Volume 220, number 3 PHYSICS LETTERS B 6 April 1989 STRING FIELD THEORY FROM WEYL INVARIANCE ~ Sanjay JAIN and A. JEVICKI Department of Physics, Brown University, Providence, RI 02912, USA Received 5 December 1988 We consider the quantization of strings in general massive background fields. Imposing Weyl invariance leads to equations that generalize the fl-function equations of the massless theory. Considering in detail the open string we show how the requirement of Weyl invariance produces the linearized string field theory equations of motion possessinga partial gauge invariance. Weyl anom- aly contributions of the correlation functions are important for this correspondence. We also display how ghost type vertex oper- ators can couple in the background 2D a-model. A fruitful approach to string dynamics has been to study their quantization in the presence of background fields. For massless backgrounds one obtains a 2D a-model with renormalizable interactions. It was found that the demand of conformal invariance leads to backgrounds that satisfy the effective field equations ~ and also relates the a-model string scattering amplitudes [ 2 ]. This means that string classical vacua and on-shell scatter- ing amplitudes are given by the fixed points of the 2D renormalization group (RG). This realization has resulted in attempts to generalize the 2D RG and to include other string modes [ 3-9 ]. In this approach, the a-model with fixed backgrounds is reinterpreted as a point in a dynamical theory space [ 10,5 ], the infinite dimensional space of all possible couplings. The tachyon and massless backgrounds sufficiently close to the mass shell are the relevant and marginal perturbations about the gaussian fixed point in theory space. Therefore, close to this point (and near their mass shell) only these renormalizable interactions need to be considered. Away from it, however, the RG flow produces all other operators to which these modes couple, e.g., all the massive modes, and this necessitates the inclusion of all possible couplings. It is not clear at this moment what one means by the set of all possible couplings or the most general background. Among other things, an ill- understood question is how the 2D RG would give rise to spacetime gauge invariance in the space of couplings. In this letter we consider backgrounds corresponding to all the massive modes of open string theory and use Weyl invariance to derive fixed point equations to first order in the couplings, generalizing the work of ref. [ 4 ]. We obtain agreement with linearized string field theory (SFT) equations. Our equations possess a partial gauge invariance that remains after an incomplete gauge fixing of the SFT equations [ 11 ]. We determine the Weyl anomaly coefficients of the higher derivative 2-point functions required for the correspondence between Weyl invariance and SFT. We also generalize the space of 2D quantum field theories to include operators in which the scalar fields couple nontrivially to the 2D ghost fields. To begin with, we discuss the a-model at the first massive level and using Weyl invariance to obtain the linearized equations of motion of SFT. Subsequently, we generalize to all mass levels. At the first massive level Witten's string field contains five background fields: I ~ = [E~(x)a~- ~ c~~ ~+ iB~(x)ot~_2 + iS~(x)a~ j cob_ ~ - 2~(x)c_ ~ b_ ~ +S(x)cob_2] J - ~ • Q lqb ~ = 0 gives the linearized SFT equations Work supported in part by the Department of Energy, contract DE-AC02-76ER03130.027-Task A. ~ For reviews see ref. [ 1]. 379
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Volume 220, number 3 PHYSICS LETTERS B 6 April 1989
STRING F I E L D T H E O R Y F R O M WEYL INVARIANCE ~
Sanjay JAIN and A. JEVICKI Department o f Physics, Brown University, Providence, RI 02912, USA
Received 5 December 1988
We consider the quantization of strings in general massive background fields. Imposing Weyl invariance leads to equations that generalize the fl-function equations of the massless theory. Considering in detail the open string we show how the requirement of Weyl invariance produces the linearized string field theory equations of motion possessing a partial gauge invariance. Weyl anom- aly contributions of the correlation functions are important for this correspondence. We also display how ghost type vertex oper- ators can couple in the background 2D a-model.
A fruitful approach to string dynamics has been to study their quantization in the presence of background fields. For massless backgrounds one obtains a 2D a-model with renormalizable interactions. It was found that the demand of conformal invariance leads to backgrounds that satisfy the effective field equations ~ and also relates the a-model string scattering amplitudes [ 2 ]. This means that string classical vacua and on-shell scatter- ing amplitudes are given by the fixed points of the 2D renormalization group (RG) .
This realization has resulted in attempts to generalize the 2D RG and to include other string modes [ 3-9 ]. In this approach, the a-model with fixed backgrounds is reinterpreted as a point in a dynamical theory space [ 10,5 ], the infinite dimensional space of all possible couplings. The tachyon and massless backgrounds sufficiently close to the mass shell are the relevant and marginal perturbations about the gaussian fixed point in theory space. Therefore, close to this point (and near their mass shell) only these renormalizable interactions need to be considered. Away from it, however, the RG flow produces all other operators to which these modes couple, e.g., all the massive modes, and this necessitates the inclusion of all possible couplings. It is not clear at this moment what one means by the set o f all possible couplings or the most general background. Among other things, an ill- understood question is how the 2D RG would give rise to spacetime gauge invariance in the space of couplings.
In this letter we consider backgrounds corresponding to all the massive modes of open string theory and use Weyl invariance to derive fixed point equations to first order in the couplings, generalizing the work of ref. [ 4 ]. We obtain agreement with linearized string field theory (SFT) equations. Our equations possess a partial gauge invariance that remains after an incomplete gauge fixing of the SFT equations [ 11 ]. We determine the Weyl anomaly coefficients of the higher derivative 2-point functions required for the correspondence between Weyl invariance and SFT. We also generalize the space of 2D quantum field theories to include operators in which the scalar fields couple nontrivially to the 2D ghost fields.
To begin with, we discuss the a-model at the first massive level and using Weyl invariance to obtain the linearized equations of mot ion of SFT. Subsequently, we generalize to all mass levels.
At the first massive level Witten 's string field contains five background fields:
I ~ = [ E ~ ( x ) a ~ - ~ c~ ~ ~ + iB~(x)o t~_2 + iS~(x)a~ j cob_ ~ - 2 ~ ( x ) c _ ~ b_ ~ + S ( x ) c o b _ 2 ] J - ~ •
Q lqb ~ = 0 gives the linearized SFT equations
Work supported in part by the Department of Energy, contract DE-AC02-76ER03130.027-Task A. ~ For reviews see ref. [ 1 ].
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The last of these equations is an identity that follows from the first four. These equations possess the gauge invariances
8Eu,=--r/~E, 8Bu=20~e, 8S~,=0, 6q/=3~, 8 S = ( D - 2 ) e , ( l a )
8E~=0(uE,) , 8Bu=2e u, 8 S ~ = - ( [ ~ - 2 ) ~ u, 8~=-0~Eu , 8 S = 0 . ( lb )
We can obtain field equations with a smaller gauge invariance by partially fixing the gauge S = 0 using the first of these transformations. The equations then become
( [-']- 2 )E.,, + O(,,Su) = 0 ,
( I-']-2 )B¢, + 2Su=O ,
O~E~ -B~ + S~ + Ouq/=O ,
~EPpT ~OPBp-F~=O .
(2a)
(2b)
(2c)
(2d)
Eqs. (2) have a gauge invariance only under ( lb ) . These are the equations we shall reproduce using Weyl invariance.
The bare a-model action that we start from is S= So + SB, with So (x) = ( 1 /4na ' )fd2~ x/~g~PO,~xUO~x,, and
e l d d SB((~,x)= f dt e [ l ~c~l ic~Eu,(X)+ e-~tt ( l~t t x )Bu(x) ] . (3)
In our notation gap (~) = exp [ 20 (~) ] 5,~a, ~= (~t, ~2 ) spans the upper half-plane - oo < ~1 = t < + oo, 0 ~< ~2 < + oo, and e(t) =exp [O(t, 0) ] is the einbein on the boundary.
The effective action to first order in the weak field expansion is given by F(O, X ) = [ ( 2 6 - D ) / 48tC]SL(0) +So(X) + Fa ( O, X) , where SL is the Liouville action, FB ( O, X) = ( SB ( ~, X + x ) ) + Sc~ ( O, X) , and Set contains counterterms to cancel the divergences in (SB). The equations of motion are obtained by setting 8FB, the variation of FB under a Weyl transformation 0--'0 + 80, equal to a total derivative,
8YB (0, x) + A(O, X) =0 . (4)
The total derivative term is A(O, X) = ( A(O, x + X ) ), where, for the first massive level, the choice is
A(0, x ) = f d t e l d ( s c ) l : f u X u ( x ) + 1 8 ~ ( x ) ) " e (5)
The LHS ofeq. (4) is evaluated straightforwardly after expanding the backgrounds in terms of Fourier modes. The result is
f dnk f dt e x p [ - (1 + ½kZ)0+ ik' X] (AS0+ BS~+ C8~'+ D~80+ E~'80 + F ~ 8 0 + G~6~) = 0, (6)
with the coefficients A to G which contain the background fields given by
The numbers c~ and fl are the Weyl coefficients appearing in the coincident 2-point function ( . fu( t ) • v
Xx (t))Odepe,de,,part -- C(r/U"(C~¢'+fl¢¢). In (7) we have set 2oF = 1. o~ and flcan be computed in the 2D field theory by choosing an appropriate regularization scheme• We shall see that they are also fixed completely by string field theory.
Weyl invariance implies that each of the coefficients A - G is zero Let us concentrate on A, B, and C, the coefficients of the "linear" pieces 8¢, 8~, and 8¢" in eq. (6). Setting A equal to zero gives (2a) and (2b), B = 0 gives (2c), and C = 0 gives
½¢xEap+ ½ ( 1 - c Q & B a + V = 0 . (8)
This also coincides with (2d) provided ¢x = 1. Thus, the "linear" part of (6) already produces the SFT equa- tions (if 0< = } ). It remains to consider the "nonlinear" terms D, E, F and G in (6). It is found, using (2), that they all vanish provided f l = - ~ and we get no new condition. Therefore we find that provided (c~, f l )= (},
- ~ ), the Weyl invariance condition (4) is equivalent to the partially gauge invariant SFT equations (2). What happens if o~ and fl do not have these values? Using ( 2 a ) - ( 2 c ) and (8), G = 0 reduces to
( c ~ + 2 f l ) ( E P - & B a ) = 0 . Hence if a + 2 f l ¢ 0 , we must have an additional condition on the backgrounds, EP-0ZBp=0 . Equations of this type were obtained in a recent investigation [9 ] of the same problem. This equation cannot be a gauge condition because the transformation ( l b ) leaves EPp-0ZB a invariant; it is rather like an additional constraint on the original configuration. We take c~ + 2fl= 0 which gives agreement with SFT. Our specific values turn out to be self-consistent. It will be seen later that these values are necessary if Weyl transformations are to generate the Virasoro operators L..
Note the nontrivial internal consistency in fixing the value of ¢x. A priori, the coefficients in (8) might have been such that agreement with (2d) was impossible for any value of ¢x. It has turned out that Weyl invariance produces coefficients in (8) in such a way that if one of them, say the coefficient of EPp, is chosen to agree with the SFT equation (2d), the coefficient of OPBp also agrees. A similar consistency is observed in the fixing of ft.
We remark that in (6) one may impose the Weyl invariance condition after shifting the time derivatives from 8¢ on to other factors so that (6) looks like f dt H8¢. If this is done, all terms containing Su and ~u cancel and we get just eqs. (2a) and (2b), with S~ and ~u replaced by their expressions in terms of E~, and B~ as given by eqs. (2c) and (2d), i.e.,
These equations possess the same gauge invariance as (2), namely ( lb ) . To obtain gauge invariance in this formalism it is not necessary to add boundary terms; shifting time derivatives is equivalent. That is not the case on a flat world-sheet.
We now turn to the general case of all mass levels in the a-model. In setting up the formalism we follow the original work of Ichinose and Sakita [4]. The general background coupling is SB = f dt e~J(x(t), x~ (t), x2 (t), ...), where x,,(t) is simply the covariantized version of ( l /n ! )O ' /x ( t ) , x , ( t ) -K , , (O , ( d / d r ) ) x ( t ) , K , - ( 1 / n! ) (V)", V - ( 1/e) (d /d t ) . ~ i s obtained from the Hilbert space state
I'/"}=Y~ ty~,,~ tx~a~' c~ ~ I0} kt I /22. . .1 /k \ 1 - - h i . . . . n k {n}
of string field theory by replacing au_. --,x#.
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For the purpose of obtaining Weyl transformation properties, it is useful to have a momentum variable rep- resentation ~(Po, P~, P2 .... ) for the wave functional ~U(x (t), x~ (t), x2 (t), . . .) . Then the background action reads
SB= f d t e f ~ p e x p ( i ~=oPnXn(t))~(p) , (10)
with X o ( t ) - x ( t ) and ~p=dpodp~dp: .... In this representation, the first few wavefunctions are ( 3 ( p ) - t~(pl )t~(p2)...)
When, e.g., ~lev~ 2, is substituted in (10), we obtain (3). Here i0/0pn plays the role ofxn. In this representation the total derivative term which is a generalization of (5) has a simple form, A = f d t e (1 /e) ( d / d 0 (f ~ P ×exp(iP.X) ~(P)) . Here we have introduced a 27th coordinate, denoted v(t), with v~ ( t ) - -K~v(t )= 80 and
_ /, . p ~I / t . __ c ~ v,,(t)=K,,v(t)=(1/n)K,,_~(80). Thus X~/=(x, , v,,), ;, =(p,,,q,,) and e X=~,,=o(p,,x,, +q,,v,,).~(P) isgiven by
5(qo) ensures that there are no terms oc v ( t ) ~ f 80, and the fact that ~" has only one derivative ensures that we have only terms linear in 80. For example, with ~ (p) = S u (Po) i (0/0pf ) 5 (p), ~2 (P) = 2~(po) 5 (P), A reduces to ( 5 ). Performing the q integration yields
f f ( ~ ( m + l 0 ) A= d te ~pexp( ip . x ) n=l ~ - - V n m ~ = O )pm~m+ l -}-(n+l)Vn+l ~n(P)" (12)
We are now ready to perform the computation of Weyl anomalies and derive the condition that these vanish. Part of our computation is essentially the open string version ofref. [4]. The bare vertex in (10) can be written as the product of the normal ordered vertex and a factor containing the coincident 2-point functions of the x . :
S a = f d te f ~ p : e x p ( i p ' x ) : e x p ( - ½ ~=oPnpm(Xn( t ) xm( t ) ) )~ (P)"
The Weyl variation of this then consists of two parts, one coming from the explicit dependence of the x. on 0, and the other from the dependence of coincident propagators ( ( d m / d t m ) x ( t ) ( d n / d t n ) x ( t ) ) on 0. Consider first the explicit dependence on 0 which is present through the covariant derivative V = ( 1 / e ) ( d / d t ) . It is easy to show from the definition ofx~ that 8x. ( t ) = Z ~= ~ A.mXm ( t ) , A~., = - [ n / ( n - m + 1 ) ] K._m ( ~(9 ) , m= l, 2, ..., n; A.m = 0, m > n. This results in the variation (using a change of variables p,. --, ~ o~= ~ p , (Onto + A ~ m ) ) :
8, SB = f dt e f ~p exp ( ip" x ) ( 8 0 - D Tr A - (pA ) ., O-~m ) q)(P ) .
The first term in the bracket comes from the variation of dt e, the second term - D Tr A from the jacobian of the change of variables, and the last term from the shift in the argument of ~ due to the change in variables. Using the expression for A.m above,
8, S , = dte ~pexp( ip . x ) 80 I + D m=, ~ m + m=, ~ mpm-~--~+Opm] n=ln..l_l 1 ~0) m=, ~ mPn+m ~.t.
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Making the transcription Pm= ( 1 / ~ )ot,~, m = 0, 1, 2, ...; (O/Opm)=- (x/~'/m)oL_,,, m = 1, 2, 3, ... and noting that the Virasoro generators L~ are given by
Z n = ½ :O[_mOLn+m;.-~-~ m - - "Pn+m"F PmPn--m m= -oo m= l OPm m=0
we can recognize in 6~SB the appearance of Lo in the coefficient of 60 and L, in the coefficient of Kn(80). However, Lo, L, do not completely appear; p 2 and ProP,,- m type terms are absent. We shall find this contribution coming from the anomaly present in the correlation functions ( (d"/dt" ) x (t) (d m/dt ra )X ( t ) ) . This anomalous contribution indeed comes in the form
8~S.=-l f dte f ~pexp(ip.x) ~ ~2(x,,(t)x,,,(t))p,,p,,,~(p), 11,111 ~ 0
and the final total Weyl variation 5S~ = ~tSB + 52SB is given by
6SR=fdte~f~pexp(ip'x)I~¢(1-L°)-~ ( 1 )1 n=l ~ - ~ K n ( 8 ¢ ) t ' t 2 1 - Y n ~ l ( p ) , ( 1 3 )
The equation of motion following from Weyl invariance is 8Ss + A = 0. Combining (13) and (12) we get
~ y ~ ÷ g ~ i ( l _ L o ) ~ + L _ l ~ l l + ~ 1 n=l n= l ~ t t T t K n ( ~ ( ~ ) [ - L n ~ f l q - L - l ~ n + l + ( n + l ) ~ , ] = 0 . (15)
In Witten's string field theory if we introduce a truncated set of fields
I q ~ > = l ~ > + ~ C _ n + l b - l ] ~ n ) , (16) n=l
where I ~ ) and I~.) are built out of the ot oscillators only acting on the vacuum I - ) = c l l 0 ) , and demand QI q~) =0 we obtain the equations of motion
( 1 - L o ) ~ + L _ ~ = 0 , -L.~+L_~.+I+(n+I)~.=O, n = l , 2 . . . . . (17)
The Weyl invariance condition ( 15 ) gives (17) provided Y. = 0. Thus with the proviso Y. = 0 we have derived the SFT equations from Weyl invariance.
From (14) we see that Y~=0 is equivalent to the condition that 32 (x,,,x._,.) = [2ot' / (n+ 1 ) ]K.(~q~). This equation can be used to determine the Weyl coefficients (just as for the first massive level case) in the following way: x. - K.x = ( 1 / n! ) exp ( - n¢i) Y ~= ~ bp x ~p), where x ~p) -= (d p/dt p)x (t). The b"p depend upon derivatives of
and satisfy the recursion relations
b " . = l , b"t=b"-'l-(n-1)b"-~t~, n > l ,
b",.=b"-',.-(n-1)b"-'m~+b"-',._,, m=2, 3,..., n - 1. (18)
By 82 (x,, ,x._,~) we mean the variation due only to the dependence of ( x ~P)x ~q)) on q~; we exclude the varia- tion due to the explicit dependence of the b's on q~, which is already taken care of by 6~. Thus ~z <x.x., )=- (1/n!) ( I / m ! ) exp[ - ( n + m ) ~ ] Z~,=~ Zr~=~b"pb"qS(x~P)x~q)). Using this in the equation Y.=0 gives the re- cursion relation
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p - - I g - - I p - - I q - - I ) -- E bPrn~(x(m)X(q)) q- E bqn~(X(P)X(n)) q- E E bPmbqnt~(x(m)x(n)) •
m = l n = l m = l n = l • • • • F
(19)
(19) is valid for p>_-2 and q>~2. For p>~2, q<2, (19) is valid after dropping B and C, for p < 2 , q>~2 after dropping A and C, and for p < 2 , q < 2 but (p, q ) ~ ( 0 , 0) after dropping A, B and C. For p = l = 0 , 5(x( t )x( t ) ) =2a'Sq~(t). This together with (18) can be used to determine the ¢ dependent part (denoted ( ) ~ ) of all the 2-point functions ( x (P)x (q)). We display the first few:
levell : ( . f ( t )x ( t ) )o=a'~ ,
level2: (2 ( t ) x ( t ) )o=o~ ' ( ]~ '+~) , ( , f ( t ) ~ ( t ) ) = a ' ( ] ~ ' - ~ )
There is a closed form expression for the first term in all these correlations, which is linear in 0. Denoting ( x(P)x (q) )u . . . . part = DpqO (p+q), we have Dpq = 2a' [p!q! / (p+ q+ 1 )!].
The SFT equations (17) are obtained from the Weyl invariance condition (15 ) by just looking at the part linear in ¢, i.e., containing only 8~, 80~, etc. (just as for the first massive level where we obtained (2) by looking only at A, B and C). However, ( 15 ) also displays how the terms nonlinear in 0 appear in general and we see that we get no new conditions.
Several comments are relevant concerning these equations and their derivation. First, the Weyl coefficients have been fixed by requiring the L, structure - which is present in the SFT equations - to emerge from the Weyl transformation. We have seen that this completely fixes the coefficients for all the 2-point functions. The coef- ficients obtained are such that the variation of the part linear in ~ itself produces the SFT equations. The non- linear terms in ~ combine in such a way that no new conditions are obtained. Each coefficient appears in a nontrivial internally consistent way (see the discussion for ot below (8)) . Finally, the coefficients obtained in this manner imply the Leibnitz rule for the 2-point functions, i.e. (d/dt)(x(P)x(q))o= (x (p+l )x(q) )o+ (x(P)x (q+~))o. In the 2D field theory the coefficients depend upon the renormalization scheme. It would be interesting to find the scheme which yields the above coefficients (e.g. heat kernel, which has been used in ref. [ 12 ] for closed strings).
Secondly, our procedure yields partial gauge fixed equations (identical equations have been obtained from conformal invariance on a flat world-sheet in ref. [ 7 ] ). The partial gauge condition is evident from the trun- cated field expansion (16). In the ghost sector it omits states containing b_ 2, b_ 3 .... oscillators (e.g. at the first massive level it omits 8¢ob_21 - ) , thus working in the S = 0 gauge), In SFT, these states are responsible for gauge invariance associated with L_ 2, L_ 3, etc. (e.g. at the first massive level eqs. (17) are modified to (1 - Lo ) ~ + L_ 1 ~1 + L_ 2 S = 0 , - - L~ ~ + L_ 1 ~2 "Jr" 2~1 = 0 , - - L 2 ~ q - 3~2 + 4S= 0, which possess the full invariance ( l a ) and ( l b ) ) . It would be of interest to derive the most general gauge invariant equations. They are likely to be associated with a variation more general than Weyl and a candidate is a generalized BRST transformation. This idea was also mentioned recently in ref. [ 13 ] but the mechanism of how this works is unclear.
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To implement a general ized BRST t rans format ion one needs to generalize theory space to include ghost cou- plings. One mot iva t ion for this is the preceding discussion of string field theory gauge invariances. Another reason is the possibi l i ty of nontr iv ia l dynamica l effects on theory space [ 10 ] via the c-theorem. We present here the construct ion of an addi t iona l ghost te rm S g to the a -model lagrangian.
This te rm is most easily constructed using a bosonized ghost y (~) . We note that the ghost number current j ( z ) = bz~c z is p ropor t iona l to Ozy. Hence the o p e r a t o r s j _ n ( 0 " ~ ~o : z - n -
_ z y = z. . . . . jn ), acting on I - ) generate all the ghost states with the same ghost number as I - ) - In the a-model the ope ra to r j_~ is replaced by y~ - K , y . The bosonized ghost is incorpora ted into the a -model as a 27th coordinate . We in t roduce (as for A) X , u = (X~ ,y, ), pM = (p~, q, ). The ghost background act ion is
S g = f d t e ° f c y P e x p ( i P . X ) ~ ( p ) , (21)
where ~P= ~(Po, P~, Pz . . . . qo, ql, q2, ...) and f ~ P = Y~qof dq~ dq2 ... dpo dp~ a l p 2 . . . . The sum over qo is over ghost numbers. Here ~ i s not restricted to a specific form like in ( 11 ). For example, qg=f(po)[i(O/Oq~ ) i (0/0qt ) + i ( 0 / 0q2) ]O(qo)O(q)O(P) gives S~ = f dt eO(e - ~pe- '3)+e -~ ( d / d t ) (e-13~) ) f ( x ) . The ghost act ion s g = s g + S g + S g includes also the KE par t S~, and a par t S~ conta ining addi t ional couplings not related to spacet ime fields, (e.g., the 2D curvature coupling that gives the negative central charge) . The full act ion is
S(q~, x, y) = S o ( x ) + SB(q~, x ) + s g ( 0 , x, y ) . (22)
This is a 2D field theory in which the mat te r field x and the bosonized ghost y are coupled in a nontr iv ia l way. It is of interest to de te rmine the cri t ical proper t ies of this model as well as the flows of couplings and Zamolodch ikov ' s c-function under RG. This work and its general izat ion to the closed string is under progress.
The next quest ion concerns interact ions. In general one can recover the S-matr ix [6,7 ], bu t it is not clear i f an off-shell extension exists. In recent invest igat ions o f t a c h y o n equat ions in the r - func t ion approach problems have appeared at the level of the q~5 in terac t ion [6 ]. It would be of interest to investigate these quest ions in the present approach.
References
[ 1 ] C.G. Callan, TASI Lectures, Brown University ( 1988); S. Jain, Intern. J. Mod. Phys. 3 (1988) 1759, and references therein.
[2] B. Fridling and A. Jevicki, Phys. Lett. B 174 (1986) 75; C. Lovelace, Nucl. Phys. B 273 (1986) 413; A.M. Polyakov, Phys. Scr. T 15 (1987) 191.
[3] S.R. Das and B. Sathiapalan, Phys. Rev. Lett. 56 (1986) 2664. [4] I. Ichinose and B. Sakita, Phys. Lett. B 175 (1986) 423. [5] T. Banks and E. Martinec, Nucl. Phys. B 294 (1987) 733. [ 6 ] H. Ooguri and N. Sakai, Phys. Lett. B 197 ( 1987 ) 109; Tokyo preprint TIT- 123 ( 1987 );
R. Brustein, D. Nemeschansky and S. Yankielowicz, Nucl. Phys. B 301 (1988) 224; 1. Klebanov and L. Susskind, Phys. Lett. B 200 (1988) 446; N. Nakazawa, K. Sakai and J. Soda, Hiroshima preprint RKK 88-16; V. Periwal, Princeton/Santa Barbara preprint ( 1988 ) ; U. Ellwanger and J. Fuchs, Heidelberg preprint HD-THEP-88-10.
[ 7 ] J. Hughes, J. Liu and J. Polchinski, Texas preprint UTTG- 13-88. [8] C.M. Hull and P.K. Townsend, Nucl. Phys. B 274 (1986) 349;
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[ 10 ] A.B. Zamolodchikov, JETP Lett. 43 ( 1986 ) 731; S,R. Das, G. Mandal and S.R. Wadia, Tata preprint TIFR-TH°88/33; C. Vafa, Harvard preprint HUTP-88/A034; D. Gross, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986 ); V. Periwal, Princeton preprint PUPT- 1079 ( 1987 ); D. Friedan, unpublished.
[ 11 ] E. Witten, Nucl. Phys. B 268 (1986) 253; B. Zwiebach, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986 ).
[12] E. D'Hoker and D.H. Phong, Phys. Rev. D 35 (1987) 3890. [ 13] A.N. Redlich, Phys. Lett. B 213 (1988) 285.
