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, be- cause 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.
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 con- structed by Hata e t al .4p6 Their gauge-invariant open- string action is given by
which has an invariance under the local gauge transfor- mation
Here, the open-string field @ as well as the gauge transfor- mation 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 conse- quence 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 coor- dinate transformations should have a closed-string func- tional 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 sym- metry (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 transforma- tion), and the closed strings on the RHS's are transfor- mation 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 pa- rameter A, cannot be arbitrary but is subject to the con- straint
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 transforma- tion 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 de- formation 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 con- straint (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 formula- tion of type-I superstring field theory.' In this theory also, the closed-string field cannot be introduced as an ele- mentary field, and the gravitino as well as the graviton arises as a bound state. The usual supersymmetry trans- formation, which has a closed Ramond string functional as its transformation parameter, should be given as a superstring generalization of (1.5). [Because of the con- straint (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 transfor- mation 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 trans- formation 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 con- ventions 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 transi- tion 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)) .
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