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Volume 241, number 3 PHYSICS LETTERS B 17 May 1990 EQUATIONS OF MOTION IN NON-POLYNOMIAL CLOSED STRING FIELD THEORY AND CONFORMAL INVARIANCE OF TWO DIMENSIONAL FIELD THEORIES Ashoke SEN Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India Received 30 January 1990 We show that given a solution 7Jdof the classical equation of motion in the non-polynomial closed string field theory proposed recently, we can construct a BRST charge 0B such that (0B)2=0. This shows the connection between solutions of the equations of motion in closed string field theory and conformal invariance of two dimensional field theories. It has been known for sometime that equations of motion in string theory correspond to conformal invariance of two dimensional field theories [ 1,2 ], or, equivalently, nilpotence of the BRST charge in the combined ghost- matter conformal field theory [ 3 ]. One hopes that a better understanding of this result may be obtained once a proper closed string field theory is found. Recently, a version of closed string field theory based on non-polynom- ial interactions has been proposed [4 ]. This theory correctly reproduces all the tree level on-shell S-matrix of the string theory. Some progress towards understanding higher loop contributions has also been made [ 5,6 ]. A gauge invariant version of this theory was written down in ref. [ 7 ]. The string field ~takes a value in a subspace of the Hilbert space of the combined ghost-matter conformal field theory (CFT), and the string action S(~u) can be regarded as a map from the Hilbert space to the space of complex numbers. In this paper we show that if 7vd is a solution of the classical equations of motion derived from the action S(~), then it is possible to construct an operator QB in terms of ~ud, acting on a subspace of the Hilbert space of combined matter-ghost CFT, such that (QB)2=0. QB may be interpreted as the BRST charge of the two dimensional field theory describing the propagation of the string in the presence of the background field ~c~. This, in turn, establishes the connection between the equations of motion in string theory and the nilpotency of the BRST charge in the corresponding two dimensional field theory. A similar result for open string field theory was established by Witten [ 8 ]. We shall work with string field theory, not in the flat space-time background, but in an arbitrary background corresponding to a c=26 matter CFT and the CFT of the free ghost fields b, c, b-, c [9], using the formalism developed in refs. [ 10,1 1 ]. In order to do this we first need to state the results of ref. [ 7 ] in the conformal field theoretic language, without any explicit reference to background fields. We define the ghost number in such a way that the SL(2, C) invariant ghost vacuum, annihilated by the bn and the b-n for n>~ - 1, and by the cn and the C, for n >/2, has ghost number 0. The fields b, b- have ghost number - 1, and the fields c and J have ghost number 1. In this convention the string field ~Fis taken to be a state with ghost number 3 in the Hilbert space .~ of the combined matter-ghost CFT. In order to write down the action we need to define two operations. For a set of states A,, ..., AN~.~, [A~..dN_, ] denotes another state in .~ given by ~1 ~ We shall use the words state and local operator interchangibly since there is a one to one correspondence between them [ 12]. 350 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

Equations of motion in non-polynomial closed string field theory and conformal invariance of two dimensional field theories

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Volume 241, number 3 PHYSICS LETTERS B 17 May 1990

EQUATIONS OF M O T I O N IN N O N - P O L Y N O M I A L C L O S E D STRING FIELD T H E O R Y AND C O N F O R M A L INVARIANCE OF T W O D I M E N S I O N A L FIELD T H E O R I E S

Ashoke SEN Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

Received 30 January 1990

We show that given a solution 7Jd of the classical equation of motion in the non-polynomial closed string field theory proposed recently, we can construct a BRST charge 0B such that (0B)2=0. This shows the connection between solutions of the equations of motion in closed string field theory and conformal invariance of two dimensional field theories.

It has been known for sometime that equations o f motion in string theory correspond to conformal invariance o f two dimensional field theories [ 1,2 ], or, equivalently, nilpotence of the BRST charge in the combined ghost- matter conformal field theory [ 3 ]. One hopes that a better understanding of this result may be obtained once a proper closed string field theory is found. Recently, a version of closed string field theory based on non-polynom- ial interactions has been proposed [4 ]. This theory correctly reproduces all the tree level on-shell S-matrix of the string theory. Some progress towards understanding higher loop contributions has also been made [ 5,6 ]. A gauge invariant version of this theory was written down in ref. [ 7 ]. The string field ~takes a value in a subspace of the Hilbert space of the combined ghost-mat ter conformal field theory (CFT) , and the string action S(~u) can be regarded as a map from the Hilbert space to the space of complex numbers. In this paper we show that if 7vd is a solution of the classical equations o f motion derived from the action S ( ~ ) , then it is possible to construct an operator QB in terms of ~ud, acting on a subspace of the Hilbert space of combined matter-ghost CFT, such that (QB)2=0. QB may be interpreted as the BRST charge o f the two dimensional field theory describing the propagation o f the string in the presence of the background field ~c~. This, in turn, establishes the connection between the equations o f motion in string theory and the nilpotency of the BRST charge in the corresponding two dimensional field theory. A similar result for open string field theory was established by Witten [ 8 ].

We shall work with string field theory, not in the flat space-t ime background, but in an arbitrary background corresponding to a c = 2 6 matter CFT and the CFT of the free ghost fields b, c, b-, c [9], using the formalism developed in refs. [ 10,1 1 ]. In order to do this we first need to state the results of ref. [ 7 ] in the conformal field theoretic language, without any explicit reference to background fields. We define the ghost number in such a way that the SL(2, C) invariant ghost vacuum, annihilated by the bn and the b-n for n>~ - 1, and by the cn and the C, for n >/2, has ghost number 0. The fields b, b- have ghost number - 1, and the fields c and J have ghost number 1. In this convention the string field ~Fis taken to be a state with ghost number 3 in the Hilbert space .~ of the combined matter-ghost CFT. In order to write down the action we need to define two operations. For a set of states A,, ..., AN~.~, [A~..dN_, ] denotes another state in .~ given by ~1

~ We shall use the words state and local operator interchangibly since there is a one to one correspondence between them [ 12].

350 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

Volume 241, number 3 PHYSICS LETTERS B 17 May 1990

2N_~6 ~I2N--6 022) ] [ [AI'"AN-I]~: ~r f 1-I (0~'i)\1 i~=l (~ [~122b(Z)'Jff~izZ'6(Z']

x f I N) o qbc~(0)f: (N) o [ b f f e A l (0 ) ] . . , f (N) o In0-PAN_. (0 ) ] }bb-PI q)r ) , (1)

where

1 bff = ~ (bo-b-o) , (2)

and P is the projection operator into the Lo =/So state:

P=dl , o-f_o,o . (3)

{ I q)r) ) denotes a complete set o f states in the two dimensional CFT and { ( (b~ J } denotes the dual basis such that ~2

< ¢,~ I ,:t,~ > = , ~ . (4)

The ri denote the 2 N - 6 real moduli of the moduli space of a sphere with Npunctures, and the qi are appropriate Beltrami differentials. The integration over ri is restricted to the range required to reproduce the correct elemen- tary N string vertex of ref. [4 ]. In the Strebel representation [ 13 ] the N punctured sphere is represented by a polyhedron with N faces. The map f ~u) takes the unit disk to the ith face of the polyhedron. Given any local field A (z, i ) and a conformal map f f o A denotes the conformal transformation of the field A under the map f ( ) denotes the correlation function in the combined matter-ghost CFT.

We also define a quanti ty {AI..'AN} a s

{ A , . . . A N } = ( - - 1 ) m + ' ( A ~ { [ A 2 . . . A N ] ) , (5)

where n~ denotes the total ghost number carried by the state I A~). Using completeness of the set of states we can bring eq. (5) into the form

f2N--6 /['-2N--6(f )] {AI"'AN}=-- E (dr')~L E [rljb(z)+OiS6('z)ldiz

x f I N)° [5 5 PAl (0) ]f -~ N) o [bo PA2(O) ].,.f (N N)° [bo PAN(O ) ] 1 " (6)

After integration over ri, the expressions ( 1 ) and (6) become symmetric in the A, up to - signs due to the interchange of terms inside the correlation function. In other words, if n, is the total ghost number carried by the field A,, we have

[AI ..'Ai_tAi+lAiAi+z...Ai_l] - - ( - 1 )(,,+ 1)( .... +l) [A 1 ...AN_I] , {A, .. 'A,_ ,Ai+ ,A,Ai+ 2..'AN} = ( - 1 )(hi+ I )(ni+l + l ){m ' ..,AN} . (7)

It was shown in ref. [ 7 ] that the state [mb.,A N_ I ] satisfies the following identity:

~2 According to our convention, if I A ) = A ( 0 ) I 0 ), then (A I = ( 0110 A ( 0 ), where I denotes the SL ( 2, ~ ) transformation 1 (z) = - 1 / z.

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Volume 241, n u m b e r 3 P H Y S I C S L E T T E R S B 17 M a y 1990

N--I QB[AI'"AN-I]--

i=1 v j=, (,j+) [AI...QBAi...AN_ 1 ] ( - - 1 ) - ' 1

+ ~ ~. ~. [Aj,...Aj,._:cff[A,,...A,._~ll(-1)"(I"JkI)=O, (8) m.n>~3 {jk} {tl}

m + n = N + 2

where

1 cC = - ~ (Co-go) • (9)

( - 1 ) ~<~j'jk/) takes into account the extra - sign that appears due to the rearrangement of the terms inside the correlator ~3

Although eq. (8) was derived for string field theory in flat space-time background, the only ingredient used was the generalized gluing theorem [ 14] which also holds for arbitrary conformal field theory [ 11 ]. (See also refs. [ 15-17 ]. ) Thus eq. (8) holds in the case of any background corresponding to an arbitrary two dimensional conformal field theory with c = 26. Using this result, and eqs. (5), ( 7 ) we can easily prove the invariance of the action

gN--2 S( 7 j) = ½ ( 7Jl QBbff PI ~ + { 7J N }

N = 3

under the gauge transformation

gN--2 gN--2 c~( bff P~J) =QBbff PA + N= ~3 (U-2)--~--~ [ TjN- 2A ] = --bff PQBA + N= ~3 ( N - 2 )! bff cff P[ ~N-2A ] '

where A is a field of ghost number 2 i¢4

(lO)

(11)

Proof From eqs. (10) and ( 11 ) we get

OS=(TJIQnbffPQBbffPIA) + ~ QB [ ~'tm-2A] ~.. gN--2 - - -- ~,7( (EIN--I~ A] N = 3

gM--2 gX--2 +

M = 3 N = 3 m~, ( N - ~ I c f f [ ( I JN-2A]} " (12)

The first term vanishes since {Qn, bff P} ac (Lo- /So)P= 0, and Q~ = 0. Since ~ carries ghost number 3, we may express ( ~1QB as ( QB~I in the second term. On the other hand, since A carries ghost number 2, we may write, by repeated use of eqs. ( 5 ), (7) and (8),

~3 Our equations differ from that of ref. [ 7 ] by some signs due to a different choice of the location of the bo inside the correlator. ~4 As we can see, only the combination bff ~appears in the action. Thus we could also have taken the string field to correspond to the

state bff ~, which has ghost number 2, and is annihilated by bff.

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Volume 241, number 3 PHYSICS LETTERS B 17 May 1990

{ ~N-~QBA}= ( Q B A I [ ~./N- 1 ] ) ~--- _ ( A I QB [ ~[/N-, ] )

= - - ~_./N-- 2 ] > ..1_ Z ( N - I ) ( A I [(QB5 v) m,t/>/3

rn+n=N+2

= ( N - 1 ){A(QB ~) ~/N- -2} - Z m,n>~3

m+n=N+2

= (N--1 ){A(Qn gJ) giN-Z}_

( N - l ) !

= ( N - 1 ) {A (QB gJ) ~/N- -2} -

( m - 2 ) ! ( n - 1)!

= ( N - 1 ){A(Qn ~) ~[/N--2} "al-

(AI W'-2c6- [ W.-l] )

( N - l ) ! {A~m_Zcff[ga,_,] } ( m - - 2 ) ! ( n - 1)!

( N - l ) ! ( m - 2 ) ! ( n - 1)' m,n>~ 3

m+n=N+2

2 m,n>~3 rn+n=N+2

(cff[ ~ ' - l ] I [A ~m -z ] )

( N - l ) ! ([g.,,_,]lcff[A7.,,,,_2]) ( m - 2 ) ! ( n - 1)!

( N - 1)! ( m - 2 ) ! ( n - 1)! m,n~3

m+n=N+2

=--(N--1){(QBga)gjN-2A}+ Y.

(Co-[A ~m-2] I [ ~ n - ' ] I >

( N - 1)! { ~¢'ln-lc~ [A ~'tm-2] } . . , . . >_- 3 ( m - 2 ) ! ( n - 1 ) !

m+n=N+2

(13)

Thus we may express eq. ( 12 ) as

gN--2 gN--2 ~ S = -- N=3 ~ (U- 2 ) ~ { ( Q B ~ ) ~-'tN-2A}+- N=3 ~ (U-2 )~ { ( Q n 7 " ) t/'tN-2A}

gU-2 g M - 2 gN--2

- ~ • (m-2)!(n-1)! {ga"-lcff[Aga''-2])+ ~" ~ ( M - 1 ) ! ( N - 2 ) ! N=3 m,n>~3 ~/=3 N=3 m+n=N+2

= 0 .

This proves the gauge invariance of the action.

- - { w " - ' c ; [ WU-2A]}

(14)

The classical equations of motion derived from the action (10) have the form

gN-2 Q B b j P I 7 " ) + N=3 ~ (N--1)~ [t/'tN-l]=0"

We shall now show that these equations imply that

(O.)~=O,

where (~B is a new operator in the Hilbert space .g of the combined ghost-matter CFT ~5, given by

gN--2 gN--2 Q~bff PIA)=Q.bff PIA) + N=3 ~ (U-2)~ [ ~N-2A]=-bff PQBIA) + N = 3 ~ (N-Z) ~ b f f Pcff[

(15)

(16)

(17)

for any state IA ) in ~g.

~5 Actually QB is not defined on the full Hilbert space ,g, but on the subspace defined by (Lo-/So) I ) =0=66- I ).

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Volume 241, number 3

Proof From eq. ( 17 ) we get

gN--2 (OB)2b°p[A>= N=3 ~" (N-2)!

PHYSICS LETTERS B

gN--2 - - QB[ ~'IN-2A] - - N=3 ~ (N--2)! - - [ ~N-Z(QBA) ]

17May 1990

gM+N--4 + M=3 ~ N=3 ~ ( M - - Z ) ! ( N - - 2 ) ! [~-'tM--2CO[~JN--2A]]' (18)

where we have used the equations (QB)2=0, {QB, b~- }P=0 and {cff, bg } = 1. Using eq. (8), we get

gN-2 gN--2 (O-")2bff PIA>= N=3 ~ (U--2)~ [ ~N-2Q"A]+ N=3 ~ (U-2)~ (u - z ) [ (Q"ga) gaN-3A]

gN--2 ( ( N - 2 ) ! - u=3 ~ (N--2)! ,,,m~3~ (m--2)!(n--2)! [gj'-zcff[~"-zAll

m+n=N+2

( N - 2 ) ! ) + m.n>~3~ ( m - 3 ) ! ( n - 1 ) ! [gjm-3cff[g"-']A]

m+n=N+2

- ~ [ ~ N - 2 ( Q B A ) ] + u=3 (N--2) w. M=3 u=3 (M--2)!(N--2)! [ ~uM-Zcff [ ~pN-2A ] ]

gU--2 gU--2 ~--- N=3 ~ (N-3) -~--~.[(QS~)~N-3A]- N=3 ~ ,mn>~3~ (m_3)!(n_l)![~"-3cff[ga"-']A]. (19)

m+n=N+2

If we now use eq. (15), we may write

g,~/-- 2 [(QB~)gjN-3A]= ~ (m-1 )~ [cff[ ~M-~] ~.tU-3A] . (20)

M= 3

Substituting this on the right-hand side ofeq. ( 19 ), we get

(O.B)2bffPIA) = 0 . (21)

This proves the desired result.

In order to compare the answer with the corresponding result for open string field theory, let us note that the action for open string is given by

S(~)=½(gJ lQul ~ ) + ~ J ~ , ~ , ~ , (22)

where the precise definition of * and f has been given for the flat space-time background in ref. [8 ], and an arbitrary background in ref. [ 11 ]. The action is invariant under the gauge transformation

~ = QB ~ - A • }/-I+ ~ , A . (23)

The equations of motion are

QB 7*+ ~ , g-'= 0. (24)

If we define

O_.,A=QuA+ 7.',A- ( - 1 )"~A, ~, (25)

where n~ is the ghost number of the state I A >, then eq. (24) implies that [ 8 ]

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Volume 241, number 3 PHYSICS LETTERS B 17 May 1990

(Qa)2=0 . (26)

Before we conclude let us note the following two facts which support the hypothesis that Qa defined in this way is indeed the BRST charge of the two dimensional field theory in the presence of the background string field ~/~:6.

(i) Note that the gauge transformation law eq. ( 11 ) [ eq. (23 ) for open string theory ] may be written as

3(bo PT) = O_B (bg PA) for closed strings,

c~T=Q_BA for open strings, (27)

where QB is given by eq. (17) [ (25) for open strings]. This has the following consequence. Let Td be a back- ground string field satisfying the equations of motion. Then if we quantize the theory around this new back- ground, we can write

~=~c,+~, (28)

and treat ~ as the new dynamical field. Eq. (27) shows that the gauge transformation law of ~P may be written as

a(bo-Pg')-a(bo-PW) =O_.~°)(bKPA)+O(~') for closed strings,

6tP-aT=O_.~°)A+O(tP) foropen strings, (29)

where

O.(s °) = Ou( T= ga~,) . (30)

From eqs. (11), (23) we see that this is precisely what we expect if 01~ °) is the BRST operator of the two dimensional field theory corresponding to the background T¢1.

(ii) It can he easily verified that if ~ is defined as in eq. (28), we may express the action as

S ( ~ ) =So + ½ ( ~l ()~o)bo- PI ~ ) +O(~t3) for closed strings,

=So + ½ ( ~'1 (~°) I ~ ) +O(~,3) for open strings, (31)

where So is independent of ~'. A comparison of this result with eqs. (10), (22) again shows that ()~o) may indeed be interpreted as the BRST operator of the two dimensional CFT corresponding to the new background Td. Thus we see that despite the apparent difference in the form of the action for open and closed string field theories, they have many features in common.

Note added. If Tcj is a solution of the equations of motion, then we can define a new composition law

gM-2 [ A I " ' A N - i ] ' = [AI" 'AN-I ]~- A'/= ~ 3 ( M - 2 ) ~ [ Tc~-2A'"'AN-~]

such that the action, expressed in terms of the shifted field ~= T - Td has exactly the same form as given in eq. (10), and ~ u n d e r a gauge transformation has exactly the same form as in eq. ( 11 ) with QB replaced by (2B, T replaced by tfi, and [ ] replaced by [ ] '. Also, (~B and [A~..dx_ 1 ]' satisfy an equation similar to eq. (8) with QB replaced by (~B and [ ] replaced by [ ]'.

~6 Some related results for open string field theory have been discussed in ref. [ 18 ].

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Volume 24 I. number 3 PHYSICS LETTERS B

References

17 May 1990

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