On the “covariantized light-cone” string field theory II

Volume 196, number

PHYSICS

1

ON THE “COVARIANTIZED S. UEHARA

LIGHT-CONE”

LETTERS

B

24 September

1987

STRING FIELD THEORY II

’

Department of Mathematics, King3 College, Strand, London WCZR 2LS, UK Received

7 May 1987; revised manuscript

received

8 June 1987

We present a full non-linear BRS transformation which corresponds to one of the generators of the OSp(1, 112) or OSp( 26,2 )2) algebra using the covariantly extended joining-splitting type interaction vertex functions in open string field theory.

We have two definite proposals of covariant field theory of the interacting string [ 1,2] so far: one of them [ 1] seems to be consistent according to recent investigations [ 31. However, it is difficult to straightforwardly extend such a theory to the closed string case and hence to the heterotic string case (c.f. ref. [4]). The other [ 21 has no such a difficulty of extending to the closed string case, however, unitarity is uncertain because of the unphysical “string length” parameter cx in spite of recent investigations [ 51. If the joining-splitting type interactions are used in the covariant string field theory, the parameter (Yis introduced and it should be guaranteed unphysical. Recently two attempts have been made to make the parameter (Yactually an unphysical parameter: one [ 61 is making the parameter gauge and the other [ 7-91 is using the Parisi-Sourlas mechanism [ lo]. As to the free parts, both of them seem to be consistent, however, interaction parts are obscure so far. In both cases the following BRS charge QB plays an important role, at least in the free part #I: QB=Qao+ic,dlacu~Q~o+A,,

(1)

where QE” is the BRS charge by Kato and Ogawa [ 111, c, another ghost zero mode than c0 and a the “string length” parameter. Note that (i) c, is anti-commutable with every ghost c, and anti-ghost c,, in Qg” and (ii) this BRS charge corresponds to one of the (fermionic) generators of the OSp( 1, 112) or OSp( 26, 2 12) algebra [ 6,7]. In this letter we present a full non-linear BRS transformation 6, of the open string field @, Bg~~(B~+gS~+g2B~)~-QB~+g~2V~,I+g2~3~~v,

(2)

using the three-string and four-string vertex functions Vi,, and Viv, respectively. Since the non-linear BRS transformation whose O(g”) part is Q E”@ takes a similar form [ 121 and Sg in (2) corresponds to one of the generators of the OSp( 26, 2 12) algebra, we can easily find the candidates for Vu, and V,, [ 6,7]. Hence we give the result first and then prove the nilpotency of the transformation. We follow the notations and conventions in refs. [ 12,4] with slight modifications if necessary. A string field @ here has NFp = 0 (and hence is Grassmann-even) and is a functional of the bosonic string coordinate X@(a), the Grassmann Faddeev-Popov (FP) ghost coordinates c(a), C( CJ)and c, and the so-called string-length parameter (Y#2, ’ On leave from Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima 725, Japan. ” This is slightly different from the one in ref. [ 91, however, we shall see that the BRS symmetry of a gauge-fixed action derived the gauge invariant action in ref. [ 91 is generated by this BRS charge. ” @ is a function of the “time T” as well in the second case [ 7-91, and the corresponding equations can be easily written down.

0370-2693/87/S (North-Holland

03.50 0 Elsevier Science Publishers Physics Publishing Division)

B.V.

from

47

Volume 196, number 1

PHYSICSLETTERSB

24 September 1987

0 = O[X"(a), c(a), g(a), ca, a] ,

(3)

or a functional of X"(a), y(a), ~](a) and a, • = O[?"(a), ?(a), 7(a), a] ,

(4)

where the Grassmann variables 7(a) and ~(a) have the following oscillator [13] representations: y(a)=

70+

(?. +~,_.) cos na

,

(5)

7o+ _~l(7~+f_. ) c o s n a

,

(6)

tt=l

f(a)=

and each mode ~,. or 7. is related with those of c(a), e(a) and ca [12], yo=C~, y,,=a-~e,,

~n=inac.

(7)

(n~O),

(8)

.

7. and ~. satisfy the following properties: 7~=r-~,

{7.,~.}=ind.+,.,o,

;*.=7-~.

(9)

Notice that ~o is anti-commutable with energy ~,. and f . , as is expected. The hermitian conjugate 0* is defined by O*[X"(a), c(a), e(a), ca, a] = O [ X u ( ~ - a ) , - c ( r t - a ) , g(Tr-a), - c a , - a ] .

(10)

This condition is expressed in the bra-ket representation as 2 (O(X2, e62), c~.2~, a 2 ) l = f dx, de~ I) ~dal dc~t) (R(1, 2) 10(x,,

6 6 1 ) , C (1) , Og I ) ) i f 2

(2) ,

(11)

with the two-string vertex function (R (1, 2)1 (R(I, 2)1 =d(c~.~ +c~2)) (R(1, 2) in ref. [1211 =6(c~ '~ +c~.2))d(x~ - x 2 ) 6 ( ( ~ ~ -g~2))2~z~(C~l +a2) × j ( 0 1 2 ( 0 ] exp

-

[(1/n)a~

l}.,v(2)

fi(l)r(2)--fi(2)r(l)]

(12)

l

Note that the BRS charge (1) is converted by (R(1, 2)1 as [12] (R(1, 2)I (Q~~) + Q~2))=0.

(13)

Now we give the main result: the full non-linear BRS transformation of the open string field • ( 3, 4) is given by dB0=-d°0+gd~0+ged20

,

(14)

~°10) =iQBIO),

(15)

d~ [ 0 ( 3 ) ) = i ~ dl d2 ( 0 ( 1 ) I ( 0 ( 2 ) I I Vm(1, 2, 3 ) ) ,

(16)

d

48

Volume 196, number 1

PHYSICS LETTERSB

24 September 1987

8~ [ q~(4) ) = - - i f dl d2 d3 ( qS(1 ) [ (q)(2) I ((/)(3) ] [ Vw(1, 2, 3, 4) ) ,

(17)

with dl-dxldg~)l)(da~/2n)dc~ l) , etc.,

(18)

I Vm(1, 2, 3) ) =/.L(al, OL2,~3)G(O'l)IE0, 2, 3) )3(1, 2, 3),

(19)

a+ J V~v(1, 2, 3, 4) ) = j d a o f ( a o ) G ( a , ) I E ( 1 , 2, 3, 4) )3(1,2, 3, 4 ) ,

~(/~l, OL2,0/3)

= IZl

(3)

-Z2[ [Z2 -Z3[ IZ3--Zl I exp --r~l JV~r0 ,

G(a,)-x/-~oG1H " e(') ( a ,(') )=

x/niH~°(a{°)

(r: arbitrary),

( a{ r) : interaction point of the rth string),

(20)

(21)

( 22 ) (23)

11 ZI

Z 2 Zrl/

j ( a o ) = d e t 1 Z2

Z 2 Z~]

Z3 Z4

Z 2 Z!l' Z24 Z4]

[ Z'~= ( d / d a o ) Z ( ao) ] ,

(24)

a(1, 2, 3, (4)) = (2a)a+ l 6 ( ~ p r ) 8 ( ~ o # ) a ( ~ y(~r)) 8 ( ~ y~)r)) ,

(25)

IE(I, 2, 3, ( 4 ) ) ) =exp[(Ex +Evp)(1, 2, 3, (4))] 10),

(26)

( E x + E v v ) ( 1 , 2, 3, (4)) = r,~ l \

(r)-(s) L-~tlru(r)n "Od(--S)m-~-i )~--n~--m) .....° %rr',,nmk2 ).

(27)

Note that the differences of these equations from the corresponding ones in ref. [ 12] are that (i) yo=C, here, while 7 o - 0 in ref. [ 12], and (ii) the ~-function with respect to ~o is involved in 3 (25) here. Because of this the connection conditions (for ghosts) with IE(I, 2, 3, (4)))3(1, 2, 3, (4)) here are different. In fact, they are, for example, OiZ(1)(Ol)-~O2Z(2)(o2)-Z(3)(63)

=0

on IE(1, 2, 3))3(1, 2, 3) ,

(28)

with z(r)(O'r)-~x(r)(O'r),~(r)(O'r),fl(r)((~r)

,

c~71P(r)(ar), O~.rl~2(r)(ar) , Ogrl~(r)(O'r) ,

(29)

where fi and fl are the conjugate variables to 7 and ~, respectively, and

49

Volume 196, number 1

PHYSICSLETTERSB

O l ( a ) ~" O(7~Ogl -- 0") ,

02(a)

~-O(a--~al)

,

al ~o'/al

,

a2 ~ (0"-- 7~al)/a2

24 September 1987 ,

0"3 ~ (Tr ] a3 [ --0")/I 0~3 [

(30) (we are considering the case a l , a2 >0 and a3 <0). We prove the nilpotency of the BRS transformation (14) in the following. First we see that if we could show the conditions [ 12 ] 3

QI~) I Vm(1, 2, 3)) = 0 ,

r~l

(31) a+

4 Qg) I vw(1,2, 3, 4 ) ) = - ~ -

r=l

dao

[f(ao)C(zo)C(z*)lE(1,2,3,4))3(1,2,3,4)],

(32)

~r

where [ 12 ] C(zo) C(~o) = -2(O~r)2C(r)(a~r))iH~°(a[ 0) ,

(33)

then the nilpotency 6~ = 0 holds according to the same arguments as in ref. [ 12 ]. The condition 6~ = 0 is equivalent to the following set of conditions: (60) 2 = 0 ,

(34)

{6°,61}=o,

(35)

(61) 2+{6 o, 6~} =0,

(36)

{6~, 6 2 1 = 0 ,

(37)

(6~) 2 = 0 .

(38)

The first condition (34) is satisfied because Q2 =0. The second one (35) is satisfied if the condition (31) holds because {6°,6L}1~(3))=

f

d l d 2 (45(1)1 (45(2)1 ~ Q ~ r ~ l g . i ( 1 , 2 , 3 ) ) . r=l

(39)

Note that eqs. (11 ), (13) and (15) lead to 6°(45(3)1 = - i ( 4 5 ( 3 ) I Q ~ 3) .

(40)

As to the third condition (36), we find {6°,6~}145(4))=-

I

d l d 2 d 3 (45(1)1 (4~(2)1 (45(3)1 ~4 Qg) l g w ( 1 , 2 , 3 , 4 ) ) ,

(41)

r=l

( 6 ~ ) 2 1 4 5 ( 4 ) ) = - I d l d2d3 (45(1)1 (45(2)1 (45(3)1 × [ G(a,125)G(a,334)[A,v(5)(1, 2, 3, 4) ) + G(a~37)G(al TM) [A[~)(1, 2, 3, 4) ) l , where

50

(42)

Volume

196, number

I&‘(1

92 >3 94))

=

I

2’, 5) 1(p&(1’,

d5 d2’ dl’ (E(l’,

l&‘(l, =

PHYSICS

1

LETTERS

B

2’, 5)@)(~&(5,3,4)

24 September

lE(5,3,4))

1987

]R(l’,

1))

(R(2’, 2))

,

(43)

]R(l’,

3))

]R(2’, 2))

.

(44)

2, 394)) I’, 7) I($)(2),

d7 d2’ dl’ (E(2’,

l’, 7)L?‘)(/&(l,

7,4) ]E(l, 7,4))

Following the same arguments as in ref. [ 121, the integration be performed and (42) becomes

+G(a:34)G((T174)(1/&7)[det(l

-@88~77)]-‘2~(~2,

and the contractions

in eqs. (43) and (44) can

(Y’, 04) I V&l, 2, 3,4))},

03, -cx~)P(cx,,

(45)

where I vdl,

2, 3,4))

= lEtI> 2, 3,4))&1,2,

Rn=&TA(%~3,

@‘J&c

~‘i,7,=~N7,7(~,,CY7,LY4)~,

334)

(46)

Rfn=(-)“J@A(%Q2,

-%)&(-Y

m~n=(-,mfim~&f2,a3,

-cY7)fi(-)“.

(47)

>

(48)

These equations, (4 1) and (45), take the same form as the corresponding ones in ref. [ 121, especially the pieces between the bra and ket vectors are exactly the same ones. Thus they cancel out if the condition (32), which also takes the same form as the corresponding one in ref. [ 121, is satisfied according to the same arguments as those in ref. [ 121. (It may be said that in ref. [ 121 no use was made of the explicit expressions of integration measures, string fields and the d-function part ) V,( 1,2, 3,4) ) of the vertex functions in proving the condition (36) with their BRS transformations.) Hence we can straightforwardly apply their arguments to our case. Similarly we can find that the remaining conditions (37) and (38) hold according to the same arguments as those in ref. [ 121. So we prove the conditions (3 1) and (32) in the following: Proofof(

First we find that our three-string

vertex function

I V,,,) (19) can be rewritten

as

(49) where 1V,$3,)KK0 ) is exactly the same three-string v3

=i

i f [iV&y6’)r?!, )‘.s=,Ml=,

Then the condition

vertex function

--IV& cos( rnoj’))r6’)r6”)]

.

as in ref. [ 121 and (50)

(3 1) becomes

51

Volume 196, number 1

PHYSICS LETTERSB

24 September 1987

QBIVIII>~(~r a~r) ]Vlll>)--~-g(~r ~)~r))eV3{aB-~-[aB, V3]+l[[aB, U3]U3]-I-...}IV(31)KKO> =(~(~r ~r) )eV3{~B+[QKO, v3]+[ZIB, V3]+½[[QKO, Vz]V3]}l l/(31)KKo>=O,

(51)

where we have used the fact that [2,12] --mKKO ) --

r=l

(52)

Q~°(r) I Vh3kKO) ----0.

From eq. (2.53) in ref. [5], (53)

0= E .~, rgrrs .v(s) ATrs trCtrtl" Om/-m--~* Omcos(matS))g6s)] , r.s=1m= 1

(54)

we find

(~(~r ~)6r))/IB[V(H3)KKO>--~---g(E~)6r))[QKO, V3]IV(H3)KKO>•

(55)

Hence the condition (31 ) or (51 ) becomes g ( ~ ),~r))eV~{[A., V3] + ½[[QKO, V3] V3]), V~3kKO > = 0 .

(56)

By making use of the formulas [ 12] (Y?= ~6c~,=0) -rs

1 3

~.

gNoo =grsZ t=lEgOLim=l~ fi(~O cos(ma} ~)) ,

(57)

1~-I g/~-~s =,=1 ~ got, (~-/~ ,.= ~ 1N~SN~ocos(mat s) ) -~--~ k__~l ( n - k ) N ' C ~ _ k N ' s ~ - g r s ~ N 1~ o )- ,r t '

(58)

we find [under the existence of 6()~#~r))]

r.s=l m=l .

.

± .

.

.

r,sd=l n = l \ m = l

r,s=l m=l

)

. n N o . N m ~o c O s ( m a l ~s) ) + . ~ -m- ~ o. . . . ~r~t o ~)~~) ? OLs m=l O/s

~r)~(s)•

(59)

And through a straightforward calculation of [[Q~°, v3], v3] we find [A,, v31 +½[[Q~°, v3], v3l =0 under the existence of 6(57~y~~)). Thus we have proved (31). P r o o f o f (32). Similarly to the three-string vertex functions, IViv> (20) is rewritten as

52

(60)

Volume 196, number !

PHYSICS LETTERS B

x+

24 September 1987

x+

IV, v > = - f d x ~ ( ~ r

Y~r))e~'4l'/'(4)" x,H[KKO ) = f d x l V x . , v > ,

x-

(61,

x_

where we have taken a gauge o f fixing Z~, Z2 and Z3 to some constants for (4) the projective transformation [ 5 ] , I v(4)rx,HIKKO/X is exactly the four-string vertex in refs. [12,5] and v4 is given by ( 50 ) withr, s=l, 2 , 3 , 4 and Nm, - rs being the four-string N e u m a n n functions. F r o m the information on Q ~ ° I v(4), x,mKKO/\ in refs. [5,12], we find

x+ QBIV~v>=

dx ~

4, ] +[&,x]lV.,iv>lx=~ .... e~4{Q.+[QB,V4I+½[[QB,V4]V4]+...}IV~x,HKKO>

y~)

x

=

]

[i~(~r ,~r))eV4_~O~ON~Og.rC(r)(17[r))l V~.H,KKO (4) > + [A., x] I Vx,,v > ....

•

(62)

x=x_

Note that the equation corresponding to (60) also holds in this case and similar equations to (55) hold here [51,

( -- [ Q B ,

AB I I"/'(4) ---x'HIKKO /X --

V4]

+idv4dxx/~OGC(r)(a[r)))lr/'(4) ~

d~o

\

v x,HIKKO / •

(63)

We should be careful in evaluating the first term on the RHS of (62) since [12] dx dao

----+00

,

c(r)(o'[r))IV(4)Vx,HIKKO/ \ ~ 0

whenx-+x+ .

(64)

Considering that

C(zo) - C(z*) = - 2a~c~r) (a} ~)) ,

(65)

the first term can be evaluated as 1/"( 4 ) XqX=x+ [--~--~(~(~r '~r))eVa((Zo--Z~)d~o)C'(Zo)l--x,H,KKO/]x=x

=

-½ix/~(Zo-Zg)

{C'(Zo)-~-[V4,

C ' ( Z o ) ] } I Vx, IV>

x=x

•

(66)

Then we find if+

QBIV, v > = - ' ~

dao[f(ao)C(zo)C(z*)lVo>]

cs "~[([Z~B,X]--½i~(Zo--Z~)~[v4,

Ct(zo)])'Vx, IV>II~I +

(67)

Finally we see that the second term on the R H S o f (67) vanishes. The commutator [v4, C'(zo)] is evaluated using equations (A.23) and (F.8) of ref. [12] as dpdCl=limdPFi ~ . . ~,so,,,~o ~,~,)v(s) [va, C'(zo)]=lim [ v4,-52-5-7, * ..... --I"7 - z.. ~(r) ~,, ~n~rl '~ | u~w,.j .... dzL t,s=l 4 f z~zo m=l OLr~/7~ n#O ] dp ~ ~ =lim~zz~lm=,~-

z~zo

i m7 2~sr e,<,.,~ ~ o,. ~o = i

i 1 s=t ~Zo --Is

7~,~

"

(68) 53

Volume 196, number 1

PHYSICS LETTERS B

24 September 1987

F r o m ( 5 . 4 8 ) o f ref. [12] a n d ( 2 . 1 2 0 ) o f ref. [5] we h a v e lira ( Z o - Z * )

.....

[ & , x+ ] -

d x = l i m 2i -x)(z*-x)=(2i/oq)(z~-x+) dao x~x± a--~(Zo

(zg

-

From (68)-(70)

~

X ~

0(4

)2

4

~

2

(69)

iy~r)

r : l (Zg

--Zr)

(70) "

we see that the s e c o n d t e r m a c t u a l l y v a n i s h e s , a n d h e n c e we h a v e p r o v e d ( 3 2 ) .

We h a v e c o n s t r u c t e d the n o n - l i n e a r B R S t r a n s f o r m a t i o n ( 1 4 ) - ( 2 7 ) w h o s e l i n e a r part (15 ) is g i v e n by using the ( f i r s t - q u a n t i z e d ) B R S charge (1) [7,9] in o p e n string field theory. O u r n e x t p r o b l e m is to i n v e s t i g a t e a c t i o n ( s ) c o n s t r u c t e d w i t h this n o n - l i n e a r B R S t r a n s f o r m a t i o n . T h i s will be r e p o r t e d in detail elsewhere.

Referenes [ 1] E. Witten, Nucl. Phys. B 268 (1986) 253. [2] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 172 (1986) 186, 195. [3] S.B. Giddings, Nucl. Phys. B 278 (1986) 242; S.B. Giddings and E. Martinec, Nucl. Phys. B 278 (1986) 91; S.B. Giddings, E. Martinec and E. Witten, Phys. Lett. B 176 (1986) 362; D.J. Gross and A. Jevicki, Nucl. Phys. B 283 (1987) 1; E. Cremmer, A. Schwimmer and C.B. Thorn, Phys. Lett. B 179 (1986) 57; S. Samuel, Phys. Lett. B 181 (1986) 255; K. Itoh, K. Ogawa and K. Suehiro, Nucl. Phys. B 289 (1987) 127. [4] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 35 (1987) 1318. [5] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 35 (1987) 1356. [6] W. Siegel and B. Zwiebach, Nucl. Phys. B 282 (1987) 125; B 288 (1987) 332. [ 7 ] A. Neveu and P. West, Nucl Phys. B 293 (1987) 266. [8] J.G. Taylor, Phys. Len. B 186 (1987) 57. [9] S. Uehara, Phys. Lett. B 190 (1987) 76. [ 10] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. [ 11 ] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [ 12] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 34 (1986) 2360. [ 13] A. Neveu and P.C. West, Phys. Lett. B 168 (1986) 192.

54