeed the difference between the original exponent operator F;C in (6.19) and the purely

3 5 - COVARIANT STRING FIELD THEORY. I1

transverse one in (6.25) is the following part containing the p = i- modes a_,, +:

The last terms vanish by the identity (6.26) under the choice a, =2p',. Then the first and second terms turn

out not to contribute to (ext 1 eFi 10) in (6.191, since their contributions are always accompanied by the + oscillator a_,, + in the most left by virtue of the absence of the last term and hence vanish because of (ext a?:,': = O noted above in (6.18). Once the - modes a_,, _ disappear from the exponent F i completely, the D D F operators A:"' defined in (6.17) also can be replaced by the simple transverse operators ah"' since the azk mode dependent terms in A:"' cannot survive. Thus (6.25) follows.

The result (6.25) for J2', is just identical with that in the light-cone gauge string field theory. Therefore we have shown at the tree level that the on-shell physical amplitudes in our covariant theory agree with the correct dual amplitudes in the light-cone gauge string field theory aside from the total a-conservation factor:

VII. DISCUSSION

We have presented the covariant string field theory in its full detail for a bosonic closed string. Thus, together with paper IV for an open string, we have completed our covariant string field theory for bosonic strings. For both cases of open and closed strings, we have established the BRS invariance and its nilpotency for the gauge-fixed action on the one hand, and the gauge invariance and its group law for the gauge-invariant action on the other. It should be emphasized that the gauge invariance or the BRS invariance had deep connection with the duality property of the theory and held only at the critical dimension d = 26.

Now there remains no doubt that our string field theory is a consistent and satisfactory one. We have shown that the on-shell physical amplitudes in our theory correctly reproduces the usual dual amplitudes for general AT-string scatterings with arbitrary external states.

One may, however, worry about the point that our theory contains the string "length" parameter a which is clearly unphysical. But, with respect to this problem, we have shown that the on-shell physical amplitudes are in fact independent of the choice of the length parameters a, of the external states at the tree level aside from the total conservation factor 6( x:=l a, j. This was also the case for open string as shown in paper IV. Further we computed there the zero-slope limit of our open-string field theory, based on which we conjectured that the on-shell physical N-string amplitudes of L-loop diagrams would have the following particular form of a dependence:

X T (a independent )

with T ( a independent) standing for completely a - independent amplitudes. Actually we have recently suc- ceeded in proving this con jec t~ re . '~ We have found a certain (nonlinear) transformation of string field 4 [containing p p ( a / a a ) d ] which leaves the action invariant in the physical subspace. From this symmetry, it follows immediately that the form (7.1) is true at least for N 5 2 6 . This proof is valid both for the mopen and closed-string theories. The divergent factor ( da/2.rr)l in (7.1) can be factored out and can be absorbed in the overall multi- plicative parameter of the action such as fi or g (i.e., mul- tiplicative renormalization of the loop-expansion parameter). Thus the a-independent part T ( a independent) gives the true physical amplitude for which unitarity holds as was discussed in IV. Incidentally, as another bonus of this proof of a independence at any loop order level, we can take all the external strings a, to be zero from the start, and then it becomes manifest that the 1-loop (as well as tree) amplitudes reduce directly to those expressions of the so-called operator formalism. Therefore it is now already verified explicitly even at the l-loop level that our covariant theory reproduces the correct amplitudes. Furthermore this fact also confirms the previous expecta- tion in IV that the closed-string pole is generated dynami- cally at the 1-loop level in our covariant theory even in the case of pure open-string system. These results will be reported in more detail in a separate paper.10

These facts might suggest the following possibility: there may exist yet another formulation of covariant string field theory which has larger gauge symmetries than the present one, in particular, the local gauge symmetry which enables one even to gauge away the parameter a . In another gauge choice for this local symmetry, however, one could retain the a parameter in the field and the theory would reduce (or become equivalent) to the present formulation of ours. Indeed this type of possibility has recently been suggested in an interesting paper by Siege1 and zwiebach,'%n which they discussed a possible way to construct such a covariant theory from the known formulation in the light-cone gauge.

As an example of covariant string field theory which is free from such an unphysical parameter a , there is Witten's formulation of open string.25 However, we think that Witten's theory needs further precise c ~ n f i r m a t i o n ~ ~ as for its gauge invariance and its reproducibility of usual dual amplitudes with such an accuracy as presented in this paper and in IV. In addition it seems difficult to ex-


tend his theory to closed string and hence to heterotic string2' too. Indeed, a natural extension of the Witten- type vertex to the closed-string case looks like Fig. 14 as is discussed by Lykken and ~ a b y . ~ ~ If one constructs the closed-string vertex by a product of two open-string vertex functionals of the right-moving and left-moving modes, then the resultant vertex takes the form of Fig. 14. However this vertex does not satisfy the Jacobi identity (nor the associativity), and therefore the gauge invariance is violated at the order g2. In fact we have drawn in Fig. 15 the three-string configuration corresponding to the product (@"'a Q ' ~ ' ) * @ I 3 ' =ah4' which we called the P term in our case in Sec. IV. The Q and R terms are obtained by cyclic permutations of 1, 2, and 3. In Fig. 15 we have indicated the twisting angle 0 of the intermediate st"ng, with respect to which the integration J:d@ must FIG, The configuration of ( m , ~ ~ * m ~ ? 8 1 * m ~ ~ j - - ~ ~ The probably be performed as an inevitable nature of closed twisting angle 0 is integrated from 0 to 2T, string. Then we immediately notice from Fig. 15 that the half-circle of the resultant string @y' in this P configuration is fully supplied by the string 3 but the another half of a?' is partly by string 1 and partly by string 2. Clearly such P configurations asymmetric between 3 and (1,2) can never coincide with those in the Q and R terms for any twisting angles 0 ( f 0,n-1 and therefore the necessary cancellations for the Jacobi identity (or the associativity) do not occur contrary to our case in Sec. IV.

ACKNOWLEDGMENTS

The authors would like to thank M. M. Nojiri and K . Suehiro for careful reading of the manuscript. The work of one of the authors (T.K.) was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture (No. 61540206).

APPENDIX A: THE NEUMANN FUNCTIONS OF CLOSED STRING

Generally the two-dimensional surface, the p plane, of the N closed-string diagram is conformally transformed onto the whole complex z plane through the Mandelstam mappings (Fig. 16):

.\-

p(z )= 2 a i ln (z -Zj) , (A 1) ! = I

FIG. 14. The configuration of the closed-string product @'"*@'" when the Witten-type three-string vertex is used.

with string length parameters a, satisfy~ng x;\=, a, =O. Here 2, ( i = 1-N) are the points onto which the external ith string (at T= T cc ) transform and are called Koba- Nielsen variables (complex). Each portion of the surface corresponding to the rth string in the p plane ( p = T + i a ) is parametrized by a complex coordinate c r :

where a, is the intrinsic coordinate of the rth string defined modulo 27r and a:' is the ar coordinate at which the rth string interacts. Here 7:) and Or are the "time" ( = R e p ) and "a coordinate" ( = I m p ) of the interaction point cr = jay ' , respectively, and are given by

for a suitable z;' satisfying

With a fixed set of aj 's , each choice of a set of Zi3s corresponds to a closed-string diagram such as Fig. 13. However the variables Zj are not uniquely specified by the string diagram, since we may combine the Mandel-

FIG. 16. A typical closed-string diagram to which the whole complex plane is mapped by the Mandelstam mapping (Al) with 1v = 5 .

3 5 - COVARIANT STRING FIELD THEORY. I1 1347

stam mapping (A 1) with a one to one conformal transformation of the complex z plane onto itself. The most general form of such a transformation is known as projective (or Mobius) transformation and is given by

with complex constants A, B, C, and D. (Since the multiplication of a common factor to A, B, C, and D does not change the mapping, it is conventional to require AD -BC = 1.) Therefore, if all the Zi's are simultaneously transformed by (A5), they remain to correspond to the same closed-string diagram as the original Zi9s. We can use this projective invariance to fix three of the Zi's

at arbitrary values, e.g., Z, = w , Z2 = 1, Z , = 0. The Neumann function N(p,p3 on the p plane corre-

sponding to a closed-string diagram such as Fig. 13 is defined by

with p = r + i u and 62(p-p3=6(~-?)6(a-6) . This Neu- mann function is simply given by In 1 z-TI in the z plane. The functions N Fm corresponding to that string diagram specified by parameters ( a i , Z i ) are given as the Fourier components of N (p,p3:

with pr =a,(6, + i u r ) + r [ ' + i ~ , and ps are assumed to lie on the rth and sth strings, respectively. Note that N rm are generally complex contrary to the open-string case.

It is convenient to separate the Neumann function (A71 into the part Sln(z -F) analytic in z and its complex conjugate as

Actually this equation is obtainable from (A71 by considering its derivative (a/azIz, with keeping p(z) and p(F) on the rth and sth strings, and then integrating it. There actually the integration constant Crs, which may depends on r and s, remains undetermined other than the constraint Crs +C,*, = N $+% g* which follows since the real part has to coincide with (A7). We can fix the imaginary part of R$, which is in fact arbitrary in (A7), by C, = N & and obtain (A8).

We have the following expressions for ym:

and similarly for (T). Equation (A91 is derived from (A81 by putting Y+Zs

(implying gs+- a ) first and then by taking the limit z+Zr (cr+- rn 1. In case of r =s, we need to substitste expression (A121 for 6, coming from the 6,8(6, -gs ) term of (A8) before taking the second limit.

To derive (AlO), we differentiate (AS) with respect to C r , keeping 6, 2 cs :

I

+ nR "r"C.+mis (n 2 1 ) , (A101 (cr2Ss) 9

n , m > O

- 1 dz d,?f 1 -nc , (z ) -mEx[?~ (A 13)

N y m = = $ -6 -- nrn ' r 2rri Zs 22.i (z - ~ ) 2 which yields, in the limit Y-z,(~"+o),

(n,m 2 1) , ( A l l ) srs+ 2 nNY@,"= ( w , = e S r ) . (A141 where gr(z) is given from ( A l ) and (A21 by n > 1

1348 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA 3 5 -

E u a t i o n (A10) is obtained by performing $doro;"-l x (A141 around a contour enclosing or =O ( z =Zr).

Equation (A1 1) f@lows similarly by differentiating (A13) with respect to f,:

- (Gs=eCs) and then considering $ d o r $d&w,"-' ~ w , ~ - ~ ~ ( ~ 1 5 ) .

Since the Mandelstam mapping (A l ) and the formulas (A9)--(All) take the same forms as in the open-string case (except for the imaginary part of N ro), we can derive the other formulas in the same way:

The formulas (A23) and (A241 in IV hold as they stand. Formulas (A28)-(A30) in IV also hold giving the changes of 8 ;m under the (infinitesimal) projective transformation

remain unaltered. Only (A271 in IV is replaced by

which follows from 6fr = 0 (4, is the intrinsic coordinate on string r and hence should be invariant under the projective transformation).

Despite the same forms in these formulas for N rrn (except for 1m#&), it should be kept in mind that Nym are generally complex valued in this closed-string case since Zr's are complex, as noticed before. Only for the three- string vertex, for which Zlp3 can be taken real (e.g., 0, 1, a ) by using the projective invariance, N yrn become real and- identical with the open-string's ones (except for ImN &).

Finally in this appendix we prove Eq. (6.211, which is also of the same form as in the open-string case. It follows from the fact that the quantity (A81 [= ln (z -F)] is invariant under the change of parameters ai ( i = 1-N),

FIG. 17. A typical closed-string diagram, In which the shaded region corresponds to the effective vertex A' '' ) .

keeping 2:=, ai = 0,

ai-ai +6a i tiffi = O ] , (A201

since it changes neither z nor .Z Under this change (A201, we have, from (A121,

while (A161 and the limit i-Zs (5,- - a ) of (A8) give, respectively,

Substituting (A221 into (A21), we find

Now the change of Eq. (A8) under (A20) yields, for the case 6,2 cs,

O = 6, 2 (6 f r -6~s )en"spCr ' I n +a t r 1 + 2 6#;meni '+mlx

n , m 2 0

+ 2 . (A241 n , m > O

We note that the first terms proportion_al to 6, is written as follows by using (A231 for 6fr and 6f,:

where 2' denotes the summation excluding n = m =O. The constant terms (i.e., terms independent of f r and f , ) of (A241 give


Substit~ting (A23) also into the third term of (A241 by taking account of (A26), and comparing the coefficients of en5r+m5, (n +m 2 1) we obtain

SN:m -R ~ m ( n S R & + m S N & )

This gives the desired equation (6.21) for NFm defined by (6.12).

APPENDIX B: EVALUATION OF N-STRING EFFECTIVE VERTEX

The N-string effective vertex I A'"') is defined as a generalization of the N = 4 one I A,) in (6.1): The N-string tree amplitude Fly is written in terms of it as

where [ d ~ d e ] = n r = 7 3 d ~ i d 6 i and the parameters 7; and 8, come from (6.3) and (6.4) applied to the propagator and the projection operator of the ith intermediate string, respectively. General explicit form of A"') would be clear from the following example for the diagram of Fig. 17:

We prove in this appendix that the effective vertex In the same way as in IV we write 1 A'"") is generally given by the form (6.5b) with the

Fourier components N rm of the Neumann function for the corresponding string diagram. The proof proceeds IA (2y) ) - - e P ( ~ + + ~ _ l ( e - P ( ~ + + ~ p 1 1 A'2v1 ) ) , quite similarly to the open-string case given in Appendix H of IV. We use the OSp(d/2) momentum P"(u) and .\'

(B4)

coordinate ~ " ( o ) by which the Lagrangian of first quan- P(L+ + L - ) = 2 <:(L';'+L'T') , tized theory is given by $ a , ~ " a , , ~ " ~ ~ , ~ ~ ~ . p"(u) has r=l

the mode expansion -gO'L++Lp ' P " ( ~ ) = ( P ~ ( U ) , - a c ' ( ~ ) , a - ~ ~ ( u ) ) and first evaluate the latter factor e 1 hi.") by

1 " the path-integral technique, and then multiply the former -- - 2 (ajl+)Me;nu+a;-)Me-ino ) factor to take the limit {:+- m . Since A'.')) corre-

2V'G , ,= -= sponds to the string diagram with external strings ampu- 1 2 (P;+ )Meino+pA- 'Me - i n u

tated, for instance, to the shaded region in Fig. 17, the -- - 1 , 2~'; n 2 0 quantity e - P ' L + + L - ) 1 h i# ' ) corresponds to the diagram

-(a;* !p I * 1 - i + ) ( i ) M -

with external strings of "time" length <: attached, like the , ~ n ,Y n- ) ( n f 0 ) , whole of Fig. 17. Denoting by Nc the Neumann function

( i ) M - ( a b & ) p - I p p 0 ~ ) _ ~ b i ) M - 2 , *

(B3) of the finite region of such string diagram bounded by <:, a 0 - we obtain through the usual path-integral procedure the (i)M- ( i ) M ( F ) M p, -an +a_, (n >O) . following expression for the second factor of (B4):

The Neumann function Nc here is different from the previous one N defined by (A7) in Appendix A, which corres onds l' to the infinite region, and is subject to the boundary condition that the normal derivatives at the boundaries <, ={, vanish. As explained in Appendix H of IV, Nc is well approximated if / <: I >> 1 by taking into account only the first re- flection waves from the boundaries {: and hence is given by


Note here that this N, indeed goes in the limit <:- - cr; to the Neumann function N over the infinite region. Substituting (B6) into (B5) we find

2 (r11 + I \

7 ~ ~ 1 p n + 2 60% r

Since we have from (B3)

and hence

we see that the first term in the exponent of (B7) gives a functional expression of the ground states of oscillator modes of all the N external strings:

( t i M a exp

Therefore we can rewrite (B7) as

-

To obtain A"") we have to operate exp[ 2, ~ ( L Y ' +L? ' ) ] on this, as a result of which the term 2, cCpr2/4 in (BIZ)

disappears and the oscillators a ~ " " in P A r l ' i i are replaced by n:"* ' enr ; ) :

Taking the limit ,tO- - m, we finally arrive at the desired expression:

APPENDIX C: PROOF OF THE GENERALIZED CREMMER-GERVAIS IDENTITY

The proof proceeds quite similarly to the original Cremmer-Gervais one for the open string.9 We estimate the four- string effective vertex / A,) defined in (6.1) directly in terms of oscillator language and compare the result with the previous expression (B 14) obtained by path integration, in particular, with respect to their zero-mode parts.

As was cited already in IV, we use the formula

3 5 COVARIANT STRING FIELD THEORY. I1 1351

which is valid for the bosonic oscillator Bn satisfying [Bn,/3,] =n6, +, o. A similar formula holds also for the fermionic oscillators (FP ghosts) i f the determinant factor is replaced by ( d e t ~ ) ~ ' ~ . Applying these formulas to the oscillators o f the intermediate string 5 in the definition (6.16) o f A,) , we obtain, with the help o f (5.29) and (3.331,

+(terms quadratic or linear in oscillators ) ,

in agreement with the form cited in (6.5a). Here have calculated only the ( r n o m e n t ~ m ) ~ part in the exponent F;~ ' explicitly by using the formulas (C1) and (3.14). The exponent given in (C3) has to coincide with that o f (B14) obtained by the path-integral method. In particular, the equality

4

moment urn^^ terms in (C3)= $ 2 ( N GirS+N $,'"* )pr.ps r,s = 1

should hold up to the terms which vanish by the conservation law 2 ; = , p r = 0 , where RE'" are the n =m = O components o f the Neumann function o f the relevant four-string diagram (Fig. 12-P). Remember that the n =m = O components N g generally have such an ambiguity since they indeed change under the projective transformation by the amounts 6 N g satisfying 6( 2 p r ) Z,,, 6 N G r . p s =O.

T o avoid this ambiguity we choose P, Q , and p =p5 as three independent momenta and express all the p,'s by using them:

~ 3 = ( a 3 p + Q ) / a 5 , p4=(a@- Q ) / a 5 , p5 =P .

Then we have


for any symmetric coefficient CrS=CSr, which we take in the present case as

c r s = p (41lrsi oo + H . c . , pgI!rsi - l - (4lrs - (4)sr

(C9) = y ( N o o +Noo 1 .

Comparing the coefficients of P 2, Q 2, etc. on both sides of (CS), we obtain

(C l la)

- F+ (imaginary constant)= ( r a ' + i F 3 ) - ( ~ b 1 I + i f i ~ )

Note that the RHS's all give projective invariant combinations of P & since 6N f 0 = 6 f i + 6 ~ ( Z r +Zs ) [Eq. (A281 in IV] under the projective transformation. Equations (C10) are the relations giving a, b, and c in terms of the N&'s and Eqs. (C11) are the identities for 8 & The latter identities in fact can be proved directly from (A16), the relation T-=T~~'-~L", and the fact that the interaction points T( r I +i/3, are common between the strings 1 and 2 (or 3 and 4).

Now Eqs. ((210) are the relations between only the real parts of a , b, c, and z&. However, we note that a, b, and c are the analytic functions of a complex variable 9-= T + i a 5 6 which is also an analytic function of Zr 's since

where z + =zb3'=zb4' and z- =zt '=zb2 ' are the two interaction points determined by

On the other hand, R z s are also analytic functions of Zr's. Therefore each of Eqs. (C10) in fact gives two separate equalities for the analytic part and the antianalyt- ic part (up to an undetermined purely imaginary constant denoted by iK generically); thus

a l a2a52a -iK, = a l a 2 ( % ~ J 1 1 + N ~ 4 0 ) 2 2 - 2 N ~ i J 1 1 2 ~ ) 126

-asro , iC 14a)

We can see the equality a l a z a =a3a4c from the definitions (C4) of a and c and the relations [(3.11) in IV]

by comparing the expressions

nrn in 1 - a l a 2 a = 2 (AnBn 1- 1 n + m

(BmAm )- (AIBl) . . . (BkAk ) , m + I 1

[A, - 1 / T 2 1 1 i I : , B, r v ' z 4 ( i ? % , o ) n ]

nm ml - C Z ~ C L ~ C = 2 (B,A,)---- (AmBm 1- iBIAl ) . . . (AkBk)

n + m m + I

The equality of the RHS's of (C14a) and (C14b), up to an imaginary constant, can also be proved directly from Eqs. iA3), (A9), and


[Note that oy' in (A21 is a constant independent of the Zi9s.] Therefore it is convenient to rewrite (C14a) and (C14b) as

a l a 2 a =a3a4c=(2a52)-1[sum of RHS's of (C14a) and (C14b)+iKa +iKb] . ((217)

Now we can express a, b, and c in terms of Koba-Nielsen variables Z, by using (C171, (C14c), and (A9). Since they are projective invariant we fix the "gauge" by (4.281, i.e., Z 1 = C C , Z 2 = 1, and Z 4 =O. Then we find

T;34 - $6 - 7 a la la = a3a4c = - t l n ~ , - ~ 2 a 4 1 n ( l - ~ 3 ) + i ~ ,

2% 2a5

~ 1 5 ~ 6 =In( 1 -Z3 ) + iK' (K,Kf are real constants)

with .7 denoting the "complex time interval" T + ia50 given in ((212). On the other hand, we can prove the relation

from the definitions of a, 6 , and c in (C4). The proof goes in quite the same way as in the open-string case of Cremmer and ~ e r v a i s . ~ We first define

zjj = a I a 2 ( f i 1 cifi ,;:(I -fi 66fi;:)-1~~ ( fi ') , - (C2O) bjj-(ala2a3a4)1/2(fi,;- / c i ( l - f i 6 6 & , F ) - I ~ ~ ( f i 6 ) ,

by inserting the powers of the matrix ( C),, =n6,, , as generalizations of a and b:

Then, by using the relation (C15) and the equations

etc., it is easy to derive 6m=601(alo- 1) , - -

l n d e t ( l - ~ " ~ , ~ ~ ) = - F i ~ ~ , I which, together with the symmetry property bij=bji following from (C151, lead to the above Eq. (C19).

(C23) Now we are in the final step of the proof. We have from (C12), (C131, and (C16) in this "gauge" Z 1 = C C ,

In particular we have ~ = a 2 4 ~ ~ 3 ~ + 2 ( a 2 a ~ + a ~ a ~ ) ~ ~ +a342 , (C25)

- = -b , By using this and (C181, we obtain

a 2a3a4 z ~ ~ ( I - z ~ ) -- - - - A2

9

as-- Hm=2bl0 a5- blo [ ] - [ $ 1 - a r 2 a g 2

a b - z 3

=2610611(alo - 1) , ((224) a 7 a 5 2 ~ 1 / 2 '


and hence by (C19) 4 -tc)= N g ) ~ ~ - 2 ( N ~ ' i 1 2 1 + , ~ $ ' 1 3 4 1 )

a ala2a3a4~32( 1 -z' l 2 J = I ----lndet(1 -A@ 66E ;:)= - a,yP2 A" -126

- ((228) (C29)

r=1,2 a r r=3.4

This gives the desired estimation of the determinant. On the other hand, Eqs. (C14a), and (C14b) give and hence we have in this gauge

We compute the second derivative of this quantity; from alone, we see

together with (C26) and a l a z a =a3a4c we find it to coincide with (C28) times - 12. Thus we obtain

The integration constants y and 6 can be fixed by examining the behavior around Z3=0. Integrating (C251, we have

with iK being an undetermined imaginary constant. Here the constant part has been determined in fact from (C18) by using the knowledge a = o ( z ~ ~ ) . [The sign choice for ( c ~ ~ * ) ' / ~ = ? a ~ is due to the fact that T =Re3--+ cc as Z3 -+0.] Indeed, from the leading behavior .7-= -a51nZ3

- ( 7/ajl since the time development factor e = 0 ( Z 3 ) is contained at least twice in the defining expressions (C4) of a and c and in Indet( 1 -fi 66& t5). Substituting (C33)-(C35) into (C32) and considering the O ( l n Z 3 ) and 0( 1) terms we find

Comparing ((232) and (C36) with (C30) we finally obtain

The "Cremmer-Gervais" identity (4.31) used in the text is of course the product of this equation and its complex conjugate.

'Present address: Department of Physics, Kyoto University, Kyoto 606.

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5For the open-string case, the same gauge-invariant action as ours was found independently also by Arefeva and Volovich


aside from the quartic interaction term. See I. Ya Arefeva and I. V. Volovich, Teor. Mat. Fiz. 67, 486 (1986).

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]We use the metric qp,=diagi -, +, +, . . . , + 1. 12The a parameter in 2 here is -a instead of n--u for the pre-

vious open-string case. Note, however, that this difference n- is meaningless for the closed-string field since it is invariant under the a translation [see the constraint (2.10) below].

I3Note that the Hermiticity of the ghost ze~o-mode variables o t and their integration measures are To=TO, nc = -$,

( d r o ) + = -dTo, and idn-:)+=d~:. 141t should be noticed that the present (Ri1,2) corresponds to

the previous open-string one ( R( 1,2) R ' ~ ' with twist operator in IV.

ISM. Kato and K. Ogawa, Nucl. Phys. B212, 443 (1983). 16A necessary and sufficient condition for the nilpotency

QB2=0 for QBi of (2.16) is that L$*!+L:~'" - ~ ( o ) s , , , ~ L ~ " = - ( +Q B , ~ , -'*'I satisfy the canonical

Virasoro algebra [ L A * ' , LL*'] = in - rn )LA:; without the cen- tral charge term. See, J. H. Schwarz, Prog. Theor. Phys. Suppl. 86, 70 (1986).

"Here we have used a property v [ Z ~ , Z ~ , Z , ] = v [ z , , z ~ , Z ~ ] in (3.34) in advance. See Sec. I1 of IV for the translation rules between the functional and bra-ket representations.

'8It is necessary to note here that the connection conditions of the form (3.9) or (3.11) for d w i a ) ' s are equivalent to those for the a-reversed ones JY' w~ -a ).

19This property, which was noted in (3.65) in IV, can be understood intuitively by drawing the figures of the 6 functional. It also follows directly from the more precise oscillator expression of I V:Pe" ) by using the following identity for the Neu- mann function a f,,i - a ,+ l / a , ) :

20H. Terao and S. Uehara, Phys. Lett. 173B, 409 (1986). >'In general, the Hermiticity factor €ego of QB@, i.e.,

( Q R @ ) + [ Z ] = E ~ ~ Q ( Q B @ ) [ Z ] , is given as ~ ~ ~ ~ = t ~ ( - ) I + l o , and that of @*\V by E ~ * * = E * E ~ ( - ) " l l P +' owing to (4.3) and (4.2).

22H. Hata, K . Itoh, T. Kugo, H. Kunitomo, and K. Ogawa, Phys. Lett. 175B, 138 (1986).

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28J. Lykken and S. Raby, Nucl. Phys. B278, 256 (1986).

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Covariant string field theory. II