- Email: [email protected]
Weyl and Möbius invariance in the path integral derivation of reggeon vertices
Volume 221, number 1
PHYSICS LETTERSB
20 April 1989
WEYL AND M O B I U S INVARIANCE IN T H E PATH I N T E G R A L DERIVATION OF R E G G E O N VERTICES H. D O R N and H.-J. OTTO Sektion Physik, Humboldt-Universitgit zu Berlin, PF 1297, DDR-1086 Berlin, GDR
Received 11 January 1989
We discuss some issues concerning Weyl and M/Sbius invariance in the functional integral derivation of the N-reggeonvertex both for matter fields and bosonizedghost. The usual recipe of conformaltransport of the metric scale is avoided by the introduction of conformallytransformed normal ordered exponentials describing reggeonemission.
1. Introduction
Weyl invariance or equivalently conformal invariance plays a central role in the first quantized approach to string theory based on the functional integral. In the critical dimension the dependence on the conformal factor p of the 2-dimensional metric disappears on shell for correlation functions of vertex operators. This is directly related to conditions on target space background fields resulting from Weyl invariance [ 1 ]. Taking general covariance for granted, Weyl invariance implies conformal invariance, leading for the simplest case of genuszero topology to M/Sbius invariance (fixed p) with respect to the position of vertex operator insertions, i.e., the Koba-Nielsen points. Hence a priori M6bius invariance is explicit on shell, only. To be complete, we should mention the relation between the M/Sbius gauge fixing necessary on shell and the power-like ultraviolet divergencies of the 2D field theory describing the string in general background fields [2 ]. On the other hand there is the standard operator formalism based on conformal invariance at fixed p yielding dual amplitudes by sewing reggeon vertices. In this approach off-shell M/Sbius invariance is crucial as it seems to be required generally by factorization for off-shell quantities [ 3]. The relation to the Polyakov functional integral has been clarified to a large extent by the authors of refs. [ 4,5 ]. The N-reggeon vertex was shown to be equal to the functional integral evaluated for certain singular reggeon sources and sewing rules motivated by functional integration have been proven to give the correct tree and loop amplitudes. But there is one delicate point in these considerations. The obvious off-shell dependence on the conformal factor p is eliminated and M6bius invariance restored by application of the "transport recipe" p ( z i ) --" I V~' ( 0 ) [ - 2p ( 0 ) , where V~is a M6bius transformation with Vi(0) = zi. The justification of this recipe is difficult to find within the functional integral approach where the string position integral has to be performed at arbitrary but fixed p. To overcome this conceptual difficulty the aim of this paper is to define the N-reggeon vertex at arbitrary p as the expectation value of suitable conformally transformed composite operators of the underlying 2-dimensional field theory. Since the problem under discussion is most striking for the bosonized ghost system where the coupling to a background charge Q = 3 introduces additional p dependences we are going to consider Q # 0. The vertex for one out of 26 = D matter fields is reobtained for Q = 0.
16
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division)
Volume 221, number 1
PHYSICS LETTERS B
20 April 1989
2. Generating functional for reggeon sources
We start with some comments concerning the treatment of zero modes for the Green functions of the bosonized ghost field on a surface with the topology of a sphere. The action is ( a ' = 2, conformal gauge gz~=½p, x/g g = - 40-0= log p) So[~0l= ~
d2z (O~oO=~o+lQix//~R~0).
(1)
The generating functional Z [ j ] is defined by
Z[j]= f ~goexp(_So[~o]+ f d2zp~oj),
(2)
For an arbitrary function h (z, g) we separate zero modes by
ho=A-' ~ d2zph,
h=ho+/~,
A= f d2zp.
(3)
Then we have
f d2zp~°j=Atp°J°+ f d2zPO)'"
(4)
The Green function is defined by
zr-'0_-O~G(z,~]p) =
- 8 (2) (z-z'
) +A -lp(z) ,
(5)
with the solution
G(z, z' [ p ) = - l o g l z-z'
]2+ f ( z ) + f ( z ' ) - f o ,
(6)
where i
f(z) =A -i f
d2z ,, p(z") l o g [ z - z " [2
(7)
The z independent integration constant has been choosen in a way to ensure the following crucial properties of G:
G(z, z' IP) = G(7(z),
?(z')I/~)
(8)
for a M6bius transformation 7 (z) and/~ (y ( z ) ) = [~' (z) [ - 2p (z),
G(z, z' iP) =G(z', zip)
(9)
as well as
f dZzp(z)G(z, z' [p)
(10)
Writing ~ 0 = d~0o ~ 0 and performing the trivial ~0ointegral one gets
,
)01 +
The calculation of the (? integral via the usual shift procedure has to be done preserving the orthogonality to the zero modes. The shift
(o(z) ~(o(z) + f d2z'p(z ' )G(z, z' )7(z' ) 17
Volume 221, number 1
PHYSICSLETTERSB
20 April 1989
fulfils this requirement due to the projector property (10). Hence
Z[j]=~(Q+iAj~)N~exp{ ~ f d2zd2z~ p(z)p(z') [f(z)G(z~z')f(z')--~(R(z)- ~)G(z~z')f(z')~} (11) with
No= f
~0exp(-So[~] ) .
(12)
We prefer to split offN o instead of No=o since it contains the contribution proportional t o Q2 to the conformal anomaly of the bosonized ghost system [6 ]. As a preparation for the definition of normal products we calculate the one- and two-point function of ~0 (z). From ( 11 ) and 8
~j(z)
8 -
~)'(z)
-
p(z) f d2z, 8j"(z'8----~ A -
(13)
-
we find
+iQd(Q)O(z) ] ,
<~0(z) > : N o [ i d ' (Q)
<0(z) > :iNoQO(z),
(14)
< ~o(z, )~o(z2) > =d(Q) - N o { [~(z~ ) +O(z2) ]Qd' (Q) +~" ( Q ) } , < 0(z, ) O(z2 ) > = N o [
G(z,, z2) -Q2(~(Zl )0(z2) ] •
( 15 )
The function O(z) is defined by 0(z) = - ½[logp(z) - (logp)o] - [f(z) - f o l •
(16)
It is related to G via 0(z)=-8--~
dZz'p(z')R(z ')G(z,z').
(17)
We want to point out that ~ acquires a vacuum expectation value and that G (z 1, z2 ) defined in ( 6 ) is indeed the connected part of the two-point function. Let us now specify the sourcej to the reggeon case. We follow the line of reasoning ofrefs. [4,5 ] but extend it to the case of general p. Defining the differential operator (D~i, Dei covariant derivatives at z,, a }n), ~ }n) tensors at zi) L,(z,)=ki+ f
1
~ [a}')(D~,)"+a}~)(De,)~l
(18)
and considering a source term j = Z~j, with
jl(z) = iLl(z/) d ( 2 ) ( z - z t ) p(z)
we get from ( 1 1 ), (17)
18
(19)
Volume 221, number 1
PHYSICS LETTERS B
20 April 1989
3. Conformal transport of narmal ordered exponentials Due to the 6(Q), 6' (Q) .... factors in the n-point Green functions of ~, which sum up to a 6-function with shifted argument for exponentials of ~, one should define normal products after separation of the zero mode. For
~b=O-iQO(z)
(21)
we have from (14), ( 15 )
(0)=0,
(,(O(Zl)(O(z2))=NQG(Z,,z2) .
(22)
Defining as usual :~b(Zl )~b(z2): = ~b(zl )0(z2)
--G(zl, z2) ,
(23)
Wick's theorem yields for a linear differential operator L (z) exp[iL(z)O(z) ] = exp[ =exp[ -
IL(z)G(z,
-QL(z)~(z)]
t
exp[ - ½ L ( z ) G ( z , z ) L ( z ) ] :exp [iL(z)~b(z)]:
z ) L ( z ) ] :exp [iL(z)~b(z) ]:
(24)
Together with the convention :exp (iL~o): = exp (ik~0o) :exp (iL0): and (20) this gives
( I~ :exp[iLt(z,)~o(z,) ]:) =NQ6(~k,-Q) exp(-½i~jL,(zi)Lj(zj)G(zi, zj)-Q~Li(zi)fb(z,) ) .
(25)
As a side-remark we emphasize the necessity of keeping track o f f ( z ) also to get the correct covariant transformation properties of the bosonized ghost correlators [6]. Eq. (25 ) is invariant under M6bius transformation zi--, y (zi) only if there is an accompanying transformation of the metric as it arises in considering it as a diffeomorphism. To get Mtibius invariance for fixed p we have to look for the expectation value of some modified operators. To this purpose we now define the conformal transformation of :exp [ iL (z) ~o(z) ]: in obvious generalization of the well-known case L (z) = k as (7 :exp (iL~0):) (z) = B ( z ) :exp [iL(z)~0(y -1 ( z ) ) ]: . The factor B(z) can be fixed by requiring conformal invariance of the vacuum and of all expectation values of the operator products under discussion. We find
(Q-k)L(z) ½log p ( z ) +f(x-l(z) ) - f ( z ) ] }.
(y :exp(iL~0):) (z) = :exp [iL(z)~0(7- I ( z ) ) ]: e x p { - Q L ( z ) [½ log p ( y - 1( z ) ) -
[log[ (X-~)' (z) l -f(y-~(z)
) +f(z) ] (26)
For comparison with the conformal transform of the usual ground state vertex operator we write down the flat limit explicitly (7 :exp(iL~0):) (z) = I (7-1) ' (z)l -ktQ-k~
×exp(-(Q-k)~l(o~")O~"'+6L~"'O'~)logl(y-1)'(z)l):exp[iL(z)~o(x-t(z))]:..
(27)
After these preparations we consider ( 1-Is [ ( V f ~ :exp (iL/p): ) (0) ] ). The M6bius transformations Vi are constrained by V, (0) = zi. 19
Volume 221, number 1
PHYSICS LETTERS B
20 April 1989
Using (D~)'=p'(p-'O~)"
(28)
which is valid for (D~) n acting on a scalar and making use of momentum conservation 5~ki= Q we arrive at
( l-J[(VT':exp(iL/p):)(O)])=Nod( ~ k i - Q ) ×exP~i.~j
I½kikjN°°+~mki(°t)m)N°m+6zJm'lv°m)+½~m. (Og(in)OgJ(m)N~m"l-6l(in)~)rn)lv~m)lexp n ( ~iEi) (29)
with N °° =log[
Iz~-zjl21V'i(0)
I - ' I V~(0) I - ' ] ,
1 N°'"=-~.. [p(w)]m{ [p(w)]- , Ow}mlog[Iz~--Vj(w)I21V~(w)I--1]I~=o, 1
Ng'" = n!m!p"(w~) [p-~ (w,)Ow,]"pm(wj)[p-~ (wj) 0~j] m logl V,(w~) -
Vj(wj)I z ,
w,=wj=O,
ei =k,L,(w)f(w) + ½QL~(w) log p(w) + ½k~(Q-k~)fo - Qkj[ ½(logp)o +fo] I ~=o • As usual mixed a, (e terms vanish for zi# zj. The second exponential is independent
of the Koba-Nielsen points z~, it factorizes and can be eliminated by a suitable normalization of the vertex operator ( Vf ~:exp (iLj~0):) (0). Eq. (29) is our main result. The expression is by construction M6bius invariant (V,--,TV~). It still depends on the scale of the metric and its derivatives to arbitrary order at the reference point z = 0 . This dependence drops after choosing the gauge of ref. [7]: Ozp-0~p = 0 at z = 0. Then the coefficients multiplying k~kj, k~oz)m) and o~}")a) "~ are just the standard ones of the N-reggeon vertex. This is independent of the value of Q and in complete agreement with refs. [ 8-10 ] based on conformal field theory in the fiat case with a background charge at oo. n
m
n
Acknowledgement H.D. Thanks P. Di Vecchia for a stimulating discussion.
References [ 1 ] C.G. Callan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl. Phys. B 262 (1985) 593. [2] A.A. Tseytlin, Phys. Lett. B 208 (1988) 221. [3] T. Kubota and G. Veneziano, Phys. Lett. B 207 (1988) 419. [4] P.Di Vecchia, R. Nakayama, J. L. Petersen, I.R. Sidenius and S. Sciuto, Nucl. Phys. B 287 (1987) 621. [ 5 ] J.L. Petersen and J.R. Sidenius, Nucl. Phys. B 301 ( 1988 ) 247. [6] H. Dorn and H.-J. Otto, in: Proc. 22 Intern. Symp. on the Theory of elementary Particles (Ahrenshoop, 1988) (AdW BerlinZeuthen, PHE 1989), to appear. [7] J. Polchinski, Nucl. Phys. B 307 (1988) 61. [8 ] A. D'Adda, M.A. Rego Monteiro and S. Sciuto, Nucl. PhT~. B 294 (1987) 573. [ 9 ] U. Carow-Watamura and S. Watamura, Nucl. Phys. B 302 ( 1988 ) 149. [ 10] A. Le Clair, M.E. Peskin and C.R. Preitschopf, String field theory on the conformal plane, I, II, preprints SLAC-PUB 4306/4307 (1988).
20
PHYSICS LETTERSB
20 April 1989
WEYL AND M O B I U S INVARIANCE IN T H E PATH I N T E G R A L DERIVATION OF R E G G E O N VERTICES H. D O R N and H.-J. OTTO Sektion Physik, Humboldt-Universitgit zu Berlin, PF 1297, DDR-1086 Berlin, GDR
Received 11 January 1989
We discuss some issues concerning Weyl and M/Sbius invariance in the functional integral derivation of the N-reggeonvertex both for matter fields and bosonizedghost. The usual recipe of conformaltransport of the metric scale is avoided by the introduction of conformallytransformed normal ordered exponentials describing reggeonemission.
1. Introduction
Weyl invariance or equivalently conformal invariance plays a central role in the first quantized approach to string theory based on the functional integral. In the critical dimension the dependence on the conformal factor p of the 2-dimensional metric disappears on shell for correlation functions of vertex operators. This is directly related to conditions on target space background fields resulting from Weyl invariance [ 1 ]. Taking general covariance for granted, Weyl invariance implies conformal invariance, leading for the simplest case of genuszero topology to M/Sbius invariance (fixed p) with respect to the position of vertex operator insertions, i.e., the Koba-Nielsen points. Hence a priori M6bius invariance is explicit on shell, only. To be complete, we should mention the relation between the M/Sbius gauge fixing necessary on shell and the power-like ultraviolet divergencies of the 2D field theory describing the string in general background fields [2 ]. On the other hand there is the standard operator formalism based on conformal invariance at fixed p yielding dual amplitudes by sewing reggeon vertices. In this approach off-shell M/Sbius invariance is crucial as it seems to be required generally by factorization for off-shell quantities [ 3]. The relation to the Polyakov functional integral has been clarified to a large extent by the authors of refs. [ 4,5 ]. The N-reggeon vertex was shown to be equal to the functional integral evaluated for certain singular reggeon sources and sewing rules motivated by functional integration have been proven to give the correct tree and loop amplitudes. But there is one delicate point in these considerations. The obvious off-shell dependence on the conformal factor p is eliminated and M6bius invariance restored by application of the "transport recipe" p ( z i ) --" I V~' ( 0 ) [ - 2p ( 0 ) , where V~is a M6bius transformation with Vi(0) = zi. The justification of this recipe is difficult to find within the functional integral approach where the string position integral has to be performed at arbitrary but fixed p. To overcome this conceptual difficulty the aim of this paper is to define the N-reggeon vertex at arbitrary p as the expectation value of suitable conformally transformed composite operators of the underlying 2-dimensional field theory. Since the problem under discussion is most striking for the bosonized ghost system where the coupling to a background charge Q = 3 introduces additional p dependences we are going to consider Q # 0. The vertex for one out of 26 = D matter fields is reobtained for Q = 0.
16
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division)
Volume 221, number 1
PHYSICS LETTERS B
20 April 1989
2. Generating functional for reggeon sources
We start with some comments concerning the treatment of zero modes for the Green functions of the bosonized ghost field on a surface with the topology of a sphere. The action is ( a ' = 2, conformal gauge gz~=½p, x/g g = - 40-0= log p) So[~0l= ~
d2z (O~oO=~o+lQix//~R~0).
(1)
The generating functional Z [ j ] is defined by
Z[j]= f ~goexp(_So[~o]+ f d2zp~oj),
(2)
For an arbitrary function h (z, g) we separate zero modes by
ho=A-' ~ d2zph,
h=ho+/~,
A= f d2zp.
(3)
Then we have
f d2zp~°j=Atp°J°+ f d2zPO)'"
(4)
The Green function is defined by
zr-'0_-O~G(z,~]p) =
- 8 (2) (z-z'
) +A -lp(z) ,
(5)
with the solution
G(z, z' [ p ) = - l o g l z-z'
]2+ f ( z ) + f ( z ' ) - f o ,
(6)
where i
f(z) =A -i f
d2z ,, p(z") l o g [ z - z " [2
(7)
The z independent integration constant has been choosen in a way to ensure the following crucial properties of G:
G(z, z' IP) = G(7(z),
?(z')I/~)
(8)
for a M6bius transformation 7 (z) and/~ (y ( z ) ) = [~' (z) [ - 2p (z),
G(z, z' iP) =G(z', zip)
(9)
as well as
f dZzp(z)G(z, z' [p)
(10)
Writing ~ 0 = d~0o ~ 0 and performing the trivial ~0ointegral one gets
,
)01 +
The calculation of the (? integral via the usual shift procedure has to be done preserving the orthogonality to the zero modes. The shift
(o(z) ~(o(z) + f d2z'p(z ' )G(z, z' )7(z' ) 17
Volume 221, number 1
PHYSICSLETTERSB
20 April 1989
fulfils this requirement due to the projector property (10). Hence
Z[j]=~(Q+iAj~)N~exp{ ~ f d2zd2z~ p(z)p(z') [f(z)G(z~z')f(z')--~(R(z)- ~)G(z~z')f(z')~} (11) with
No= f
~0exp(-So[~] ) .
(12)
We prefer to split offN o instead of No=o since it contains the contribution proportional t o Q2 to the conformal anomaly of the bosonized ghost system [6 ]. As a preparation for the definition of normal products we calculate the one- and two-point function of ~0 (z). From ( 11 ) and 8
~j(z)
8 -
~)'(z)
-
p(z) f d2z, 8j"(z'8----~ A -
(13)
-
we find
+iQd(Q)O(z) ] ,
<~0(z) > : N o [ i d ' (Q)
<0(z) > :iNoQO(z),
(14)
< ~o(z, )~o(z2) > =
G(z,, z2) -Q2(~(Zl )0(z2) ] •
( 15 )
The function O(z) is defined by 0(z) = - ½[logp(z) - (logp)o] - [f(z) - f o l •
(16)
It is related to G via 0(z)=-8--~
dZz'p(z')R(z ')G(z,z').
(17)
We want to point out that ~ acquires a vacuum expectation value and that G (z 1, z2 ) defined in ( 6 ) is indeed the connected part of the two-point function. Let us now specify the sourcej to the reggeon case. We follow the line of reasoning ofrefs. [4,5 ] but extend it to the case of general p. Defining the differential operator (D~i, Dei covariant derivatives at z,, a }n), ~ }n) tensors at zi) L,(z,)=ki+ f
1
~ [a}')(D~,)"+a}~)(De,)~l
(18)
and considering a source term j = Z~j, with
jl(z) = iLl(z/) d ( 2 ) ( z - z t ) p(z)
we get from ( 1 1 ), (17)
18
(19)
Volume 221, number 1
PHYSICS LETTERS B
20 April 1989
3. Conformal transport of narmal ordered exponentials Due to the 6(Q), 6' (Q) .... factors in the n-point Green functions of ~, which sum up to a 6-function with shifted argument for exponentials of ~, one should define normal products after separation of the zero mode. For
~b=O-iQO(z)
(21)
we have from (14), ( 15 )
(0)=0,
(,(O(Zl)(O(z2))=NQG(Z,,z2) .
(22)
Defining as usual :~b(Zl )~b(z2): = ~b(zl )0(z2)
--G(zl, z2) ,
(23)
Wick's theorem yields for a linear differential operator L (z) exp[iL(z)O(z) ] = exp[ =exp[ -
IL(z)G(z,
-QL(z)~(z)]
t
exp[ - ½ L ( z ) G ( z , z ) L ( z ) ] :exp [iL(z)~b(z)]:
z ) L ( z ) ] :exp [iL(z)~b(z) ]:
(24)
Together with the convention :exp (iL~o): = exp (ik~0o) :exp (iL0): and (20) this gives
( I~ :exp[iLt(z,)~o(z,) ]:) =NQ6(~k,-Q) exp(-½i~jL,(zi)Lj(zj)G(zi, zj)-Q~Li(zi)fb(z,) ) .
(25)
As a side-remark we emphasize the necessity of keeping track o f f ( z ) also to get the correct covariant transformation properties of the bosonized ghost correlators [6]. Eq. (25 ) is invariant under M6bius transformation zi--, y (zi) only if there is an accompanying transformation of the metric as it arises in considering it as a diffeomorphism. To get Mtibius invariance for fixed p we have to look for the expectation value of some modified operators. To this purpose we now define the conformal transformation of :exp [ iL (z) ~o(z) ]: in obvious generalization of the well-known case L (z) = k as (7 :exp (iL~0):) (z) = B ( z ) :exp [iL(z)~0(y -1 ( z ) ) ]: . The factor B(z) can be fixed by requiring conformal invariance of the vacuum and of all expectation values of the operator products under discussion. We find
(Q-k)L(z) ½log p ( z ) +f(x-l(z) ) - f ( z ) ] }.
(y :exp(iL~0):) (z) = :exp [iL(z)~0(7- I ( z ) ) ]: e x p { - Q L ( z ) [½ log p ( y - 1( z ) ) -
[log[ (X-~)' (z) l -f(y-~(z)
) +f(z) ] (26)
For comparison with the conformal transform of the usual ground state vertex operator we write down the flat limit explicitly (7 :exp(iL~0):) (z) = I (7-1) ' (z)l -ktQ-k~
×exp(-(Q-k)~l(o~")O~"'+6L~"'O'~)logl(y-1)'(z)l):exp[iL(z)~o(x-t(z))]:..
(27)
After these preparations we consider ( 1-Is [ ( V f ~ :exp (iL/p): ) (0) ] ). The M6bius transformations Vi are constrained by V, (0) = zi. 19
Volume 221, number 1
PHYSICS LETTERS B
20 April 1989
Using (D~)'=p'(p-'O~)"
(28)
which is valid for (D~) n acting on a scalar and making use of momentum conservation 5~ki= Q we arrive at
( l-J[(VT':exp(iL/p):)(O)])=Nod( ~ k i - Q ) ×exP~i.~j
I½kikjN°°+~mki(°t)m)N°m+6zJm'lv°m)+½~m. (Og(in)OgJ(m)N~m"l-6l(in)~)rn)lv~m)lexp n ( ~iEi) (29)
with N °° =log[
Iz~-zjl21V'i(0)
I - ' I V~(0) I - ' ] ,
1 N°'"=-~.. [p(w)]m{ [p(w)]- , Ow}mlog[Iz~--Vj(w)I21V~(w)I--1]I~=o, 1
Ng'" = n!m!p"(w~) [p-~ (w,)Ow,]"pm(wj)[p-~ (wj) 0~j] m logl V,(w~) -
Vj(wj)I z ,
w,=wj=O,
ei =k,L,(w)f(w) + ½QL~(w) log p(w) + ½k~(Q-k~)fo - Qkj[ ½(logp)o +fo] I ~=o • As usual mixed a, (e terms vanish for zi# zj. The second exponential is independent
of the Koba-Nielsen points z~, it factorizes and can be eliminated by a suitable normalization of the vertex operator ( Vf ~:exp (iLj~0):) (0). Eq. (29) is our main result. The expression is by construction M6bius invariant (V,--,TV~). It still depends on the scale of the metric and its derivatives to arbitrary order at the reference point z = 0 . This dependence drops after choosing the gauge of ref. [7]: Ozp-0~p = 0 at z = 0. Then the coefficients multiplying k~kj, k~oz)m) and o~}")a) "~ are just the standard ones of the N-reggeon vertex. This is independent of the value of Q and in complete agreement with refs. [ 8-10 ] based on conformal field theory in the fiat case with a background charge at oo. n
m
n
Acknowledgement H.D. Thanks P. Di Vecchia for a stimulating discussion.
References [ 1 ] C.G. Callan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl. Phys. B 262 (1985) 593. [2] A.A. Tseytlin, Phys. Lett. B 208 (1988) 221. [3] T. Kubota and G. Veneziano, Phys. Lett. B 207 (1988) 419. [4] P.Di Vecchia, R. Nakayama, J. L. Petersen, I.R. Sidenius and S. Sciuto, Nucl. Phys. B 287 (1987) 621. [ 5 ] J.L. Petersen and J.R. Sidenius, Nucl. Phys. B 301 ( 1988 ) 247. [6] H. Dorn and H.-J. Otto, in: Proc. 22 Intern. Symp. on the Theory of elementary Particles (Ahrenshoop, 1988) (AdW BerlinZeuthen, PHE 1989), to appear. [7] J. Polchinski, Nucl. Phys. B 307 (1988) 61. [8 ] A. D'Adda, M.A. Rego Monteiro and S. Sciuto, Nucl. PhT~. B 294 (1987) 573. [ 9 ] U. Carow-Watamura and S. Watamura, Nucl. Phys. B 302 ( 1988 ) 149. [ 10] A. Le Clair, M.E. Peskin and C.R. Preitschopf, String field theory on the conformal plane, I, II, preprints SLAC-PUB 4306/4307 (1988).
20