Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

T H E B O S O N I C O P E N S T R I N G F I E L D T H E O R Y W I T H W E S S - Z U M I N O T E R M

Yu-liang LIU Physics Department, Fudan University, Shanghai, P.R. China

and

Guang-jiong NI Center of Theoretical Physics CCAST (World Laboratory), P.O. Box 8 730, Beijing, P.R. China and Physics Department, Fudan University, ShanghaL P.R. China

Received 17 October 1988

The motion space of strings is assumed as a direct product of d-dimensional Minkowski space Md with a dG-dimensional intrin- sic group manifold G, MuXG. We investigate the bosonic open string field theory with Wess-Zumino term by using the method of conformal field theory and prove the reparametrization invariance and BRST symmetry of the theory constructed. The critical dimension of Minkowski space turns out to be d= 26 - kdU (k + CA ).

1. Introduction

Recently, the problem of constructing a self-con- sistent and gauge-invariant interacting string field theory has attracted much attention. There are two different kinds of covariant string field theory. The main difference lies in the different form of the inter- acting vertex of strings. One is of the joining-split- ting type while the other is of the mid-point type. For the former type, the bosonic open string and closed string field theory was constructed by Hata, Itoh, Kugo, Kuni tomo and Ogawa ( H I K K O ) [ 1 ]. On the other hand, Witten constructed the open bosonic [2 ] and super [ 3 ] string field theory by using the string vertex of the mid-point type. The same problems have been studied more comprehensively by Gross and Jevicki [4]. They proved the gauge invariance by means of the conformal field theory method.

In this paper, basing ourselves on the Hilbert space with first quantized creation and annihilation oper- ators of the bosonic string model with WZ term, we are going to construct a bosonic open string field the- ory, also using the method of conformal field theory. The organization is as follows: In section 2 the over- lap equations for different fields are constructed.

Then in section 3 the identity operator and the three- string interaction vertex are constructed. Finally, the reparametrization invariance and BRST symmetry are proved in section 4.

2. The overlap equation

Before discussing the overlap equation o f fields, let us first describe the background string model for con- structing the string field theory. We consider the space M where the strings move as a direct product o f flat d-dimensional Minkowski space Ma with an intrinsic group manifold of dimension dG: M a x G. The basic fields characterizing the string motion are the bo- sonic fields X~'(a, z) ( # = I .... , d) and the group val- ued field U(~). The action in this theory is the sum of the action o f the bosonic fields S ( X ) and the WZ action S( U):

S( X, U) = S ( X) + S( U) , (2.1)

S ( X ) = - ~ daaq'~O,~X~'O~X~,, (2.2) S

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

K r J d 2 a t / ~ T r (U-tOo~U U-~OpU)

16n s ( u ) = -

S

K f ~ijk + ~ d3x Q

(S= ~Q)

× T r ( U-~O~U U-~OjU U-~OkU) . (2.3)

Here K is a positive integer while Q is a three-dimen- sional manifold with the two-dimensional word sheet S as its boundary.

Because of the local symmetry in the WZ action S(U) , it is invariant under the following transfor- mation:

U--+A(a_)UB-~(a+ ) . (2.4)

Accordingly, there exist two conserved currents

& ( a _ ) = (iK/2rO U-~0_ U,

JR(a+)= -- ( iK/2n) U0+ U -I , (2.5)

0 + J L ( a _ ) = 0 . 0_JR(Or+)=0 , (2.6)

where J = J~ T% ~r+ = z +_ a. Hence the basic field U(a) can be replaced by the conserved currents JL and JR, the study on S(U) is equivalent to the study on JL and JR together with their equations of motion. Ow- ing to the noncovariance of JR and JL under the transformation (2.4), they satisfy the commutation relations of the affine Kac-Moody algebra [ 5 ]. Ac- tually, the commutators of their modes read

[ J L ( R ) m , b ~ ( a b c t c a JL(R) n] --'a "-'L(R)m+n+Km~m+n.O ~ab , a b [JLm, JRn]=O , (2.7)

where f ~t,~ is the structure constant of the group G. On the other hand, the commutation relations of the bosonic fields are simply

P ~zP [oe~, ce~] =m~m+.,og • (2.8)

For constructing a covariant string field theory for the bosonic string with WZ term one has besides these two basic fields X"(a, z) and J~(a, r), to introduce a pair of canonical conformal ghost fields c(a, z) and b (a, z) with their mode commutators as

{c,,,, b,}=a,,,+,.o,

{bm, b,}={c, , ,c , ,}=O. (2.9)

We are now in a position to investigate the overlap

equation for the basic field. A field q/j is endowed with a conformal weight J if it transforms under the con- formal transformation Z--,Z' (z) as

V,'a(Z) = (OZ' /OZ)@j(z ' ) . (2.10)

Therefore, for the general string field, we can get the overlap condition operating on the interacting vertex and identifying its value at the point z ( z = e p, p=z+ic r ) with that at the point z' = l / z , this corre- sponds to setting a - , 7t- a. A simple calculation leads to the overlap condition for a general field to be

~ j ( a ) = ( - 1 ) J ~ j ( z r - a ) , O<~a<~½rc. (2.11)

Then the overlap equations for the D-string vertex IV D) read

x]~r(~)__~x~tr-l(7~-G), p~r(Cr) = - -pur - I (~ - -a ) ,

C r ( a ) = - - c r - ' ( ~ - - a ) , br(a) = - b r - l ( ~ - ~ ) ,

J°r(a)= - J a r - ' ( ~ - a ) ,

0~<a~ ½7c, r = 1, 2, ..., D . (2.12)

When r= 1 one obtains the overlap equations for the identity operator [ ~u°(a) = qJ(a) is understood ].

One has for the Virasoro generators

1 L ' ( a ) = ½ :p2(a): + : J2(a) : , (2.13)

2 ( K + CA)

where cA6ab=f a(af bed. Using the basic overlap equa- tions (2.12 ), one easily proves

L'~(cT)=L'r-l(Tg-a) , O<~a<~ ½;g. (2.14)

We have obtained the overlap equations for the fields as well as for the Virasoro generators. These re- lations will enable us to construct the explicit expres- sions for the identity operator and the three-string vertex operators.

3. Identity operator and three-string interacting vertex

We begin with the overlap equation of a single string for constructing the explicit expression of the iden- tity operator I I ) . The basic overlap equations (2.12) can be recast in the following form via their mode expansions:

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Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

[a~, + ( - 1 )"o~5,~1 I U ) = 0 ,

[J,~, + ( - 1 )'"J~_,,,] IIS) = 0 ,

[c,. + ( - 1 ) ' " c , .] [ Igh) -=0 ,

[ b , , , - ( - 1 )"b_,n] I lgh)- -0 , (3.1)

where m> 0 and 11 ~'5 with ~u=x, gh, Ja re the "bare" identity operators for the corresponding fields, respectively.

The simple appearance of (3.1) enables us to write I P ' ) (where ~u= x or gh) in a gaussian form of crea- tion operators for the corresponding field so that the overlap equations (3.1) are satisfied explicitly:

- oe , o ~ ~ I 0 ) , II"5 =exp ~ ,,=,

) [/gh) =exp ( - 1 )"c_,b_~ I0) ~ /2 , (3.2a) tl 1

where I 0) ~/2 = I co = 0) . However, we failed to ex- press the [I J) in the gaussian form of creation oper- ators. This is because J ( a ) obeys the Kac-Moody al- gebra (2.7) which is different for (2.8) in spite of the similarity between the overlap equations of J ( a ) and p(a ) . What we can do is to e x p r e s s II J) in terms of a path integral with respect to J(cr),

I IJ) = f ~ J ( a ) I J (~ ) )

× [ ] d ( J ( a ) - J ( n - ~ ) ) , (3.2b) O~<o-~n/2

where [J(o ') ) is an eigenstate of J (a ) . To construct the three-string interacting vertex op-

erator, one needs a conformal mapping which can map the overlap onto the upper half plane of z ( z= e p, p = ~+ ia). Actually, for a basic six-string mapping

p=ln[ (z3--i ) / (z3 + i) ] -- ½in, (3.3)

which maps the overlap of six strips onto the unit disc. The three-string rearrangement corresponds to the identification z--, - z . It is equivalent to the full disc with one additional mapping co=z 2. The mapping is then given by

p = l n [ (CO3/2--i)/(CO3/2 + i ) ] -- ½i7~ . (3.4)

Its inverse transformations are

z i r . (a )=z~ [ ( l + i e i ~ ) / ( 1 - i e i ~ ) ] '/3,

r '=_+ 1, _+2, _+3, (3.5)

or

colt(a) =cot[ ( 1 +iei~) / ( 1 - i e i~) ]2/3,

r = l , 2, 3, (3.6)

where coi=exp(in/3) , co 2 =ex p ( - i n /3 ) and 0)3= exp( - in). The most important basic property of the inverse functions z l r and co I r given above is that they obey the overlap equations:

Z [ r ' ( O ' ) = Z [ r ' - 1 ( 7 / 7 - - 0" ) ,

colt(a) =co[r_t ( n - - a ) , 0~a~<n/2 . (3.7)

These functions are the basic ingredients in the con- struction of the Neumann functions (correlation functions) which will obey the same overlap equations.

In the co-plane, according to the conformal field theory, the basic correlation functions (Neumann functions) read

K~( CO, co,) = ( p ( co )p( co, ) ) = ( co_co ) -2 ,

KJ(CO, CO')=(J(CO)J(CO'))=(CO-CO') -2 , (3.8)

which transform into

K(p, p' ) = (Oco/Op)K ~v') (co, co' ) (Oco' /Op' )

= (Oco/Op)(co-co' ) -2(00. / /Op') (3.9)

under the conformal transformation co-+p (co). Here ~/stands for x or J.

From (3.7), one can easily prove that the Neu- mann functions in (3.9) satisfy the overlap equa- tions for corresponding fields specified by super- scripts r and s:

Kr'(o, c r ' ) = K ' - l ~ ( n - ~ , a ' ) , O~<a~<½n. (3.10)

With this relation, we manage to construct the "bare" three-string vertex operators as follows:

IVY) =exp ~ au_r.Krfl,,azL% IO) • r = l m , n = l

(3.11a)

The quadratic form is given by

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i d a i do' A"= ~ ~nX~[(a) [Kr~(a,a')]*X~(a'). - i t -- Tr

(3.1 lb)

As in (3.2b), we can only express the I V~ ) in the following form:

, V~>= f (~=I~ ~J ' (a) ) sH= ' ,JS(a) > VJ(J(a) )

× 1-[ 6 (JS(a) -J~- ' (n -a) ) • (3.1 lc) 0~a~<n/2

However, since both p" (a ) and J"(a) have the same overlap equation and Neumann function, their op- eration on the I V~ > and I V3 J ), respectively, is also the same. So we may infer the overlap equation of p (a ) and J ( a ) via their mode expansions

,,r , , : ,~= m . . . . . . . } I z ~ > = o ,

F o r D = I, r=s= 1, (3.12) reduces to (3.1). As for the conformal ghost fields, since b(a) and

c (a) have different conformal weights, they obey dif- ferent overlap equations. To construct the vertex op- erator for them we must recast one of them into

C(a) = i 0 ~ c ( a ) = ~ c,.n.exp(-ina) ,

so that b(a) and c(a) satisfy the same overlap equations:

b"(a)=b~-~(n-a), ~ (a)=~r-~(n-a) . (3.13)

Thus the Neumann function o fb ( a ) and c(a) can be written as

~'(p, p' ) = ½ (z/z' + z' /z)

× (Oo9/Op) (09-09)-2(0o9 '/Op' ), (3.14)

where (z/z' + z ' /z) is the symmetrized factor which does not alter the singularity in N(p, p' ),

zlr(a)----- [OgI~(O) ] '/2

=zr[(l+iei°)/(1-iei~)] w3, r = 1 , 2 , 3 . ( 3 . 1 5 )

If we choose

z~=exp(in/6), z 2 = - e x p ( - i n / 6 ) , Z3=--i,

then z I r satisfies the following overlap equation:

Z I r ( O ) = - - z l r - , ( n - o ) , 0<o~<½n. (3.16)

Hence the Neumann function shown in (3.14) sat- isfies the same overlap equation as that for b(a) and ¢(a) :

N ~ S ( a , a ' ) = / V r - l s ( n - a , a ' ) , 0-..<a-..<½n. (3.17)

Accordingly, the "bare" string vertex operator for conformal ghost fields may be expressed as

( 3 ~br~rs'l~Cs~l IV~ h ) = e x p =~l r _ . . . . . . j 10>~/2,

(3.18)

where 10>3/2 = [e l=0> ]C2=0> 1C3~-~-0>. The qua- dratic form is given by

da i ~--~ brr(a)[~rs(O'a')]'~sr(ff')" # ' c = i ~ -- 7r -- ~z

(3.19)

Notice that, however, the identity operator II gh> just constructed for the conformal ghost field has a ghost number ½, whereas the correct ghost number should be - 3 . To cure this, we add an extra factor b+(½n)b_(½n) [3], so that

II'gh)=b+(ln) b_(½n)llgh> (3.20)

has a correct ghost number - 3 . As the three-string vertex operator I V~" ) already has the correct ghost number 3, no extra factor is needed. Therefore, we have found the identity and three-string vertex oper- ators completely:

II> =b+(½n) b_(½n) IV> [igh> iiJ> ,

Iv~>=Iv~> IvY> IVY">. (3.21)

4. The reparametrization and BRST invariances

In this section we are concerned with demonstrat- ing the reparametrization invariance and BRST sym- metry of the vertex operators constructed above, i.e.:

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Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

K , , I V D ) = 0 , Q I V ~ ) = O ( D = l , 3 ) , (4.1)

where 1(,, = £~ - ( - 1 ) ~ L _ ,, and Q is the BRST charge which takes the following form in ref. [ 6 ]:

r = 1 - - c o

- ½ ~ ~ ( r n - n ) : c L , , , c 2 , , b , r , + , : +Dac , (4.2) r = [ - - ¢ ~ ,

where a = 1 and

L ' ; ;= 1 ~ 1 g-, . jar lar - , : °~¢~2 '"a~: + 2 ( K + C ~ ) _~," " . . . . . . . . . .

with D representing the number of strings. Actually, these two kinds of generator are mutually

related as follows:

L r = l

D

= Z [ L f , - ( - 1 ) " L r , ] = L , - ( - 1 ) " L - , , r = l

(4.3)

where

L,, = L ',, + ( n + m ) . . . . . . c,,, : - Dad,,.o r = I - - , : m

= L ',, + L {h - Dad,,,o . (4.4)

Thus K,, invariance follows from Q invariance plus the overlap equation for ghosts, On the contrary, K . invariance also implies Q invariance. But their equivalence ties up crucially with the overlap equa- tions for ghosts, as will be seen below.

For the identity operator I 1), we can prove that

K~A. , ] I-'-) = -- ½ d N ( - 1 ) x l U ) ,

K~N [I s ) = -- ½ [Kd~j/ ( K + CA ) ] N ( -- 1 ) x [ I J ) ,

K~, I I 'g") =-~ N ( - 1 )x] i ,g , ) . (4.5)

In calculating K~Pv for VD, we have used the relation

[K2x, b+ (~zt) ] = 8N( - 1 )Xb+ ( ½zr).

Combining (4.3) with ( 3.21 ), one obtains

K2N I I ) = ½ [ - d - K d G / ( K + C A ) +26] N( - 1 )Nil ) . (4.6)

The condition that the identity operator has repara- metrization invariance leads to the well-known result

d = 2 6 - K d U ( K + C A ) . (4.7)

For the three-string vertex I I/3), one finds similarly

K~,vl V~) = [d. 5 / ( 2 . 3 2 ) ] N ( - 1)x} V ~ ) ,

K'~xl V J ) = [dG '5 / (2"32 ) ] • z---fU-~ N ( - 1 )N[ V~)

K~hl V~ h) = - [26 '5 / (2 '32 ) ] N ( - 1 ) x [ V~h).

(4.8)

The combination of (4.8) with (3.21 ) leads to

K2NI V3 ) = [5/(2"32 ) I N ( - 1)x

X [ d + K d U ( K + Ca) - 2 6 ] [ V3 ) . (4.9)

Obviously, the critical value ofdgiven by ( 3.8 ) guar- antees the reparametrization invariance of the three- string vertex [ V3).

Now we proceed to demonstrate the BRST invari- ance of the vertices. The basic idea is to write the BRST charge as

o= L n = l t C - - n .1

+ c ~ ( ~ ; - 1 ) ) , (4.10) /

where

. . . . + L J a- l t g h (4.11 ) 2P~ - L,, - ~ ,

which is not the total Virasoro generator L,,. Then one can evaluate the operation of Q on the vertices, mak- ing use of the overlap equations for C,r,, and one fi- nally arrives at

Q I I ) = ~, c_,,K,, [ I ) , n = l

1 3 Q] V3 ) = 3 r----~, ,,=~1C~-nKn]V3>" (4.12)

Therefore, we have accomplished the proof of BRST invariance of the vertex operators constructed above.

In conclusion, by means of the method of the con- formal field theory, we have constructed the open bo- sonic field theory with WZ term, and demonstrated that this theory does have the reparametrization as

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well as BRST invariance. It would be useful for fur- ther construction of the four-dimensional string the- ory and the quan tum open string theory.

Acknowledgement

We thank Wei-Dong Zhao for helpful discussions, This work is supported by NSF of China under Grant No. KRI2040. Finally, we wish to thank the referee whose kind comment pointed out a mistake in our manuscript .

References

[ 1 ] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 172 (1986) 186, 195; Phys. Rev. D 34 (1986) 2360; D35 (1987) 1318.

[2] E. Witten, Nuel. Phys. B 268 (1986) 253. [3] E. Witten, Nucl. Phys. B 276 (1986) 291. [4] D. Gross and A. Jevicki, Nucl. Phys. B 283 (1987) 1; B 287

(1987) 225; B293 (1987) 29. [5] D. Bernard and J. Thierry-Mieg, Phys. Lett. B 185 (1987)

65. [6] J.H. Schwarz, Prog. Theor. Phys. Suppl. 86 (1986) 70.

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