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PHYSICAL REVIEW D VOLUME 36, NUMBER 4 15 AUGUST 1987

New symmetry in covariant open-string field theory

Hiroyuki Hata and Mihoko M. Nojiri Department of Physics, Kyoto University, Kyoto 606, Japan

(Received 1 April 1987)

We construct a new symmetry transformation in a covariant formulation of open-string field theory. It has a closed-string functional as its transformation parameter and consists of open-closed and open-open-closed string vertices. The structure of the transformation and its algebra are those of the (stringy) general coordinate transformation known in the closed-string field theory. However, because of the constraint on the transformation parameter necessary for the action invariance, the present transformation is restricted to represent only the global part of it.

I. INTRODUCTION

One of the most interesting findings in the recent study of the covariant field theory of the (pure) open string is that the closed string is concealed in it as a bound state of the open In this respect there is a striking difference between the covariant f o r m ~ l a t i o n ~ ~ ' and the light-cone gauge one9-" of the open-string field theory. In the latter case, the pure open-string theory is incon- sistent (with Lorentz covariance) and a closed-string field must be added as an elementary field. On the other hand, in the covariant formulation, the closed-string field itself and its interaction arise effectively as loop effects of the open-string field.I2

Then this fact naturally raises the following question on the covariant formulation of open-string field theory: Since the closed string and, in particular, the graviton is effectively contained in it, does the covariant open-string field theory have in fact a larger gauge symmetry which includes general covariance?

The present paper was motivated by this question, and is a step toward its understanding. Here we point out that the covariant open-string field theory has a stringy symmetry with a closed-string functional parameter which includes the translation and the Lorentz transformation.

Let us consider a covariant open-string field theory constructed by Hata e t al .4p6 Their gauge-invariant open- string action is given by

which has an invariance under the local gauge transformation

Here, the open-string field @ as well as the gauge transformation parameter A is a functional of the open-string coordinates X u c u (0 5 u < n-1 with the boundary condition (X;(u ) ,c ( u ),T '(u)),=o,,= 0:

It should be noted that @ (and also A ) depends on anoth-

er (unphysical) variable a ( - cc < a < cc ) which is called the string length parameter. This is an inevitable consequence of the form of the interaction terms @3 and @4 (see Fig. 1). The action (1.1) is constructed so as to con- serve a, and the interaction point in the parameter space of the "longest" string 3 in a3 is specified by a ' s as

1 a2/a3 n-. Precise definitions of various quantities ap- pearing in (1.1) and (1.2) are given in Ref. 6.

The local gauge transformation (1.2) with an open- string functional parameter A is a string extension of the Yang-Mills gauge transformation. On the other hand, as we have seen in the covariant formulation of the closed- string field t h e ~ r ~ , ' ~ . ~ , ' ~ the string version of general coordinate transformations should have a closed-string functional A, as its transformation parameter:

where the string coordinates (X,(u ),c ( u ) ,T (u 1) are those of a closed string with a periodicity (X,(u + 2n-), c ( u + 2 n - ) , T ( u + 2 n - ) ) = ( X , ( u ) , ~ ( u ) , T ( u ) ) . NOW, what we would like to show in this paper is that the open- string field action (1.1) has, besides the local gauge symmetry (1.21, an invariance under the following transfor-

FIG. 1. Open-string interaction terms @' and @"

36 1193 - @ 1987 The American Physical Society

1194 HIROYUKI HATA AND MIHOKO M. NOJIRI 3 6 -

(1.2), an invariance under the following transformation 6, with a closed-string functional parameter A, in (1.4):

where S:O1@-- A,, and S~"@-@A,, and they are given schematically in Fig. 2. In Fig. 2, the dashed lines on the right-hand sides (RHS's) represent the open string on the LHS's (i.e., the one which suffers the transformation), and the closed strings on the RHS's are transformation parameter A,. More precisely, 6:OJ6, (6L1'@) is A, (@A, ) multiplied by the 6 functional of the string coordinates (X, , c , T ) represented by the RHS's of Fig. 2 and then integrated over the coordinates. (See Sec. I1 for details.) The string connections of the form of Fig. 2 are already familiar in the light-cone gauge string field theory of the open-closed mixed system.' In order for (1.5) to be a symmetry of (1.11, the transformation parameter A, cannot be arbitrary but is subject to the constraint

Q6Ac = o , (1.6)

where Q i is the Becchi-Rouet-Stora (BRS) operator for the closed string. [Both Qj and A, in (1.6) should be understood to be the primed terms defined in (5.15) of Ref. 14 (see Sec. III).]

If it were not for the constraint (1.61, the transformation 6, would be regarded as a string extension of general coordinate transformation on the open-string field since the homogeneous term S:", in particular, generates a deformation of open string by a closed one at each string point a. (see Fig. 2) . In fact, the effect of 6;'' is locally the same as the homogeneous term of the local gauge transformation in covariant closed-strlng field theory:" '"

where @, is the closed-string field and the * product on the RHS is that for the closed-string fields (see Ref. 14). In particular, we find that the transformation (1.5) forms the same closed algebra as (1.7) does. However, the constraint (1.6) severely restricts the transformation (1.5). As seen from the fact that (1.6) implies the vanishing of the inhomogeneous term of ( 1.7), the transformation 6, ( 1.5) with A, at the massless level, for example, is essentially the translation and the Lorentz transformation although it is certainly wider than these due to the degree of freedom of a .

Therefore, the new symmetry 6, (1.5) is a very small one compared with our original motivation of finding the

FIG. 2. Schematic expression of the transformations 6l"' and 8:". 8:' contains the integration over the interaction point a n .

general covariance in open-string field theory. However, it should play an important role in the covariant formulation of type-I superstring field theory.' In this theory also, the closed-string field cannot be introduced as an elementary field, and the gravitino as well as the graviton arises as a bound state. The usual supersymmetry transformation, which has a closed Ramond string functional as its transformation parameter, should be given as a superstring generalization of (1.5). [Because of the constraint (1.6), it is also restricted to a global supersymmetry transformation.]

The rest of this paper is organized as follows. In Sec. I1 we present a precise definition of the transformation 11.5) in oscillator representation. The invariance of the open-string action ( 1.1 ) under this transformation is shown in Sec. 111. In Sec. IV we study various properties of the transformation. In particular. the gauge transformation algebra formed by (1.5) and (1.7) is studied there. The final section (Sec. V) is devoted to the discussion, where we construct a closed-string field explicitly as an open-string composite field and study its property under the BRS transformation.

11. THE FORM OF TRANSFORhlATION

As stated in the Introduction our new symmetry transformation 6, (1.5) consists of two terms: 6:' and 61'" In this section we present their explicit expressions i11 the 0s- cillator representation. The invariance of the open-string action ( 1.1) under this transformation will be shown in the next section.

In this and the following sections, we follow the conventions and the notations of Refs. 6 and 14, which will be referred to as I and 11, respectively.

The inhomogeneous term 6h0'@ of the transformation is constructed by making use of the open-closed transition vertex familiar in the light-cone gauge formulation of string field theory.' In terms of bra-ket notation it is given by

Here and In the following r and r, ( r = 1,2, . . ) denote the set of open-strlng zero-mode variables r ~ ( p , , ~ ; " , a , ) and the closed-str~ng one r, = (p , ,~ : ' l ,~~ ' r ' , a , 1, respective-

ly. Accord~ngly we define d r =dp,dTg d a , / i 2 ~ ) ~ + ' and dr, ~ d p r d ~ ~ ' d ~ ~ r ' d a r / ( 2 ~ ) d + ' . In (2.11, A,i2, 1 ) IS

the closed-str~ng field valued transformation parameter, the same thlng that appeared In the local gauge transfor- matlon In closed-strlng field theory.'"n particular, A, 1s anti-Hermltlan (i.e., A:[z] = - A, [z] ), carrles a Faddeev-Popov (FP) ghost number NFp= -2, and obeys the constraint

expressing the lnvarlance under the rigid a translat~on. Note that, owing to the presence of the sr: " factor In (2. l ) , only the A: ) component of total A, ),

36 - NEW SYMMETRY IN COVARIANT OPEN-STRING FIELD THEORY

~ A : ) + T : ,A , ,> (2.3)

[cf. (5.15) of 111, contributes to (2.1). As stated in the In- troduction, A, must obey another important constraint. Now its precise expression is given by

In terms of 1 A: ) of (2.3) and Q'$ defined by [see (5.15) of 111

Q; = ~ ' $ + i ( a / a ~ ; ) ( L + -L- I - ~ ~ T : ( M + - M - , FIG. 3 . The string connection of the 6 functional I Uo(1,,2)) .

(2.5)

the condition (2.4) is restated simply as The vertex / U(1 , ,2 ) ) in (2.1) is given as

Q'S A: ) = o . (2.6) 1 ~ ( 1 , , 2 ) ) = P " ' 8 ~ 1 2 Uo(1,,2)) , (2.7)

The necessity of condition (2.4) or (2.6) will become clear where Uo( 1, ,2)) is the oscillator representation of the 6 in the next section. functional for the string connection of Fig. 3:

for ag)=ag*l=ag!(~l- 1

-yPr 9

o ' ~ ' ( ~ ) = x ' ~ ' ( u ) , a r - ' ~ y ! ( o ) , (2.9) ~ ~ r l - - ~ r ( + I - -- I - ( r !

2~~ IT^ , (2.15) a r C g ' ( a ), ar -2C %'(a) .

= c ~ l ! - l = + ~ [ ! + , T ~ r l [Quantities on the RHS of (2.9) are defined by Eq. (2.3) of I for the open-strlng case r = 2 and by Eq. (2.2) of 11 for [See Eq. (2.3) of 11. As in (2.15) we use the same symbol the closed-strlng case r= 1.1 The front factor G12 in (2.7) F $ ' for two different meanlngs for a closed strlng r. How- ls the FP ghost coordinate at the interaction polnt ever, thls wlll cause no confusion slnce T ~ ' ' c T ~ ' " + ' is al-

- ways accompanied by the summation symbol zr . ] G12 = ~ n - a , ~ ~ ~ ' ( a ~ ' ) ( r = 1 or 2 ) (2'10) Therefore, 6( 1,,2) (2.12) can also be written as

( a i 1 ' = a i 2 ' = 0 in this case), which also appeared in the previous construction of various v e r t i ~ e s . ~ ' ~ . ' " . ' ~

The 6 functional Uo( 1 , ,2) ) is given by 6 ( 1 c , 2 ) = S ( p l + p 2 ) 6 ( 2 a l + a 2 ) ~ L ~ b 1 r + L ~ ~ 2 1 . I Note that Fb" in (2.16) is the same one as on the RHS's of

6( 1 , , 2 ) = ( 2 ~ ) ~ + ~ 6 [$.ab"]s [ ? a r ] S [ x a r l ~ b " , (2.15). The definition of the Neumann function P- 1 R". 1 I,, 2 1 is given in Appendix A.

(2.12)

E ( l C , 2 ) = + 2 z A ~ F m ( l , , 2 j a ' L k a ? ' m B. 6:'' n ,m 2 0 r , r

l r l - (s! The second term 6;" is a homogeneous transformation

+ i z z N F m ( l , , 2 ) y - , , y - m . (2.13) constructed by using the open-open-closed vertex. In n 2 1 r , r bra-ket notation it takes the form m 2 0

In (2.12) and (2.13) the summations over r and/or s run 6;" ( ~ ( 3 ) ) = ~ d l , ~ ~ " ' d 2 ( A ~ ( l ~ j 1 ((D(2) / 1, 1 *, and 2 with the convention

x [ 1 U ( 1 , , 2 , 3 ) ) + l U(1 , ,3 ,2 ) ) ] ,

a, * =ar , where the open-open-closed vertex U ( 1, ,2 ,3 ) is given

for a closed string r (here r = 1). In particular, we have by

1196 HIROYUKI HATA A N D MIHOKO M. NOJIRI

I n 2 - 1 ~ ( 1 , , 2 , 3 ) ) =P"' lo d o o g ( o o ) G 1 2 i

X ~ o ( 1 , , 2 , 3 ; u o ) ) . (2.18)

Here, GU3 is the ghost fact at the interaction point, G123 = V a a , ~ a ~ ' ( u : " ) ( r = 1, 2, or 3). The 6 functional

1 Uo(1, ,2,3;uo)) is given by T ; 3 o=, 0

/ U0(1, ,2,3)) =6(lC,2,3)exp[E(1,,2,3;u~)] 0 ) , - 0 2 t T

where 8( 1, ,2 ,3 ) and E ( 1, ,2, 3 ; a o ) are defined similarly to (2.12) and (2.13), respectively. [In this case r , s summations run 1, I* , 2, and 3. See Appendix A for the definition of the Neumann functions in E ( 1, ,2,3;uo).] In Uo( 1,,2,3), a, ( r = 1,2,3) must satisfy the sign relation

as well as the conservation rule

a r = 2 a l + a 2 + a 3 = 0 . r=l,l*,2,3

The 6 functional Uo( 1, ,2, 3;uoi ) satisfies the connection conditions (see Fig. 4)

[ e10~1~(u1)+e20 i2~(~2) -o i3 ' (~3) ]

where

The uo-integration measure g ( u o ) will be given in the next section.

FIG. 4. The string connection of the 6 fi~nctional 1 Uo(lC,2,3:ao)).

111. PROOF OF 6 ,S=0

The purpose of this section is to show that the gauge- invariant open-string action ( 1.1) is also invariant under our new transformation 6, = 6:'' +g6L1 ' ( 1.5) defined in the previous section. This invariance, 6,S=0, at each order in the coupling constant g requires the following four equations to hold:

A. Proof of (3.1)

The open-closed two-string vertex U ( 1, , 2 ) ) given by (2.7) in fact satisfies the BRS invariance condition (for the proof, see below)

where Q;" and QA2' are the BRS operators for the closed16 [Eq. (2.18) of 111 and open [Eq. (2.19) of I] strings, respectively. From (2.1) and by using (3.5) we have

Hence, condition (3.1) holds if I A, ) satisfies (2.4) [or (2.6)] since ( ~ : " ' ) ~ = 0 . [Note that we can make a partial integration for Q B ~ ' to let it operate only on (A, ( I , ) 1 since ( QA' ',a:' ' ' 1 c L (:' - L '1' and A, ) satisfies (2.2).]

The proof of Eq. (3.5) proceeds in quite the same manner as the BRS invariance proof for the open three- string vertex given in Sec. I11 of I. The reason should be ascribed to the fact that both the open-closed transition and the open three-string interaction have the same string connection structure locally near the splitting point. First, we introduce a set of (operator-valued) analytic

functions ( A (p ), c (?(p) ) defined on the complex p plane of Fig. 5. Namely,

where the upper (lower) signs correspond to the region O < I m p < 2 a a l ( - 2 1 - a , <Imp<O). In terms of these functions, the sum of the closed-string BRS operator QA]' and the open-string one QA2' is rewritten as

36 - NEW SYMMETRY IN COVARIANT OPEN-STRING FIELD THEORY 1197

where the integration contour C, is depicted in Fig. 5. Next, we pass from the p plane to the z plane connected via the Mandelstam mapping

p (z )= 2 a r ln (z -2,) , (3.9) r = I , I * , ~

with o r , * r a l , Z l* = Z f . Here, Z 2 (2 , ) is real (complex with nonvanishing imaginary part) but otherwise arbi- h 2d0 k trary. Then, the LHS of (3.5) can be rewritten as [cf. (2.711 FIG. .._.In 5. The integration contour C, of (3.8).

where the new functions ( A (z) , c ( z ) ,C(z ) ) of the z variable are defined by

and zo is the position of the interaction point on the z plane, i.e., it satisfies

Note that C ( z 0 ) is equal to GI2 (2.10) in (2.7). Now, the rest of the calculation is exactly the same as the one for the open three-string vertex given in Sec. I11 E of I: We make in (3.10) (with the regularized Q B ) all possible con- traction of operators ( A , c , ~ ) by making use of the formula

Then, performing the contour integration (see Fig. 6) we finally find that (3.10) is equal to

where a , b, and c are the coefficients of the Taylor expan- sion of p(z) (3.9) around z =zo:

~ ( ~ o ) - ~ ( ~ ) = ~ ( z - z o ) ~ + ~ ( z - z o ) ~ + c ( z - z ~ ) ~ + . . . . (3.15)

Therefore, Eq. (3.5) actually holds at the critical dimen- sion d = 26.

B. Proof of (3.2) I: S!"( @.QB @ )

The open-open-closed interaction described by the vertex / U ( 1,,2,3) ) in 61" has the same string connection structure as the open four-string interaction locally around the interaction point. Namely, they are both

I d

j f

FIG. 6. The integration contour C, of (3.10)

1198 HIROYUKI HATA AND MIHOKO M. NOJIRI 3 6 -

FIG. 7. The integration contour C , of the RHS of (3.8) for the calculation of I;:=, Qk" U ( lC,2,3) ) .

exchange-type interactions. Hence the proof of Eq. (3.2) turns out to be almost the same as the O ( g 2 ) nilpotency proof of open-string nonlinear BRS transformation given in I. The vanishing of the first term of (3.2), 6~"(@.QB@), requires the BRS invariance of the open-open-closed vertex : U ( 1 , , 2 , 3 ) ) . However, ~ ( 1 , , 2 , 3 ) ) is not com- pletely annihilated by 2: Q:! but leaves nonvanishing pieces corresponding to the configurations of Fig. 4 at uo=O and / a2 j T , which are locally just the "horn" diagrams.4'6 We shall find that they are just canceled by the second term of (3.2).

Let us turn to the concrete calculations. From (2.17) we have

where the term containing the closed-string BRS operator QB~), which vanishes owing to condition (2.41, has been added by hand. In calculating

we first express 2; = Q t ) as the p-integration form of the RHS of Eq. (3.8). In this case the complex p plane and

-'d

FIG. 8. The integration contour of (3.20).

the integration contour Cp on it are depicted in Fig. 7. Then, we go to the z plane through a Mandelstam mapping

with a,*=al and Z , * =Z;. Here, as in (3.101, Z z and Z3 are real but Z1 is complex with a nonvanishing imaginary part. They are given by the parameter uo (position of the interaction point) modulo the freedom of (real) projective transformation. From (2.18) and by using

with zo being the position of the interaction point on the z plane (see Fig. 81, we have

where the integration contour is shown in Fig. 8. The z integration (3.20) is of apparently the same form as that we encountered in the analysis of the four-string vertex in the open-string field theory [see Eq. (4.9) of I]. Accordingly the result of integration should also take the same form and we get [see (4.19) of I]


where a, b, and c are given by (3.15) for the present p ( z ) (3.17). (Proof of Eqs. (4.16) and (4.17) of I given in Appendix F of I also applies here. In the present case, however, I m O ( z o ) in various equations in I should read (2 i ) - ' [O (zo) - 0 ( z ; ) ] since Z1 and Z , * are now complex. In particular, Eq. (F17) of I should be replaced by

for r = l , 1*,2, and 3.) Therefore, if we choose the a"-integration measure g (ao) in (2.18) as the one satisfying

the RHS of (3.21) becomes a surface integral

The reason why we need the sign factor sgn (a2 ) in (3.24) is that Eqs. (4.16), (4.17), and (4.20) in I were proved by defining the parameter a n as u o = +Imp(zo) + const, which, in the case a? <O, is opposite in sign relative to the present a" defined in (2.22). Similarly to the case of f (a,,) in the open four-string vertex, we take the following g ( u o i :

where the prime denotes d / d o o , and 'q;;0( 1, ,2,3) in (3.25) is the Neumann function for the open-open-closed vertex 1 Uo( 1 , , 2 ,3 ) ) . [ J ( a o ) (3.26) should be understood to be the absolute value of the determinant.] We can show that this

g ( o o l actually satisfies the differential equation (3.23). [In fact, this is a direct consequence of the fact (4.22) in I is a solution of (4.20) there.] Hence, from (3.16) and (3.24) we get

6 ~ " ( @ . ~ ~ @ ) = 2 J d l , . ; i ~ ' ~ d 2 d 3 t r ( A , i l , i ( Q ( 2 ) ( Q i 3 ) 1 go=- c12 1

s g n i g i u o : C i r o i C ( z j 1 1 U " ( l c , 2 , 3 ; a u ) ) g0=0 I The RHS of (3.24) is a subtle quantity as before: C (zo )C (z; vanishes at o o = O and 7; a 2 but g (ao) is divergent

there, and we get a finite result. However, we defer the actual calculation and turn to the analysis of the second term of (3.2): 6:0'@3.

C. Proof of (3.2) 11: 6L0'( f @ 3

Let us consider the second term of (3.21, 6L0'@'. From (2.1) and the definitions of the . and * products (see Sec. V G of I), we have

6 ~ ' ( ~ @ " = 2 ( ~ ~ ~ @ ) . ( @ * @ )

=2Jdl,r;:" 'd2d3 t r ( h c ( l C ) ( @ ( z ) ( ~ ( 3 ) J d 5 d 4 ( ~ ( 4 , 5 ) 0" ~ 1 5 ~ ( 2 , 3 , 4 1 ) , (3.28)

where 1 V ( 2,3 ,4) ) is the open three-string vertex (see I for details).

1200 HIROYUKI HATA AND MIHOKO M. NOJIRI - 3 6

Performing the integrations and oscillator contractions over the strings 4 and 5 by using the expression of the vertices, (2.1 I) and (3.29) [for ( R (4 ,5) 1 see (2.16) of I], 6h0'(fQ3) is further rewritten into

6 ~ 0 ' ( ~ Q 3 ) = 2 ~ d 1 c ~ ~ 1 ' d 2 d 3 t r ( ~ c ( l c ) ( @ ( 2 ) ( Q ( 3 ) 1 G15G234 A ( l c , 2 , 3 ) ) , (3.31)

with A( 1 , ,2 ,3) ) given by

In (3.321, the factor l / a 4 [ = - l / ( a 2 + a 3 ) = 1 / 2 a l ] comes from the ~ b ~ ~ , ~ ~ ' integrations, and the determinant factor arises from the oscillator contractions.

/ Wo( 1, ,2,3)) (3.33) is the 6 functional representing the connection of strings 1, 2, and 3 obtained by combining the connection conditions of I Uo( 1, , 5 ) ) and 1 V0(2,3,4) ). Corresponding to the three types of string connection represented by V0(2,3,4)) [(A), (B), and ( C ) of Fig. 91, there are three types of connection conditions of Wo( 1 , ,2 ,3) ) ; (A), (B) , and (C) of Fig. 10. These correspond to the string lengths relations

with a 4 = 2 a l . We show in the following that (i) the contribution of the (A)-type diagrams to (3.31) cancel among themselves, and (ii) (B)- and (C)-type diagrams cancel with (3.27).

First, let us consider the (A)-type diagram. For the (A)-type contribution to (3.31) (i.e., for the integration reg~on / a2 / + a 3 = 2 / a / ), we make a symmetrization for strings 2 and 3:

where we denote the intermediate strings of the second term by 6 and 7 instead of 4 and 5 [see Eq. (3.2811. The string connection structure of W[ ( I , , 3 ,2) ) in the second term I A A ( I,, 3,2 1 ) is depicted in Fig. 11. Diagram (A) of Fig. 10 and the diagram of Fig. 11 differ in the structure of the intermediate open string (dotted line) but they represent the same connection for the strings 1, 2, and 3. This implies that the 6-functional parts I W[ ) in AA( 1 , ,2 ,3) ) and

1 A A ( l c , 3 , 2 ) ) are the same, i.e.,

In addition, when d=26 , the front c-number factors also coincide between A ~ ( 1,,2,3) ) and A *( 1, , 3 , 2 ) ) :

where

Equation (3.38) is a consequence of the Cremmer-Gervais identityH for the scattering process of two open and one closed strings shown in Fig. 12:


FIG. 9. The string connection structure of the open three- string 6 functional V 0 ( 2 , 3 , 4 ) ) , for the cases (A) / a 2 / + a 3 = l a a , (B) a : , + l a 4 = a z 1 , and (C) 1 a4 + 1 az = i a, , respectively.

where

In the light-cone gauge string field calculation of the complete dual amplitude for the decay process of one closed string into two open strings, which is given as the sum of two diagrams of Fig. 12, can be written, by (3.401, as the integration over the Koba-Nielsen variables Z, ( i = 1,2,3). [ Z I is complex, while Z2 and Z 3 are real. By fixing the (real) projective invariance, the integration reduces to the one over one real variable.] Equation (3.38) follows from the smoothness of (3.40) at T=O.

Equation (3.40) can be shown in quite the same manner as in the pure open-string case of Cremmer and ~ e r v a i s " (cf. Appendix B). Note that the factors 8 and p in (3.40) had originally the projective-invariant expressions

FIG. 10. The string connection structure of the 6 functional 1 W o ( 1, ,2 ,3) ) . The dotted line represents the intermediate

string.

FIG. 11. The string connection structure of W d ( 1 , ,3 ,2 ) .

8 =

Therefore, we have found that AA( 1, ,2,3)) = hA( 1,,3,2)) (when d=26) . What remain are G s , the ghost factor at the interaction point. As seen by comparing the diagram (A) of Figs. 10 and 11, the following relations hold:

exp [ - x n L ( l c , 5 ) , J !

p ( a 2 , ~ ~ 3 , f f 4 ) =

This implies that G15G234= -G17G326 and hence the two terms in the square brackets of the last line of (3.36) just cancel each other, and we have

D. Proof of (3.2) 111: (B)- and (C)-type diagrams and their cancellation with (3.27)

n dZi i=2,3,5

V235

In order to discuss this problem, let us first of all remember that, in the light-cone gauge formulation of string field theory, the complete dual amplitude for the

exp [ - 2 8 & ( 2 , 3 , 4 ) , j I

- T

FIG. 12. Two diagrams in the light-cone gauge string field theory for the decay process of one closed string into two open ones.

1202 HIROYUKI HATA AND MIHOKO M. NOJIRI 36 -

process o f an open string and a closed one into an open string consists o f three diagrams o f Fig. 13. Diagrams ( I ) and (111) o f Fig. 13 consist o f an open three-string vertex and an open-closed one and contains an integration over the time interval T between the two interactions, while diagram (11) is made o f an open-open-closed vertex which contains an integration over the interaction position 00. Then by making use o f the Cremmer-Gervais identity (3.40) for diagrams ( I ) and ( I I I ) , and the expression o f g ( u o ) , (3.25), for diagram ( I I ) , the sum o f three diagrams

reduces to an integration over one real variable among Z; 's with an integration region which is independent o f the external a's ( = p _ 's) . The situation is exactly the same as the t-u dual amplitude for four open-string scattering which consists o f a diagram with an open four- string vertex as well as diagrams with two open three- string vertices.'-" The z plane diagrams corresponding to each diagram o f Fig. 13 are given in Fig. 14.

Bearing this fact in mind, let us show the cancellations between the following pairs:

(3 .27) at u0=0++(3.31 corresponding to the configuration ( C ) o f Fig. 10 , (3.45)

(3 .27) at uO=r a 2 / t t ( 3 . 3 1 ) corresponding to the configuration ( B ) o f Fig. 10 with strings 2 and 3 exchanged .

In fact, the 6-functional structures are the same between the members o f the pairs, i.e.,

and our task is to compare the front ghost factors at the interaction point and the c-number coefficients. At first sight one might think these terms separately vanish by themselves since they contain two ghost factors at the

coincident point. [Note that zO=z$ at uo=O and T a > / in (3.271.1 However, the determinant factor in (3.32) for the ( B ) and ( C ) configurations as well as g ( a o ) in (3.27) at uo=O and a a2 is diuergent and a careful treatment with some kind o f regularization is necessary. The situation is exactly the same as that we encountered in the analysis o f "horn" di- agrams4,6 in the 0 ( g 2 ) BRS nilpotency proof in open-string field theory (see Sec. V C o f I ) .

First, for (3.27) we have

where ( u _ ,o + )=(O,.rr / a 2 1, C ' ( z ) = ( d / d z ) C ( z ) , and z t (real) is the interaction point in the complex z plane in the limit a,-o+; i.e., limo,,+zO= lirn,,,-z,* = z , (see Fig. 14). As for (3.31) we define it by separating the intermedi-

ate strings 4 and 5 by an infinitesimal time T [i.e., inserting exp ( T L ' " / ~ ~ ) to the right o f ( R ( 4 , s ) / R'" in (3.28)] and then taking the limit T+O. Therefore, (3.31) for the (B) - and (C)-type configurations o f Fig. 10 is given precisely by

= 2 J d l C . r r ~ ' " d 2 d 3 t r ( ~ , ( l , ) ( ~ ( 2 ) / ( @ ( 3 ) / lim [ h - ( T ) . r r ~ ( z ~ " ) ~ ( z b ~ ) ) 1 U 0 ( 1 , , 2 , 3 ; u o = u p ) ) 7 - 0

where

In (3.48), zbl' = z b l ' ( T ) and zb3'=zh3'( T ) (real) are the interaction points in the z plane corresponding to G 1 5 (or G , , ) and G234 (or G326), respectively (see Fig. 14). They both approach a common point z $ [zof for the (B) term and z ;

+ for the (C) term] in the limit T-0, l i m T , 0 z ~ 1 ' 3 1 3 ' ( ~ ) = z ~ . Hence, the last term o f (3.48) is further rewritten as

lim h t ( T ) . r r ~ ( z b H ) C ( z ~ ' ) U ~ ( 1 , , 2 , 3 ; o o = u f ) ) = . r r lim [ ( Z ~ " - ~ ~ ' ) ~ ~ ( T ) ] C ' ( ~ $ ) C ( Z $ ) U ~ ( U ~ = U + ) ) . T-0

(3.50) 7-0

By comparing (3.47) and (3.50) the desired cancellation between 6L1'(@.QB@) (3.27) and ~ ; " ( f @ ~ ) , B ~ + ( c ) is realized i f we can show that the equality

NEW SYMMETRY IN COVARIANT OPEN-STRING FIELD THEORY 1203

FIG. 13. Three diagrams in the light-cone gauge string field theory contributing to the process of a closed string and an open string into an open string.

holds and that it has a finite limit. T h e uo-integration measure g (go) is given by (3.25). O n the other hand, the Cremmer-Gervais identity (3.40) also holds for h ( T ) corresponding to the scattering process of Fig. 13 (in the light-cone gauge string field theory) and we have

sgn(cr,)h+(T)=JT(lc,2,3)exp

(3.52) with Jj- given by (3.41). Note that sgn(cr4) =sgn(cr6)=sgn(a2) .

In proving (3.51), it is convenient to take the gauge

FIG. 14. The z plane diagrams corresponding to the process of Fig. 13. (The upper half plane is mapped onto the disc.) The diagrams (B) and (C) are boundary configurations at uo=u+ of (11) [T=O of (11111 and at uo=u. of (11) [T=O of (I)], respectively. Diagrams change as (I)+iC)+(II)+(B)-(111) as we move Z I along the dotted line of (I) with Zz and Z3 held fixed.

By differentiating both sides of

with respect to x and using the fact that dp(z ) /dz=O for z =zb",zb3' o r zO,zg*, we get

Z l = x + i ( X is real) ,

z 2 , z 3 =const . Then Eq. (3.51) reduces to

From (3.56) and zo-z$ = i /zo-zo* / , we find that Eq. (3.54) holds if we can show

1204 HIROYUKI HATA A N D MIHOKO M. NOJIRI - 3 6

One can easily convince himself of the validity of (3.57) by inspecting Fig. 14. The finiteness of the limits of (3.51) or (3.54) is also seen from (3.56).

E. Proof of (3.3) and (3.4)

The remaining two conditions, (3.3) and (3.41, can be proved in the same manner as before: The LHS of (3.3) or (3.4) contains various terms corresponding to the connection of one closed and some (three or four) open strings. However, each type of string connection always appears twice and is in fact canceled out. This cancellation is again assured by the Cremmer-Gervais identity and the anticommutativity of ghost factors at the interaction point. Here we omit the detailed calculation and show only that each type of string connection certainly appears twice.

Let us adopt the schematical notations of Fig. 1 for the interaction terms a3 and @4, and Fig. 2 for the transformation 6 y ' and 6;". Then, each term on the LHS of (3.3) is given by Fig. 15. Observe the following correspon- dences among the nine terms of Fig. 15:

By using the Cremmer-Gervais identity for the determinant factor arising from the oscillator contractions and

( 1 )

FIG. 15. Schematic expression of terms arising from the LHS of (3.3).

( M )

FIG. 16. Schematic expression of terms arising from the LHS of (3.4).

carefully inspecting the relative signs, we can show that the cancellations between the pairs of (3.58) are actually realized on the LHS of (3.3). Similarly, the LHS of (3.4) consists of terms shown in Fig. 16, and the cancellations occur between (J) and (K) and between (L) and (M).

IV. PROPERTIES OF THE TRANSFORMATION

In this section we discuss some properties of our new symmetry transformation 6 , ; i.e., (i) the explicit expression of the transformation for a special parameter A,, (ii) the algebra of the transformation, and (iii) how 6, is realized in the gauge-fixed theory.

A. Explicit form of the transformation

For a general A,, the transformation 6, induces a finite deformation on the open string (see Fig. 2 ) and it does not necessarily correspond to any transformation known in a local field theory without a: degree of freedom. Our transformation 6, gets the meaning of the usual "coordinate transformation" when we take A, with vanishing string length a: (i.e., A, which is independent of a, the Fourier conjugate variable of a ) and at the zero-mass level. This is intuitively understandable since for such A, the transformation 6, is an infinitesimal deformation of the open string at each string point 00.

Therefore, let us consider the following special A, :

which corresponds to the general coordinate transformation in the closed-string field theory." The constraint ( 2 . 6 ) imposes the following conditions on [,(XI:

among which only the first one is independent. Hence the allowed [,(x) are


which correspond to the translation and the Lorentz transformation, respectively. In fact, after a tedious but straightforward calculation by using the definition (2.17) and the formulas for the Neumann functions 7m given in Appendix A, we can show that isee Appendix C )

where

Note that the operator ordering is arbitrary in (4.5) due to the constraint (4.2). (The zero-mode x, of X,io) is contained with the coefficient 1 / g ~ in X,(u ) [see (2.3) of I]. ) Equation (4.5) shows that for the present A, (4.1), 6;'' reduces to the usual coordinate transformation although it is restricted to the translation and the Lorentz transformation which are manifest symmetries of the theory from the start. Of course, for a general A,, 6:'' induces a more complicated transformation.

As for the inhomogeneous part 6:'' (2.1), no simplification occurs by taking A, of (4.1) with vanishing a since the vertex / U ( 1,,2)) is independent of a l , a z aside from the factor 6 ( 2 a l +az). (Note that the Neu- mann function fl rm depends on a, only through their ra- tio a, /a,. The explicit expression of the Neumann functions is found in Sec. 7 of Ref. 18 [see, in particular, Eq. (7.22)]. We see that 6:'' generates a shift on the even mass-level component fields for A, given by (4.1) and hence the massless vector field, in particular, is inert under 6:". The pure Poincare transformation will be constructed by taking a suitable linear combination of 6, having A, of (4.1) or some other kind and the local gauge transformation (1.2).

B. Algebra of transformations

Our new symmetry transformation 6, forms the following closed algebra with itself and with the local gauge transformation 6 (1.2):

where the transformation parameters A, and 2, for 6, are closed-string functionals, while E for 6 is an open- string functional. In (4.6), the * product A, * C, is the one defined for closed-string fields (see I1 and Ref. 5) and gives another closed-string functional. In (4.7), the new

product ( A, Z ) giving an open-string functional is defined as

with 6;'' given by (2.17). The algebra (4.6) is of the same form as the algebra of

local gauge transformation in closed-string field theory [see (5.14) of 111. (Note, however, that 6, here is a transformation which operates on the open-string field.) As can be seen by drawing diagrams such as Figs. 15 and 16, Eq. (4.6) is proved by using the Cremmer-Gervais identi- ties for the scatterings having as external states ii) one open and two closed strings and (ii) two open and two closed strings. (We omit the proof here.) It should be stressed that (4.6) holds without the restriction (2.4) on the parameter A, and Z,, since QB appears nowhere in 6,. On the other hand, the algebra (4.7) is shown quite similarly to the proof of 6,S=0 presented in the previous section. In (4.7) we need the restriction (2.4) on A,.

C. The fate of 6, in the gauge-fixed theory

The transformation 6, is a symmetry of the open-string gauge-invariant action S (1.1). However, in making the perturbative analysis we have to consider the gauge-$xed BRS-invariant action. Here let us consider how our new symm_etry 6, is realized in the gauge-fixed open-string action S given by isee Sec. VI B of I )

$[&]=S[@= -Tad+$] 1 +o . (4.9)

Here, the component fields in q5 may have any ghost number consistent with the zero ,ghost-number restriction on &. The gauge-fixed action S has, instead of the local gauge sxmmetry, an invariance under the BRS transformation 6 ~ ,

where the original BRS transformation on @ = + $ is defined by

The (reduced) BRS transformation 8g is nilpotent on the m2ss shell he . , by using the equation of motion s s / s 4 = 0 , :

( ~ B ) ~ I $ = o (on shell) . (4.12)

Now we know that 6,@( = - F o 6 , & + 6 , ~ ) is a symmetry of S (1.1 ). Let us define the reduced transformation 8, on the & component by

Then, from 6 , s [@I = 0 we have

1206 HIROYUKI HATA A N D MIHOKO M. NOJIRI - 3 6

where use has been made of the formulas

valid for any variation S [see Eqs. (6.19) and (6.20) of I]. [In (4.14) and (4.16), the dot products for the d,dl fields are defined without & integration.] If we define f [ d l by

-- sf ['I - -2(6Cd)L1J , 6 4

then from (4.14) we obtain the formula

Equation (4.18) shows that 8, is a symmetry in the BRS- invariant physical subspace { p h y s )) ) defined by1"

with the second quantized BRS charge Q B which generates the BRS transformation on the string field d :

Note that in the above argument we have never used the explicit expressiorl of 6,, and hence Eq. (4.18) holds by replacing 6, with any symmetry transformation of the gauge-invariayt action S. Although the present gauge- fixed action S has never been obtained by a standard gauge-fixing procedure where the relation between the gauge-invariant action S,,, and the gauge-fixed one SRlcd is simply given by2'

the result 14.18) is of the same form as in the case where (4.21) holds. Namely, if there is a symmetry transformation 6 of S,,, (and if [6,SB ] = 01, we have

V. DISCUSSION

In this paper, motivated by the recent finding that the covariant formulation of pure open-string field theory contains the closed string as a bound state, we have presented a new symmetry existent in the gauge-invariant action of open-string field theory. This symmetry transformation has a closed-string functional as its transformation parameter and forms the same algebra as the local gauge transformation in closed-string field theory, which is a string version of general coordinate transformation. However, we must impose a strong constraint (2.4) on the transformation parameter A, and hence our new symmetry transformation is restricted to only a "global" part of general coordinate transformation.

Aside from the problem of finding directly the general coordinate invariance in open-string field theory, another question arises as to how the unphysical polarizations in the bound-state closed-string states (e.g., the graviton) are assured not to do_harm in the theory described by the gauge-fixed action S (4.9). Namely, the gauge-fixed action S has only _an invariance under the (reduced) BRS transformation 6B (4.10) and the physical subspace of the

state-vector space is specified by (4.19). Unphysical modes of the open string (e.g., the longitudinal and the scalar modes of the massless vector state) can appear in the physical subspace only in the zero-norm combinations. But, is the condition (4.19) sufficient to exclude the unphysical closed-string states?

In order to answer this question let us consider the following closed-string field which is a composite of an open-string field:

where U) ' s are the vertices defined by (2.7) and (2.181, and tr implies the trace with respect to the group index of the open-string field @. We can show, in quite the same manner as in the proof of 6 ,S=0 presented in Sec. IV, that this composite closed-string field q ) (5.1) has the following remarkably simple property under the BRS transformation (4.11) defined for the open-string field @:

Since we want to discuss the gauge-fixed theory without ghost zero modes, let us consider the reduced composite closed-string field @,

obtained from the p component of Y (5.1):

T=p+ ( terms with Fo and/or 7: I . 15.4)

Note that @ has a zero ghost number. F r o m (5.2) we find that @Ais transformed under the reduced BRS transformation hB (4.10) as

where ( + ) on the R H S is some compo_site operator, and we have used the equation of motion 6S/Sd=0. Now, if the closed-string asymptotic field @ ""evelops in the composite operator @, we have

since the second term on the R H S of (5.5) containing the closed-string kinetic operator L C will not contribute when considering the asymptotic field. Equation (5.6) in fact answers our question: I t ensures that the unphysical modes of the closed-string bound states can appear in the physical subspace defined by (4.19) only in the zero-norm combination."

Note that the appearance of Qh on the R H S of (5.2) just corresponds to the fact that we have to impose the condition (2.4) on A,. In other words, the constraint (2.4) on A, is the price one pays for the fact that the physical state condition is unifiedly specified by 14.19) for both the open- and the closed-string states. A n open-string field theory with the f i l l closed-string local gauge invariance as well as the open-string local gauge invariance may be possible. But it will be of different kind from 11.1) and will reduce to (1.1) in a special "gauge." The open-string


"pregeometrical"21 theory whose action consists only of a4 interaction term of (1.1) might be a candidate. [This purely a4 theory in fact possesses the invariances under the highest-order terms in g of both the transformations (1.2) and (1.5) with no restriction on A, . ]

In this paper we did not discuss the Witten-type open- string field theory.' There have appeared some papers2Z discussing the closed string in the purely cubic version2' of Witten's open-string field theory.

Note added. The coordinate transformation symmetries in the purely cubic version23 of Witten's open-string field theory have been discussed in Ref. 24.

ACKNOWLEDGMENT

The authors would like to thank our colleagues at Kyo- to University for valuable discussions.

APPENDIX A: NEUMANN FUNCTIONS

In this appendix we define the Neumann functions (more precisely, the Fourier components of the Neumann function) 8 Frn ( 1,,2, . . . , N j for one closed and N - 1 open strings which appear in the vertex such as (2.1 1) and (2.19). (In the following, the string 1 is a closed string and other strings with numbers 2-N are open ones.) A string diagram, Imp> 0 part of the complex p plane of Fig. 5 or Fig. 7, is conformally transformed onto the whole of the upper z plane via a Mandelstam mapping (see Figs. 6 and 8):

where a, (real) satisfies

The Koba-Nielsen variables of the open strings, Z, ( r 2 2), are real, while Z I of the closed string is complex with ImZ1 > 0. On each string strip in the p place we define the intrinsic coordinate 5, ( r = 1,2, . . . , N ) :

where 7:' is the interaction time of the rth string given by

with a suitable z g ' satisfying

In the case of Figs. 5 and 7, rg' is common to all strings. The functions 8 ym ( 1,,2, . . . , N ) are now defined as

the Fourier component of the Neumann function:

where p r = p ( z ) = a r f r + r t 1 + i B r and p; = p ( ~ ) = a s c s +rbs'+iB, are assumed to lie on the rth and sth string regions, respectively.

Various formulas for 8 Fm are obtained from (A5). In particular, we have the following expressions for 8 Fm :

1208 HIROYUKI HATA AND MIHOKO M. NOJIRI - 36

with the conventions a l*=al , Z , t =Z;, ~ b ' * ) = r b ~ ~ , and S1*(z)= [ j l ( z * I ]* . These formulas are formally the same as (A12)-(A141 in Appendix A of I and are derived in the same manner. Note that fl ro (s = 1, l * ) and 8 & ( Y , S = 1,1* 1 appear in (A5) only in the combinations N;' ,+N;' , and N g + N , $ + N ~ ' + N , $ ' * , respectively, and they are not separately defined. In (A61 and (A71 we have made a convenient choice.

Formulas (A20) and (A21) in Appendix A of I also hold for the present RFm:,. The connection conditions (2.8) and (2.21) are shown from (A5) is quite the same manner as in Appendix B of I.

APPENDIX B: THE CREMMER-GERVAIS IDENTITY

The Cremmer-Gervais identity (3.40) and (3.52) are proved in the same manner as in Appendix C of Ref. 11 (see also Appendix C of 11). Here, we show only that the leading term in the limit T--t cz coincides between both sides of (3.40) and (3.52). This analysis fixes the front nu- merical factor in these formulas and hence determines the relative weight between 6;' and 6:" in our symmetry transformation 6, (1.5).

Since both sides of (3.40) and (3.52) are invariant under the (real) projective transformation it is convenient to take the gauge

Z1 = x +iy ( X fixed, y > 01, Z2 =0, Z3 = w . (B 1

Note that the limit T--t cz corresponds to y - + + 0. (See Fig. 17 and (I) or (111) of Fig. 14, in which the upper-half z-plane diagrams conformally mapped onto the disc are shown for the cases s g n ( a l , a z , a 3 ) = ( + , f , f ) [Eq. (3.4011 and (+, f, f ) [Eq. (3.5211, respectively.)

First, let us make some preparations. The interaction points zo in the z plane are obtained as the solutions of dp(z)/dz=O with p (z ) given by

They are given, in the approximation y << / x I , by

where the positions of zb" and zb3' are indicated in Fig. 17 and diagrams (I) and (111) of Fig. 14. Then, the corresponding interaction times in the p plane, rbl' and rb3' (common to strings 2 and 31, are given as

where r o ( a 2 , a 3 , a 4 ) is defined by (3.30). From (B3), we have

~ = ~ b 3 ! - ~ b l )

with a 4 = - a 5 = -a2-a3=2a1. Then. let us consider

From (3.41) and (B4), we have, in the gauge (Bl) ,

~ ~ ~ 2 2 3 ~ j x d y / d T 1 + O ( Z 3 )

= 2 ~ 3 ~ /xy /a4 + O ( Z 3 ) . (B6)

Further, from the formula (A6), we get

FIG. 17. The z-plane diagram corresponding to Fig. 12. Dia- gram (I) turns into diagram (11) as we move Z , along the dotted line of (I).


and hence 1 2 r ~ ~ ' o a b " ~ - - - c o s ( n u ) p y ,

8 1x1 n ( B 5 ) = ---

Y-0 la4 Y Z,,, B & a b " a t ' = ~ ( ~ ) ( p 1 2 = o ) ,

O n the other hand, from (B4) and the fact that d e t ( l - f i 5 5 f i ~ 4 ) = 1 + 0 ( y 2 ) as y-+O ( , T I - m ) , we N - - - 7 ( o ) ~ e " ~ n - ~ ~ ~

find that the quantity a 2

[det( 1 -# 4: ) ] - 8p(az1a3,a4) I f f 4 I E 2 a - u - e l "

a 2 also reduces to the RHS of (B8) in the limit y--0.

~ , , , N ~ ~ ~ o s ( n a : " ) - ~ ~ ( a ) e - ' ~ ( r = 2 or 3 ) , (C8) APPENDIX C: FORMULAS FOR THE DERIVATION

OF EQ. (4.5) iV { L = o ( E ) , ('29)

In this appendix we list major formulas necessary for the derivation of (4.5). In the limit a.1 =E+O we have 1 a2 I

g(uo)-4------ . c2

(C10)

- ( - I H N i3, ----- 6,,, ( n , m 1 ) , (C1) These equations are obtained from the formulas in Ap-

n pendix A . Here, ~ ( u ) is a phase factor with 7 1 = 1.

Formulas with 1 replaced by l * are obtained by taking 2 N $$$)@- - " cos(nu)pAi u = - , (C2) their complex conjugates. Equations (C8) and (C9) are

r=l,1*,2,3 n [ " 1 formulas in the gauge ZI =re1" ( r fixed), Z2 = 0, Z 3 =

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B90, 410 (1975). I2We should also add the fact that the construction of the open-

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