Pre-geometrical field theory of the open string

Volume 215, number 1 PHYSICS LETTERS B 8 December 1988

PRE-GEOMETRICAL FIELD THEORY OF THE OPEN STRING

Mihoko M. NOJIRI and Shin'ichi NOJIRI Department o(Physics, Kyoto University, Kyoto 606, Japan

Received 10 August 1988

We propose a gauge invariant, background independent string action, which contains open and closed string fields and no kinetic terms. The kinetic term is generated through the condensation of the string fields, which is the solution of the equations of motion, we solve the equations and show that the action is classically equivalent to the open string action proposed by Hata et al.

There are several problems related to closed strings in open string field theory. We now point out two of them: (a) In the light-cone gauge formulation, a closed string field must be added as an elementary field since the pure open string field theory is incon- sistent [ 1 ]. On the other hand, in the Lorentz covariant formulation, the closed string field arises as loop effects of the open string field [4,5]. (b) Though a space-time-metric-independent formulation of closed string field theory was proposed [ 6 ], that of the open string field theory of Hata et al. [7] is not known yet ~1. The above problems have a close connection with each other. The first problem, (a), suggests the existence of an open string action which has a larger gauge symmetry which includes general covariance and contains also a closed string field as an elementary field. We will obtain the light-cone gauge theory and the covariant theory by fixing the gauge symmetry of the action in different ways. On the other hand, the second problem, (b), arises probably because the covariant theory does not contain, as an elementary field, a closed string field which determines the ge- ometry of space-time and because the theory has no stringy general coordinate invariance [ 10]. There- fore both of the above two problems will be solved by considering the system of open strings coupled with closed strings. In this paper, we give an answer to the

~ The background independent formulation of Witten's open string field theory [8] is already given in ref. [9].

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

second problem and discuss the first one ~2 Hata and Nojiri proposed a new transformation on

the open string field with a closed string functional parameter [ 11 ] in the formulation of string field theory proposed in ref. [ 7 ] :

~c~= [r] - [ rq~]. ( 1 )

Here r is a closed string functional parameter. We denote an open string field which is given by the transition from a closed string field A by [A ] and the product of a closed string fieldA and open string field q~ by [A q~] [ 12 ] ~3. The structure of the transformation and its algebra are those of the stringy general coordinate transformation known in the closed string field theory [10]. The gauge invariant open string action:

~2 In this paper, we follow the notation of refs. [7,10-12 ] and the discussion given here is based on the theory of Hata et al. [4,6,7,10]. The one-loop scattering amplitude in covariant closed string field theory of Hata et al. obtained by applying the conventional canonical quantization violates unitarity since the integration over the moduli parameter covers the funda- mental region infinitely many times. Recently, however, Hata showed that the conventional one is inapplicable since the in- teraction vertex is non-local in the time coordinate [ 2 ]. By applying Hayashi 's theory [3] of hamiltonian formulation for field theories with non-local interactions, he found that the re- sulting one-loop amplitude coincides with that in light-cone gauge string field theory. The same result was also obtained by modifying the string field theory action and the BRS transformation order by order in h to recover the BRS symmetry vio- lated by the path-integral measure.

~3 We denote open string fields by q~, h u, .4 .... and closed string fields by A, B ..... J.

99


So=~'QB~+~g~'clo*~+ 2gZclJ'~oq~ocl J (2)

is, however, not invariant under the transformation ( 1 ) and the variation of the action (2) is given by

8~.So=-QBr.{-Cqo)+½g(ClJT)}- --QBr'T. (3)

Here (@) is a closed string field which is given by the transition from an open string field @ and we denote the product of two open string fields • and T, which is a closed string field, by (q~T) [12]. Eq. (3) tells that the action (2) is invariant if Qur is proportional to hi?. This indicates that the action is invariant under only the global part of the transformation.

Recently the authors proposed a string action, which contains both open and closed string fields and has not only the gauge invariance of the open string field theory [7 ] but also stringy general coordinate invariance. We now explain this action briefly.

We start with the following pre-geometrical action:

S~'~=g2{4~.Cl)o~o~+½J.(clJ~)+J,J.A}. (4)

This action is invariant under the gauge transformations

pre. 8 o . 8~"+~=g2{-qOo~oA+~oA°~

-Ao qb° 05+ ½ [J~] },

8op,+j= 0,

8~,~A = _ ~g2 ( A ~ ) , ( 5 )

8~?"": 8~?""q)=-g[rq>],

8[?"~ J= 4rig J* r,

6~CA = 47rgA • r, ( 6 )

pre 8)~rc: 8)]r'-'(;/:)=8.1 J = 0 ,

8 q"A = 2~g2a* J. ( 7 )

Note that we only need the o-product for open string fields, the ,-product for closed string fields and the (¢bT)-product ([Jcb]-product) in the action (4) and the transformations ( 5 ) - (7).

The action (4) is background independent like in the case of the pure closed string [ 6 ]. We obtain the action with kinetic terms by considering the condensation.

The equations of motion of (4) are given by

2q)o q~o q)+ [ J~] =0, (8)

J , J = 0 , (9)

(~q)) + 2J,A =0. (10)

Let {~o, Jo, A0} be a solution of eqs. ( 8 ) - (10 ) and we express the fields q~, J, A as the solution plus fluctuation:

• --+~o+~, J--'Jo+J, A-,Ao+A. (11)

Then we define QOpC,, QC~OSe~, q~, T, (~ ) and [J] as follows:

Q°pcnc/~g2{(__ )101+10o0oo00

+ ~o o q~o o ~ + ¢bo o q~o ~o + ½ [Jo~] }, (12)

q~ , T - = g { ¢ , o T ° ¢ ' o + ( - )~ "b + ' (/'o q~o o T

+ ( _ )I,~l+ I ~'l q~o o q~ o T}, (13)

(~ ) =- - g ( ~ o ) , (14)

[J] = - g [ J ~ o ] , (15)

Qctoscdj= 2ng2jo , j . ( 16 )

Here I~l is O(1) if q~ is Grassmann even (odd). These definitions (12) - (16) reproduce the same properties as those proved in ref. [7], i.e. (5.73a), (5.73b), (5.73c) and also eqs. (5), ( 7 ) - ( 9 ) in ref. [ 12],usingeq. (5.73d)in ref. [7] andeqs. (6), (10), (30) in ref. [ 12]. We can also show the nilpotency of QOpen and Q~lo+~d:

(Q°p~n) 2 = (Q~L°~cd) z = 0. (17)

Eq. ( 17 ) allows us to regard these operators Qop~n and Qdo+~d as BRS charges. Now after the redefinition ( 11 ), the pre-action (4) and the gauge transformations ( 5 ) - ( 7 ) are rewritten as follows:

S=So +g J" T+ ( 1/70J'Q°l°SedA

+ gZJ, J.A o + gZJ* J'A, ( 18 )

80: 8o~=Q°P~nA+g~*A-A*~

_g2{ qbo ~)oA -- q~oAo q~+Ao q)° q~}

+ i g - [ J ~ ] ,

6oJ=0,

8oA=ng(A) +g2 (q)A), (19)

8c: 8~q~= [r]-g[rclo] ,

8,.J= (2/g) Q"l°~dr + 2~gJ* r,

8~A=4~g{ Ao , r + A , r}, (20)

100


8~: 8 iq)= 8 t J = 0 ,

8,A = Q~~°~aa+ 2ng~J*a. (21 )

The transformation 80 corresponds to the gauge transformation of open string field theory [ 7 ] and 8~ to the stringy general coordinate transformation [ 10 ], i.e. eq. ( 1 ). 8~ is an unfamiliar transformation which is important in the discussion of gauge fixing.

In this paper, we now solve the equations of motion ( 8 ) - (10) and, after that, we show that the action in eq. (18) is classically equivalent to that in eq. (2).

Eq. (10) is already solved in the case of the flat background [6] and in the case of the curved background [13]. Hereafter we consider a solution Jo which gives eq. (16) in the flat background. Using ( 12 ) - ( 16 ), we rewrite eqs. ( 8 ) and ( 10 ) as follows:

QB qbo - 2g~o * ~o = 0, (22)

g (~o) + (1/rc)Q~Ao=O. (23)

A solution {~o, Ao} should have vanishing string "length" parameter c~ = 0, or else, the condensation of {@o, Ao} breaks the conservation of the string "length" parameters.

There is a subtlety in the limiting procedure when the string "length" parameter ~ goes to zero [ 6 ]. We note, however, that an adequate procedure gives the correct properties of products, transition, etc. For example, the product of two string fields with vanishing string "length" parameter should be defined so that it gives the commutator of the corresponding vertex operators [ 14 ].

Let qSo be an arbitrary string functional with ghost number - 1 and infinitesimal string "length" parameter c~ = e. Analysis of Neumann coefficients give the following equations [ 14 ]:

~oo q~o ~< qs, 7~+0( ~/2),

~o*Ooc (1/e+O(1))g.(O)O+O(e),

~ , ~ o a c (1 /e+O(1) )Tr ( . ( zO~+O(e) ,

(~o O) ~c (0) +O(~'/:),

[~oJ] zc [J] +O(E'/2). (24)

Here ~c is the FP ghost on the string [ 7 ]. In particu- lar, if q3o has a form as follows:

qSo = Q j o ~o, (25)

we obtain

~o* q~=O(~), q~* ~u =O(~) . (26)

Here ~o is another arbitrary string functional with ghost number - 1 and (o is the zero mode of the FP anti-ghost [ 7 ].

Eqs. (25), (26) and the nilpotency of the BRS charge eq. (17) tell that a solution ofeq. (22) is given by

• o = lim QB(O ~to. (27) E ~ 0

We can easily confirm that this solution q~o gives the non-vanishing *-product and open-closed transition through eqs. (13 ) - (15) , by using one simple example,

~o = (N/g)c_2 [0)6(p)O(O!--E). (28)

Here N is a finite normalization constant. Eqs. (25), (26) and (27) tell q)o. q~=0 for an ar-

bitrary field q) in Fock space. Due to this property, there remain less ambiguities in the limiting procedure when string "length" parameters vanish [ 14 ].

Using the solution in eq. (27), eq. (23) is rewritten as

Ql~tglim(g°Tu°)+lA°} ,~o (29)

Here we used the following equation [ 12 ]:

(Qz, q~) = Qs(@). (30)

By solving eq. (29), we obtain

Ao = -~zg lim (Jo 7Jo) +B. (31) E -*0

Here B is a closed string functional which satisfies the "on-shell" condition

@;B=O. (32)

We now show that the action ( 18 ) or (4) is classically equivalent to the action (2) proposed by Hata etal. [7].

First we expand the closed string field J and A with respect to the anti-ghost zero mode (o into

J = - (o~j + ~./, A = -(o~.t + ~(,. (33)

Using the gauge transformation (20) and (21), we

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Volume 215, number l PHYSICS LETTERS B 8 December 1988

choose the following gauge condit ion:

qzj = ~u, =0 . (34)

Second we note that J*J'Ao=Ao*J'J in the act ion (18) diverges due to the tachyon ~4. By regularizing this divergence by lett ing the string " length" parameter oe finite (o~= E:~ 0) , we obta in

A o * j = l ~-J+O(1). (35)

Using eqs. (34) and (35) , the act ion (18) is re- wri t ten as follows:

S=So -g¢aj T v, + ( 1/7t)~sL~A

+ g2E - 2 0 j ~ j - } - g 2 ~jO,I¢)A-~ 'O( E ) . (36)

Here we expand T in eq. (3) with respect to the anti- ghost zero mode:

T= -eoT¢,+ T~. (37)

After the redef ini t ion

e - ' 0 . , - ~ 0 j (38)

and lett ing e go to zero, we obtain,

S=So + g2 ~.,~j. (39)

The action (39) is obviously equivalent to that in eq. (2) .

We proposed a gauge invar iant and background independent string action, which contains both open and closed string fields. We showed that this act ion is classically equivalent to that p roposed by Hata et al., by choosing the gauge condi t ion (34) . Another gauge condi t ion might give the l ight-cone string field theory but we need further invest igat ion [ 14 ].

A subject s imilar to that given here was discussed by Strominger and others [ 15-19] on the basis of Wi t ten ' s act ion [8] . St rominger constructed closed string states in terms of open string oscil lators [ 17 ] and proposed an act ion describing closed string field

"4 See eq. (37) in ref. [12].

theory [ 18 ]. His discussion is based on the cubic ac- t ion [9 ] and associat ivi ty anomal ies [ 15,17]. Al- though the discussion given in this paper is sometimes analogous to theirs, the relation is not clear at present.

The authors would like to thank our colleagues at Kyoto Univers i ty for valuable discussions. This work is par t ia l ly suppor ted by the Gran t - in -Aid for Scien- tific Research form the Minis t ry of Education, Sci- ence and Culture (#62790115 ). One of the authors (S .N.) is indebted to the Japan Society for the Pro- mot ion of Science for f inancial support .

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