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BRS invariance of the Witten type vertex
Nuclear Physics B289(1987) 127-156 North-Holland, Amsterdam
BRS IN-VARIANCE OF T H E w r I T E N TYPE VERTEX Katsumi ITOH, Kaku OGAWA and KazuhikoSUEHIRO Department of Physics, Kyoto University, Kyoto 606, Japan
Received 10 October 1986
It is proved rigorouslythat the Witten type 3-string vertexis BRS invariant if and only if the space-time dimension is equal to 26. We use the fermionicrepresentation for the reparametrization ghost.
I. Introduction The construction of interacting string field theory amounts to identify the proper interaction vertices between strings. Up to now there are at least two concrete examples of Lorentz covariant and gauge invariant interacting string field theory for open bosonic string [1-3] which are both based on vertices geometric in nature and have some of the algebraic properties in common, but are qualitatively different in many other respects. One approach is a Lorentz covariant extension of light-cone string field theory, whose vertices represent the intuitive joining-splitting type interactions (JS type) between strings [1, 2]. The other is the one introduced by Witten based on the manifestly symmetric decomposition of string surfaces [3]*. The detailed examinations for both of these theories would be desirable for the full understanding of the structures and the possibilities of string field theories. As for the former theory it was proved that the string interaction vertices satisfy the various key algebraic relations, such as the distributive law of the BRS-charge QB and the associative laws [2], governing the gauge properties of the theory. It is also proved that the unphysical freedom of the string-length parameter characteristic to this formulation decouples consistently [4]. Thus a self-consistent unitary field theory of covariant open bosonic string was established and subsequently extended to the closed bosonic string case [5]. On the other hand formal analyses were given by Witten for the latter formulation [3] and more detailed investigations have been developed by various groups * The same type of vertexwas also proposed by T. Goto in Extended pictures of elementaryparticles, lwanami (1978) (in Japanese). 0550-3213/87/$03.50©Elsevier SciencePublishers B.V. (North-Holland Physics Publishing Division)
128
K. ltoh et al. / Witten type vertex
[6-9]. In refs. [7-9] an important advance was made in rewriting the vertex in oscillator modes. (see also ref. [10]). In spite of their efforts, the latter formulation still lacks rigorous proofs for some of the main assumptions, especially for its algebraic axioms (see refs. [3, 7] in this respect). The purpose of the present paper is to give the complete proof of the distributive law of QB with respect to the Witten-type * product: O s ( q ~ , ~ ) = (Osq~), ~/, + (_)letq~, (asko) '
(1.1)
where the * product is defined by [11, 2] ( ~ * '/')[ Z31 = f~tz~l'I'tZE]V[Z2, Z~,23]tdZtdZ21
(1.2)
(see ref. [2] for notation). The distributive law (1.1), which intuitively means the commutativity of QB with the vertex operators, assures directly the following properties at order g level: (i) the closure of the stringy gauge transformation and the nilpotency of the global BRS transformation; (ii) the gauge invariance of the gauge un-fixed action and the BRS invariance of the gauge fixed action. Usually the Faddeev-Popov ghost part of the Witten-type vertex functional V(1, 2, 3) is defined in the bosonized form. However for convenience of the present purpose, we here employ the fermionic representation following Samuel, who also checked (1.1) up to the fourth mass-level [8]. The explicit form of V(1, 2, 3) is given in sect. 2 from the consideration of connection conditions of string coordinates. Then in sect. 3 we prove eq. (1.1) by applying the contour integration method developed in ref. [2], for the case of the JS-type vertex. We thus established the gauge invariance of the Witten-type action at order g level. There remains to be given such a rigorous proof for the O(g 2) gauge invariance as was done in ref. [2] for the JS-type case. Some discussions on this problem and others are given in the final section. The technical details necessary for the proof are given in appendices. 2. Construction of the Witten-type 3-string vertex In this paper we follow the notations of ref. [2] for oscillator expansions of string coordinates and BRS charge and keep the coordinate representations for the zero-mode variables x ~ and ~0: Xt'(o)=
x~ + i £ -(a~-a~-,)cosno
,
n>~l II
1 +~ a"+_(o) ==-P"(o) -7-X'~(o) = ¢ 7 . :_E=~~ e +-'"°' 1
+~ E cn e+-`"°,
1
+¢¢
C+_(°)=-i~r~(°lT-c(°)=-~C-_+(o) = ?(o) -T-ilrc(o ) = v~-
E
e, e -+''°,
(2.1)
K. Itoh et al. / Witten type vertex
1
129
2
Fig. 1. The structure of overlapping 8-functionals in Witten-type 3-string vertex. The strings interact at their mid-points.
and
Q , - - ¼C~ E +
d o C + - A ~ T- 2i--~-o C +I ,
(2.2)
where a ,~, c, and ~, satisfy the following properties*
ag=p"=- - i Ox'--~'
Co = - O?° .
(2.3)
The vacuum state of the oscillators is denoted by 10); (a~, c,, ?,)[0) = 0
(n >I 1).
(2.4)
As was shown in refs. [1, 2], the distributive law (1.1), which assures the gauge invariance or the BRS invariance of the actions at order g, is equivalent to the condition on the 3-string vertex [ V(1, 2, 3)), 3
E Qg')[ V(1,2,3)) = 0.
(2.5)
r=l
The Witten-type 3-string vertex has the following S-functional structure as depicted in fig. 1 [3]: V(1,2, 3) tx
l--I
I-I
~(x(r)(°)-X(r+l)(~-°))
r = 1 , 2 , 3 ~/2~< o~<~r
XC~(c(r)(o) -~ c(r+l)(eff -- 0"))
XS(?(')(o)- ?('+l)(~r- o)), * W e use the metric ~
= d i a g ( - , + , + .... +).
(2.6)
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K. ltoh et al. / Witten type vertex
Fig. 2. The 3-string scattering process,
where we define r + 3 = r for the 3-string indices. The signs of ghost coordinate connection conditions are chosen such that eq. (2.5) holds at least naively by inspecting the expression (2.2) of BRS charge. We express this &functional rigorously with oscillator modes. In order to construct the vertex I V(1, 2, 3)) using the techniques developed by Mandelstam [12], we must find a conformal mapping from the upper half complex plane into the 3-string diagram (fig. 2). However such a mapping necessarily has a cut singularity, the existence of which causes some complications in the proof in sect. 3. Gross, Jevicki [7] and Samuel [8] avoided this singularity by making use of two copies for each string and considering the conformal mapping from the upper half plane into the 6-string diagram (the p-plane) (fig. 3). By choosing a special gauge for Koba-Nielsen variables [13], this mapping is given by [7, 8]
z(z2- 3) p = ~. aRln(z-- Z R ) - - l n 3 = l n
3-797_1
,
R~I oq = or3 = 0~5 = 1,
0/2 = 0/4 = 0~6 =
z2= , Z4=
-~-3 ,
-1,
z,=o,
Z5-B-- - v @ - ,
Z6=~.
(2.7)
We use capital letters R, S... for the 6-string indices and lower case letters r, s... for the original 3-string indices. In each string strip in the p-plane we introduce a complex coordinates ~R: ~R = ~R + iOR
( ~R < O,
0 <~ 0 R <~¢t ) , R-1
P = aR~R + iflR,
fir = Ir ~
c~s.
(2.8)
S=I
Note that the interaction point z o determined by do ~=~o (z2 + 1)2 dz = (zg-3)(zZ-1)Zo
=0
(2.9)
K. Itoh et al. / Witten type vertex
131
(a)
o1=01 X
I
jo.2=ar
°3== I
IO4=o
a
3
J5
I
t,Az tlo'4=~r
t73=OI
I %=~Itzxt A
%=ol
a
l°e=° 1~6~n
(b) Fig. 3. (a) The 6-string diagram in the p-plane. (b) The connection conditions of the 6-string diagram (a). Each pair of segments labelled X, Y or Z is to be identified. These diagrams should be placed on the top of each other in the p-plane to recover the diagram (a).
is equal to i and Oo - P ( Z o ) = ½rri.
(2.10)
F u r t h e r for later convenience we define a complex w-plane connected to the z-plane via a projective transformation [7] z-i
w=
.
z+i
(2.11)
By this transformation the upper half plane is m a p p e d into the unit disk and the interaction point z o = i to the origin (fig. 4). In terms of w, p can be written as W3+1 p=lnw3_l
6 ½7ri= E ~Rln( w - W R ) - ½ ~ r i ,
w . = w ( Z ~ ) = e-"~"/~
R=I
(R = 1 - e).
(2.a2)
132
K. ltoh et a L /
Witten type vertex
Iw
Fig. 4. The 6-string diagram in the w-plane. The circle around w o = 0 indicates the contour Cwo of the integration (3.6).
Using the Fourier components of the Neumann function (see appendix A), we can write the 6-string vertex corresponding to fig. 3 as l V(6)) = (2qr)d~ a ~=1 f i r ~
expE(l_6)10 )
-"
6
e(1 - 6) =
E
E
R,S=I
n,m~0
~RsC_~a(R,. a% *'rim \2~--n
+
~R~sn~_R2g_~).
(2.13)
Here we add tildes to the 6-string oscillators. This vertex satisfies the following connection conditions
( x,.,( o ), &.,( o ). e,+.,( o ), &.,( o ) )t v,6,> .~- ( X ( R + I ) ( ~
_ O), --~(+R+l)('/7" -- O), -- C(+R +1)('77 _ 0 ) , ~ ( R + I ) ( q 7 -- ( I ) ) 1 V ( 6 ) ) ,
½~r
(2.14)
which can be shown by using the property of Neumann coefficients* -
1
aRs - - e ; , . o . + rl
] ~ -Ns ~s . e - +~. . . .
-
m>~O
8s+l,s
1 -
e;~ ..... + y ' _N, .R.+ I ' S - +¢_i m ° R + I ,
tl
m>>.O
(n>~l),
E - - RN~0cS ^ + i m ° R - - - -{"--i ( ~ R + I ' S -- a R + I ' 6 ) O R + I "~ E
+ i ( ~ R S -- ~ R ' 6 ) O R +
*'mo~R+I'Sp . . . . m>~O
m>~O mN,~,e + -
= --
R+l,Se*i .....
m>~l
1,?ll¥~n
Se+ . . . . . .
,
m>>-I
( n >1 1 ) ,
8ss_sR,6+
~
m N-Rs ~ o e -+ ira°"=-
{8R+I'S--~R+l'6
m>~l % +~ = rr - % ,
+
XyR+t,s~+~ E m'~*mO ~ - . . . . . \] ' m>~l ½~r < % ~< ~r.
* These relations can be derived from eq. (B.17). See also appendix B of ref. [2].
(2.15)
,
K. Itoh et al. /
133
Witten type vertex
In the above expression we define R + 6 = R for 6-string indices. Now we identify the oscillators of the R = rth string with those of the R = ( r + 3)th string and replace them ~-tv(r) ~. ~ s(r) and a~(r+3) . ,
~c(r) ~ ~(r) and ~(~+3) ---(r+ 3) . ¢~-~(r) (.._ c(nr) and c.
(2.16)
Then we obtain 3
E(1,2,3) =
•
Z
• ~rs (r)-(s) ( ½N~Smot(_r)n • ot(_s)m + OtrOtslVFnmnc nC_m) ,
r , s = l n,rn>~O s - - 4r + 3, s _1_/VrmS+3 N Brs n m -~- 21\(" ~' nrm - N nm
N F~" . m = ½ ( N / m - - - r + 3, s __ ~ : ~
_~_Nn m --r+3,s+3),
+ ~ + N/m--r+
~"+~)
(2.17)
With this identification 3 (E 6 = la R?(0R)) vanishes. Samuel omitted this &function and obtained a correct expression [8]
IV(1,2,3))=(2~r)
dSa
Pr expE(1,2,3)10).
(2.18)
By using (2.15) and (2.17), one can easily show that the vertex satisfies the desired connection conditions of Witten-type 3-string vertex expected from (2.6):
(X(r)(o), A(~)(o), C(+f)(o), C~)(o))[ V(1,2, 3)) = (x(~-
o),
-- C(__f+l)(~ - o), C(r+l)(~ _ o))[ V(1,2, 3)),
½7r < o ~<~r. (2.19)
Note that this vertex already has the right ghost number and need not be multiplied by the ghost factor any longer contrary to the JS-type vertex [1,2]. Further the OSp(d/2) symmetry [2] of the vertex (2.13) is violated in (2.17). Although we use this expression (2.18) in the following calculation, we could as well rewrite (2.18) into the c o representation (see appendix B). In this c 0 representation the ghost prefactor as well as the &function of ghost zero modes appears, and the vertex turns out to have manifest OSp(d/2) invariance. Therefore it may be useful in the proof of order g2 invariance. We further discuss this point in sect. 4.
K. ltoh et al. / Witten type vertex
134
3. Proof of the distributive law (2.5)
Since the vertex (2.18) satisfies the connection condition (2.19), 3
Y'. O(sr)l V(1,2,3)) r~l
is naively expected to vanish. However owing to the singularities of A +(o) and C+_(o) near the interaction point on the vertex [1,2], it is not obvious whether it actually vanishes. In order to treat such singularities more systematically, we follow the technique developed in ref. [2] similar to Mandelstam's [14]. The only difference from ref. [2] is that there are now six strings in the p plane. Fig. 5 stands for string diagram fig. 3 plus its complex conjugate. First we divide the complex p plane of fig. 5 into 6-string regions,
o( rth string region) = [ a~( ~r + iO~) + iBr
0~Imo~<~r -rr ~ I m p ~< 0,
[Otr(~r--iOr)--it~ r
o((r+ 3)thstringregion)=(ar+3(~r+ia~)+ifl~+3 ~etr+a(~r--ior)--iflr+3
r = 1,2,3,
~0,
0~
(3.1)
On the r th string region with r = 1, 2, 3, we define (operator-valued) functions by
A(p)=
c(o)
=
arA~)(°r+i~), 'af(f)(Or -- i~) Or +
,
c(P) = { U:)(Or+i r),
(3.2)
where upper (lower) equalities correspond to the region 0
=
c.
where we have divided Q~r) into two halves, one on the to (rth string region) and the other on the o((r + 3)th string region). The contour of integration is depicted in fig. 5.
135
K. Itoh et al. / Witten type vertex
3
1
T
31
2
4
6
2
4
6
(a)
0 < ImP<~
- ~ < ImP
(b) Fig. 5. (a) The 6-string diagram in the p-plane and its complex conjugate. (b) The original contour Co of the integration (3.3) representing E3=lQtar) on the o-plane. Each pair of segments labelled X, Y or Z is to be identified.
[The contribution from the horizontal part of the contour cancels between the two contributions from I m P >< 0.] Since A and C are singular at the interaction point a n d has n o singularity elsewhere, the contour C o of fig. 5 m a y be deformed to one s h o w n in fig. 6. I n order to evaluate (3.3) on the vertex (2.18), we change a variable from P to z c o n n e c t e d via eq. (2.7). Then using the new functions A ( z ) , C ( z ) and C ( z ) , which are regular even at the interaction point, defined as
dz
I (3.4)
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K. ltoh et aL / Witten type vertex
O< I m P < zr
-zr< ImP
Fig. 6. The contours around Po and pJ which are reduced from the original contour Cp.
we have 8 3 i ~ - - • O~r~lV(1,2,3)) r=l
-1
dC(z) _
=(~C:o+~C:e) dz(dp(z)ldz, ] C(z)[-A(z)2+2----dTC(z)liV(l'2'3))' (3.5) where the contours Cz0 and Cz¢ are depicted in fig. 7. By this change of variables the cuts of figs. 5 and 6 disappear and (3.5) can be evaluated by calculating the residues of the poles at z = z 0 = i and z = z~' = - i. F o r the calculation of the pole residue at z = z 0 = i, it is convenient to m a k e the projective transformation (2.11). By this transformation the contour Czo is taken
lz
Fig. 7. The contours Go and C. d, of the integration (3.5).
K. Itoh et al. / Witten type vertex
137
into CWO,a small circle around the origin w0 = 0 (fig. 4). Since the 3-string vertex (2.18) is invariant under the projective transformation as shown in the appendix C, the integration along CZ, in (3.5) can be rewritten as
$W,Odw(d$+)[
(3.6)
-A(w)2+2d+(w)],V(l,2,3)),
where A(w)
=
c(w) = C(P(W>) 2
C(w) =
(3.7)
By moving the annihilation oscillator part in Qa to the right of exp E&2,3) in 1V(1,2,3)), we obtain the following three contributions in just the same way as in ref. [2]: gWOdw (~)-‘c(w)[-A~iwi+2~~(r)]lV(1.7.3)).
$Qw(s)
(3.8a)
-&42~ k(w), v(1,2,3)),
(3.8b)
In these expressions - stands for taking contraction and the operator 0 (= A, C, c etc.) surviving after the contractions represents the quantity O’-)+
[0, E&2,3)],
(3.9)
where (-) stands for the creation and zero mode part of the operator. First we show (3.8~) vanishes by itself. From (2.12) we obtain
(3.10) and hence -A(w)~+~----
dC(w) _ dw C(w)
I ~(~2~3))
]ri w=o
K. Itohet al. / Wittentypeoertex
138
because of the following properties
A(w = 0)t V(1,2, 3)) = 0,
(3.12a)
C(w = 0)1 V(1,2,3)) = 0,
(3.12b)
dC(w)
)2 =0.
(3.13)
w=O
Indeed, since A(w) and C(w) defined by (3.7) are regular at the interaction point w = 0, eq. (3.12) can be read from the connection condition (2.19)
A(l)(w = 0)1V(1,2, 3)) = -A~2)(w = 0)l V(1,2, 3)5 = AO)(w = 0)l V(1,2, 3)5 = -A¢l)(w = 0)1 V(1,2,3)> = 0,
(3.14)
where A~r)(w= 0) means that w approaches to zero from the rth string region. Eq. (3.12b) follows in quite the same way. These can also be checked directly by using the properties of Neumann coefficients, as is done in appendix D. Eq. (3.13) is trivial since the square of Grassmann number is always zero. For the calculation of (3.8a) and (3.8b) we need the following expressions of contractions which are shown in appendix C
A ~ ( w a ) A ~ ( w 2 )_l ( l = _ (W 1
1) W2)2
')
-wl-w2 + wl+w2 - .
C(wa)&w2)=-
(3.15a)
(W 1-~ W2)2 ~/xv'
(3.15b)
Since eqs. (3.8a) and (3.8b) contain the contractions of operators at the same point, we go back to the expression (3.3) in p variables and shift the coordinate p of one of the contracted operators to p + 2& Then we have, instead of (3.8a) and (3.8b)
~c dw
ida,w,)1( dc,w, ]IV(1,2,3)), dw' C(w) -A(w)fl(w')+ 2 ~ C ( w ' )
wo
(3.16a) -I I
~c d w wo
(d0(w)] dw ]
1
C(w,)2dC(w)c(w)lV(1,2,3)), dw
(3.16b)
K. ltoh et al. / Wittentype vertex
139
where w' and w are related by (3.17)
p(w')=p(w)+23.
The quantities (3.16a) and (3.16b) are calculated in appendix E and there we obtain
(3.16a)=-i
22 - 5d d3C 21~--~ dw 3
1V(1'2'3))
(3.18a)
W=0
d3C3 w=0I V(1,2, 3)), (3.16b) = - i F1 --dw
(3.18b)
respectively. By summing up (3.18), the total contribution to eq. (3.6) from the terms with one contraction is found to be i ( d - 26) 5
d3C w=0I V(1,2, 3)) 216 dw 3
(3.19)
This indeed vanishes when d = 26. The residues of the pole at z = z~' = - i in eq. (3.5) can be calculated in just the same way by making use of the projective transformation z+ i
w=--
z-i'
(3 20)
instead of (2.11) and can be shown that it vanishes when d = 26. Thus we have finished the proof that the vertex IV(l,2, 3)) given by (2.18) actually satisfies the distributive law (2.5). 4. Discussion
We proved that the distributive law of QB (1.1), which is equivalent to (2.5), holds for the Witten-type symmetric 3-string vertex (2.18) if the space-time dimension d is equal to 26. On the vertex functional we have rewritten V3r=lQ~r) into the form of a closed contour integral on the complex z plane. Then the proof of (1.1) was reduced to the simple evaluation of pole residues, the sum of which was found to vanish if and only if d = 26. Alternatively one can prove (1.1) by direct oscillator algebras, in this case one obtains one more condition a(0) (intercept) = 1, which was implicit in the above calculation, from the terms corresponding to the reordering of QBAlong with the line of the arguments given in ref. [2], our next task is to prove the associative law of the • product: ( * * ' / ' ) * A = O* ( # * A ) ,
(4.1)
140
K. Itoh et al. / Witten type vertex
which is conjectured by Witten [3] to hold without introducing 4- and higher-string vertices contrary to the JS type vertex case [2]. Once this axiom is established for the vertex (2.18), one can readily prove that the action [3,11, 2] S = C~ . Q Bcb + Z g cb . ( cb , cb )
(4.2)
is invariant under the stringy local gauge transformation 8 ( A )cb = Q B A + g ( ~ *
A-
A * ~),
(4.3)
with closed algebra [8(A1),6(A2)] = g 6 ( a I * A 2 - A 2 , A a ) .
(4.4)
(In this section we use the notation in ref. [2].) As was done in ref. [2] one can construct a gauge fixed action S simply by deleting the +-component of
(= -~o4, + 4,): g(q~) =S(qb)l¢= 0,
(4.5)
which is invariant under the global (on-shell nilpotent) BRS transformation
= f dl d2 (if(l)[ (dp(2)lW(3)eF(l'2'3)[0>123,
(4.6)
(the proof in sect. 6 of ref. [2] applies also here without any modification), although the precise gauge-fixing procedure is not yet well understood. The associative law (4.1) may naively be expected to hold from the diagramatic considerations. Two diagrams which contribute to (4.1) seem to cancel with each other, independently of the space-time dimension. However the determinants from the contractions of two vertices contain singularities as in the case of the horn diagram contributions in the JS-type string field theory [2]. For the latter case the careful examination lead to the condition d = 26 for the validity of (4.1). Therefore in the present case a more detailed analysis is necessary for a rigorous proof of the associative law. In this paper, we used the vertex with the ghost and the anti-ghost oscillator modes in the fermionic representation, though the original vertex proposed by Witten was written [3] in terms of the bosonized ghost and an oscillator expression of the latter vertex is already given by several authors [7,10]. The precise relation between these two vertices is still to be clarified. The authors would like to express their sincere thanks to H. Hata, T. Kugo and H. Kunitomo for valuable discussions and suggestions. They also thank T. Kugo for
K. Itoh et al. / Witten type vertex
141
careful reading of the manuscript. They are grateful to their colleagues at Kyoto University.
Appendix A PROPERTIES OF THE NEUMANN COEFFICIENTS
The Neumann coefficients ~ f f s corresponding to a 6-string light-cone diagram fig. 3 are defined as the Fourier components of the Neumann function N ( p , tS):
u(o~. &)
= -~
E -e-
~.-~.~cos(,,~)cos(,e~) - 2 m a x ( ~ .
(~
n~>l n
+2
E
~22e"~"+~'c°s('°~/cos(me~) - 2 8 , 6 ~ -
2~6g~
n , r~l ~>0
=lnlz-~l
+lnlz-~*l
(= ½1n((z-~)(z-Z*)(z*-Z)(z*-~*)}),
(a.1) where PR and t5s are assumed to lie in the region of the Rth and the Sth string strip, respectively, and connected to z and ~ via eq. (2.7) PR = p( z ) = an~ R + ifln,
(A.2)
Ps = P(~) = as~s + ifls-
The last two terms in the second expression (A.1) appear since we place Z 6 at the infinity. We can make a projective transformation (2.ll) or equivalently w+l
z = -i--,
(A.3)
w--1
in (A.1). Then the r.h.s, of (A.1) is rewritten as l n [ z - ~l + l n l z - ~'1 = lnl w - r?l + l n l w • * -
11 + 2 ( l n 2 - l n l w - 11 - I n l # - ll) (A.4)
We define another Neumann coefficients N£~s as the Fourier components of the Neumann function in the w-plane:
~(0~.~s) = -8~s
E xe-°'~-~'tcos(,o,)cos(,eA ^
- 2ma~(~.g~)
~
~2.Se"~"+m~scos(.oR)cos(mes)
+2 ~ n,m>~0
=lnlw-
~[ + l n l w ~ * -
11 .
(A.5)
142
K. ltoh et aL/ Wittentypevertex
Then by taking the limits ~--~ W6=1 ( ( s O - ~ ) obtain respectively
or w ~ W 6 ( ~ R ~ - ~ ) ,
we
^
lnlw - 11 = ~R6~R "q- Z N~6e"~"cos(noR), n>~0 ^
ln[~ - 11
=
~$6~S -Jr 2 N6Semg~cos(m6s) •
(A.6)
rn~>0
From eq. (A.4) we obtain a relation between N/,,,--Rsand NAm~Rs
"~R6+ N- -~R S _- N"2-~RS ~ - ~ . o N"6S d ~ - ~moNno
(A.7)
3.oSmoln2.
The general property of the Neumann coefficients can be found in appendix A of ref. [2]. For the case of N f s we can show the following relation in addition,
=
, : 1 . s +1.
( A.8 )
To derive this property, consider a transformation W ~ W' = e-~ri/3w,
(A.9)
which is a rotation around the origin in the w-plane. Thus if w lies in the Rth string region, w' is in the (R + 1)th string region (see fig. 4), and the coordinates ~ in the p plane corresponding to w and w' coincide with each other ~R = {R + 1,
(A.10)
where
p(w) = ~R~R+ i~R, p ( w ' ) = ag+l~g+, + iflR+l. From eq. (A.10) and the fact that Neumann function b~ in eq. (A.5) is invariant under (A.9), we obtain the relation eq. (A.8). Third we show the vertex (2.18) is invariant under the projective transformation (2.11). From eq. (A.7), we can express the coefficients N~Sm and N{~,, defined in eqs. (2.17) in terms of the Neumann coefficients N f s , =
Ars
-
~' = NF~ ^" = , NF~,,
n0t- 68"
"
+
+
°0
oo2,n2,
(A.11)
143
K. ltoh et al. / Witten type vertex
where ^ ^ ^ ^rs l(~rs "~r+3,s /WrmS+ 3 N r + 3 , s + 3 ] _ _ - - r s N_r,s+3 N B n m = 2 ~ " n m q- N ~ m + + - nm ] - - NJnm + nm ,
^ S~rmS+3Ji_Nr+3,s+3)__Nn N^rs F n m = !2( k~ "r ns _m_ ~ r +-3 ,nm - nm
^
m jWrmS + 3 " __ - - r s __
(A.12)
In eq. (A.12), the last equalities hold owing to the relation (A.8). We see from eq. (A.11) the coefficients of the ghost part of the vertex are projective invariant by themselves. As for the orbital part of the vertex, it is easy to see that the momentum conservation assures its invariance under the projective transformation. Thus we obtain an expression of the vertex in terms of the coefficients IVan,, and ]V~m,
IV(1,2,3)) = expE(1,2,3)10)(2~r)a8 a /~(1,2,3) =
E
Pr ,
(1]~SmOL(r)n " ~ ( s ) _~_ ~ ~ ~ r s . , ~ ( r ) ; ( s ) "l _ _ O-r~sZ v Fnmr~t, _ n" - m ] "
(A.13)
n,m~O
r,s=l-3
Finally we point out that one can show the following relations, NB. m - N,~m + rs _ _ - - r s NFnmNnm-
+ 6~o [~*.o Nrms+3
~*.o
]
t~ [ ~ 6 s ]~06mS+ 3 ) + V n O ~ Ore-,
(A.14) (A.15)
which result from an equality Nnm-
Ndm
--~m0(Nn~6- ~R0+3'6),
(A.16)
obtained by using eqs. (A.7) and (A.8). It is easily checked that N ~ , , and N ~ , , in eq. (2.17) can be replaced by the first two terms in eqs. (A.14) and (A.15) respectively.
Appendix B 3-STRING VERTEX IN THE co REPRESENTATION In this appendix, we derive an expression of the vertex in the c o representation, by making a Fourier transformation of (2.18) with respect to the anti-ghost zero
144
K. ltoh et al. / Witten type vertex
mode Co- The resultant vertex 17(1, 2, 3)} is 4~ri
17(1,2,3)) =
-f-~-C'(zo)C'(z~) I V0(1,2,3)),
I go(1,2,3))=(2~r)~M 3
ffS(1,2,3)=
E
(a.1)
(A)(rt ) p. 8
c(or)
ee°'2'3)10 ),
(B.2) (B.3)
N Brs , , ,(l_ot(r) t 2 - , . or(S) -m "4- nc(r)c - , - m(s)s ,]
E
r,s=l n,m>~O
where 1Iio(1 , 2, 3)) is an oscillator expression of (2.6) in the c o representation and thus it satisfies the connection condition (2.19). The operator valued function C(z) in the c o representation is defined as
C(z)= ( dP )
(B.4)
The vertex I V(1, 2, 3)) has the correct ghost number 3 due to the presence of the two ghost prefactors C'(zo) = dC(z)/dzlz=~o and C'(z~) = dC(z)/dzl~=z~. Although I Vo(1, 2, 3)) is projective invariant, the prefactors change under projective transformation. Thus the overall numerical coefficient is fixed when we choose a gauge for the projective transformation. The expression in eq. (B.1) corresponds to the special gauge in eq. (2.7). Now we give a derivation of eq. (B.1). First we rewrite (2.18) as follows, [V(1,2,3)) = I-I (1
--6(o')W('))er(l'z'3)lO>(2~r)dsa
s=l
3 w(S)-~ Z E "~r~s,tZ ~ ~'Mrs ~(r) , FnO~. _n r=l n>.l
(rt)
p~ ,
(B.5)
(B.6)
3 [ F ( 1 , 2 , 3 ) = r,s=lE [ n,m>~OE ~'Bnm2~-nl~]rs L~v(r)'Ol(s)"}--
,~, ~ M rs .o(r)7~(s) ] n,m>.lE t*rt~sZ'Fnm"t'-nt~-m] . (B.7)
The vertex 17(1, 2, 3)) in the c o representation is obtained by the Fourier transformation from eq. (B.5) 17(1,2,3))
-fr~=ld?(o')exp[-s~__?(o~)C(o"}lV(1,2,3))
=
3
~
(A)
H (c o +
Pr
s~l
(B.8)
~
~
=(c(ol)+w(1))(c(o2)+w(2))8
c(o~) r=l
eVO'2'3)10)(2~r)dSd
(A)
p~ . (B.9)
145
K. Itoh et aL / Witten type vertex
In the last equality we use 3 E W(r) = O, r~l
(B.IO)
which results from the following property of the 6-string Neumann coefficients [2], 6
E asN~s=O.
(BAD
S~1
Second we rewrite the ghost part of the exponent F(1, 2, 3) in eq. (B.5) into that of E(1, 2, 3) in eq. (B.3) up to the additive terms which contain c~0~) + w ~). These additive terms can be discarded in eq. (B.9) owing to the presence of factors I-Ir~l{,C " 0(r) + W(r)). For this purpose, we use a relation between the two coefficients N~,,. and N~,,.; 3 m N Brs. , .
+ ~r~snN~Sm
- - - "n m
E
(B.12)
s to , O t r O l t N ~rtn o N B m
t=l
which is derived by using
into
=
-n
+
(B.13)
E - ~TRr~Tsr T ~ I t'tTlYnO lYmO '
and eqs. (A.16) and (B.11). From eq. (B.12) we obtain a desired relation, 3 E E iv ~ M rs la,~(r)7.(s) ~ r ~ s z • F n m , ~ _ n ~. _ m r , s = l n,m>~l
3 E r,s~l
3 E ~ATrBnm s" "'~-"T'(r)~(s)n ~ - m -~- E n,m~O r , s = l n>~l
E nNg~.o(C(o'+W(')e(5).. ( B . 1 4 )
Therefore the IV(1, 2, 3)) is rewritten as
IF(a, 2, 3)) = (C(o1) + w(,)(C(o2)+ w(2))8
C(o" e~(~'2'3)10)(2~)~8~ r=l
= (C(o~ + w (l~)(c(g) + w (2~)1VoO, 2, 3)~
( fl) Pr
(B.aS)
Finally, we show that the ghost prefactor (C(oa) + w(1))(C(o2) + w (2)) in the above expression is equivalent to the operator valued function at the interaction point C'(zo)C'(z~) in front of the vertex 11/o(1,2,3)). Let us calculate C(+r)(~r) on the
146
K. ltoh et al. / Witten type vertex
vertex (B.2), using eqs. (2.1) and (B.14), 1
+o¢
C(+~)(-i~)lVo(1,2,3)) = ~
~
1[
c~)e'~qVo(1,2,3))
3
=~
E C~),.,e-":r--E n>~O
E
"
"
~r~s
,,,,.s
a" Fnm
.~..(~)o":r
Ir~-
m ~"
s = l n,rn>~l
+ ~2 (C(o' ) + w ( ' ) ) E nNg',,oe'~ IVo(1,2,3)), s=l
n>~l
]
(B.16) where we consider only the 1st, 3rd or 5th string which a n = 1 on the p plane. In order to evaluate each term with fr placed near the interaction point in eq. (B.16), we need some relations with the Neumann coefficients which we derive in the following. By picking out the terms analytic in z and ~ from (A.1) with proper redefinition of constant part ~ s , we obtain
kn>~I n
[1
+o((~-~.) Z-e":'<:,)-(~+i~
])
n>~l 1l
-t- E ~[2SenfR+m~s--(~R'6~Rq-~S'6~S)
(B.17)
n,m>~O
(cf. eq. (A.8) of ref. [19]). Taking z in the neighborhood of the interaction point (thus in > (S), we differentiate (B.17) with respect to fn or (s, 0z
__ O~R z -
1
= ~n,S ~
Iz
e n(~s-~R) +
t .>~x -{- E
E
]
1
rl~fRSen~n+m~s-- ~ R , 6 ,
(B.18)
n>~l rn>~O
Off
1
O~s z - ~
.>~1 + E
E mN/se";R+m~-~- ~S,6.
n>~O m>~l
(B.19)
K. Itoh et al. / Witten type oertex
147
In eq. (B.18), we take the limit ~ --+ Z s ( ( s --+ - oo) Oz
+ 8R'6 +
E n N ~ se"~" = - s r ' s n>~l
1
- -
(B.20)
O~R Z -- g S "
We multiply e - " ~ (m >/1) to eq. (B.19) and integrate with respect to (s around the point on the o-plane corresponding to the Sth external string. Then we obtain,
mNRSe"~R = 8RSe-m~r -- mNgS - ~zs d~ _ _ 1 n>~l--" "nm -
2~ri z --
e_m(s,
(m >~ 1).
(B.21)
Using eq. (B.21), we rewrite the first two terms on the r.h.s, of eq. (B.16),
1[
- - . e-< ¢g Y'. ~'~
3
E
-
1[
en]
a/~ sNy.mmc(SL
s = l n,m>~l
n~>O
= ~
E
c'0'~+ w"~ +
E
%~,%
s = l rn~>l
X~
7"-7.
Z7] e 'n~s
_.g
d~
1
1
zs+3 2~ri Zr-- ~.
Z~+3-- ~
where z, and z,+ 3 are related to ~, as follows,
o( z,) = ~,~r + iB,,
(B.23)
P(z~+3) = a,+3~', + ifl~+3,
and thus z,+ 3 = - 1 / z r When ~, takes the value corresponding to the interaction point, z, and z,+ 3 coincides (z~ = z,+ 3 = i) and the terms in the curly bracket in eq. (B.22) vanish. As for the third term of eq. (B.16), we rewrite the summation by using eq. (B.20),
n>~l
"~r
Zr-
Zs
Zr-
ZS+3
zr+3 1 +
0~r ~z,+3
--
1)} Zs +
Zr+ 3 -- Z S + 3
. (B.24)
148
K. Itoh et al. / Witten type vertex
Then we can explicitly calculate the quantity in braces in eq. (B.24) by using eq. (2.7).
( 4__4_e~i/3 a, + 0(1) 17~ z-i { } = ~ 4_4__e_,,i/3 a r lOft-
( s = 1) (B.25)
z_i +0(1)(s=2) ( s = 3).
From eqs. (B.22), (B.24) and (B.25), the singular part of c(+r)(i~,.)lVo(1,2,3)} is given as
2 z Ta,.i [e~,i/,(c(ol)+w(n) 3ff~
C(+")(-i~r)IV°(I'2'3)>
- e - ~ i / 3 ( C ( o 2) +
w(2))] I Vo(1,2,3)>. (B.26)
Similar calculation of Cff)(i~ * )1Vo(1,2, 3)) leads to C!r)(i~*)l V°(1'2'3)) - ~
2
O/r z* + i [e-'~i/3(c(°l) + w(1)) -e'i/3(C(o2)+ w(2))]lVo(1,2,3)}. (B.27)
Therefore from (3.2) and (B.15), we obtain
C(p )C(o* )l go(1,2,3)> = [
4i
1
3~
(z--i)(z*
"}-i) [ il
+0
= -
-z-i"z*+i"
' [ '2
"~[C~)_~.)_W(1)}[C~.)_W(2)}
(1 1)1 (1 1)] z - i' z* + i
+0
z-i'z*+i
I Vo(1, 2, 3)) 1V(1'2'3)) (B.28)
Since eq. (B.28) diverges at the interaction point z o = i, we multiply do/dz: do
3
-dTz = -~i(z- i) 2 + O ( ( z - i)')
(B.29)
K. Itoh et al. /
149
Witten type vertex
and its complex conjugate to eq. (B.28). We then differentiate it with respect to z and z * and take the limit z ~ z o = i and z* ~ z~'. The result is
C'(zo)C'(z~)1V0(1,2, 3)) = -3¢3-_ ~ - 1 V ( 1 , 2 , 3)).
(B.30)
Appendix C CONTRACTION FORMULAS
We show eq. (3.15) by using the expression (A.13) of the vertex. The contraction of two operators O1(0) and 0205) implies the following quantity
Ol(P)b2(p) ~ 0 ( ( - - ~)<0101(P)O2(P)lO>c ~[Ol(p),[O2(/~),E]]:g
-}-
0(~
--
,()= (C.1)
,
where the suffix c denotes the connected part and - ( + ) sign in the r.h.s, should be taken when both of 01 and 0 2 are fermionic (otherwise). For example, from (2.1), (3.2) and (A.13) we have
c(o)c(p)
=
[(O((S-'R)( ~-"en(~R-fsl+l)-O(~R-ffS)n>l y'~ x(~g's+sR+3"s)
+
,v ,~, ..... ~ l O FR,S, , "~ n ~ n ÷ m ~ s /]' z., ~R~'s
n~>0 m~>l
(C.2)
]
where P and t5 are assumed to lie in the regions of the Rth and the Sth string, respectively. Recall that the Rth string and the (R + 3)th string have the same oscillators. We have used the following properties, ^R,S_ ~R+3,S OtROtSNFm n -- OlR+ 3OlslYFm n
~ ^, ~R,S+3 ~ tlRuS+31VFm n
^ 7trR+3, S+3 = OIR+3OIS+31¥Fm n
(c.3)
^RS (R, S = 1 - 6) are defined as where the coefficients NF,,, ^RS
"-~RS "R+3, S
NFn m = Nnm - Nnm
(C.4)
150
K. ltoh et al. / Witten type vertex
(cf. eq. (A.12)). Then (C.2) can be rewritten as
-O(l~,-~s) E e"'gs-~)} 8"s+ E mT~fse"~+"~s n>>.l
n>~O m>~l
aSI{O(~s-liR)( hal
-O(~.-(s) E e"'gs-~') B"+''s+ n>l
E
(c.5)
S e n~a+m( s m N "R+3, ~m
n>~O mNl
The first term of the r.h.s, can be read from eq. (A.5) as
1 2_~Oa~(p,p) = Iras ass I~-pan
1 d~
1
(C.6)
rr d r w - ~ '
since (A.5) can be rewritten as
kn>~l
[1
])
+O(~R-- (S) Z -~n(e-n~"+e-n~Y)(en(s+en~s*)--2~R n>~l -OARS n~R "* +51 E Nnm(e +en~1)(emfs+em~s) •
(C.7)
n, m>>.O
The second term of (C.5) is obtained from (C.6) by the following replacement
P= erROR+ iflR--+ 0/R+3~R+3 -1- iflR+3,
~R+3 = ~R
w--, (e-'i/3)3w= -w.
(c.8)
Thus we have & p ) e {
-
P)
ld#(
~r d~
_
1
1)
_
_
_
w-~
+w+~
"
(C.9)
151
K. ltoh et aL / Wittentype vertex
Eq. (3.15b) is an immediate consequence of (C.9) and (3.7). Derivation of (3.15a) is quite similar.
Appendix D PROOF OF EQ. (3.12) In this appendix we prove (3.12) directly from the properties of Neumann coefficients. First we show (3.12a). By using (2.1) and (A.13) we can write A~+R)(or-- i~r) on the vertex as A~+R)(o, -- i~r) I V(1,2, 3))
1(
=
f~
3
E a(R2e - ' L + E n>~O
S=I
E ,.~~j , R S ~(S).,L I V(1,2,3)), ,Bnm~ rnl.,
(D.1)
n>l rn>~O
where
(D.2)
~ = ~ + ion, n m - - S" n m
- - - "rim
o • ( R__)~ ( R + 3 ) n
----n
(D.3)
:'
(D.4)
•
Since ~ f s has the following properties (see appendix A of ref. [2]),
E
n_n~,se,~R= ( d p ( w ) ] - 1 1
.>~10IR
~ dw
.0
nm
n---~
n ~ -~>,E--NfSe'~R o a r w~
O l s UI=
]
W--
1 W S
8 Rs,
(D.5)
ol R
(D.6)
UtllVnOlVraO~
(dp(w)]-l~ a, m_* dw ] 1 = 1 W0 -- W, a s NS~' (m>/1),
(D.7)
the coefficients of a(_s~[V(1,2,3)) in eq. (D.1) multiplied by d p ( w ) / d w can be
K, ltoh et al. / Witten type vertex
152
evaluated at the interaction point as
dp(w)(
~"nN~Sme'¢') n>~l m
) O~R Z
w'-*wo
dp(w)
/
dw
\
I=1 WO-- W I
°ts
mO
(m>/1)
O/ ~'m0 S+3
18 Rs + 8R,s+3 +
1)
- - + n>~l
/ w-'*Wo
\ Wo-
(D.8)
=0,
W 0 -- W s + 3
Ws
(D.9) where we used the following properties
W S +~ 3-- Ws~
Wo=O~
°ts+3 =
-°ls~
~mS+3,1__~S,I+3 (D.IO) 0
- - S"m0
"
Thus we obtain
A(w = 0)1V(1,2,3)>
(D.11)
= 0,
from the definition in (3.7) and (3.2). Similarly from eqs. (2.1) and (A.13) we can write
c~+R)(o~- i~r)l V(1,2, 3)) =
1(
~R~'S..... r,,,,,'- -., "
n~>l
I V(1,2,3)).
(D.12)
n>~O m>~l S=1~3
In order to prove (3.12b) it is sufficient to show the coefficients of i.e.,
1
( _ 8R,S+3 + 8 R S ) _ _ e - , ~ , _ ~ m ,~0
~'F.,."~RS Ofl~r
(m >~1)
aRasme~ (D.13)
vanish at w = 0. To this end we pick out the pieces analytic in w and ~ from (A.5) with proper redefinition of constant part No~s (see appendix A of ref. [19]),
ln(w-w) =-SRs O(~R--(S) E-e"(~s-~")-~'. n ~ l FI
+
E n, m>~0
N'2mSen~R+m~s-
(D.14)
153
K. Itoh et al. / Witten type vertex
As is explained in appendix A, when ~ lies in the region of the Sth string with the string coordinate ~s, - ~ corresponds to the point in the (S + 3)th string strip with the same string coordinate ~s. Therefore when ~R > ~s we obtain ln(w
_
=
~)-ln(w+ff)
_
f n ~ (8 R s - 3R ' s+3 )1 X - e ( ~ s - ~ R ) - t ' R ] \ .~i n ]
+
~,
(~s_
~f,;s+a)e~R*m?s"
(D.15)
n,m~O
In the above equation when w equals to zero (~R --- ½~ri), the 1.h.s. becomes constant. Thus comparing the coefficients of e mrs (m >~ 1) we obtain ^
--(~RS--sR'S+3) -e-mf"+
m
(m>~ 1),
E N~Sme"~R=0
n~>0 ~'R = ½~ri.
(D.16)
This leads to (3.12b). Appendix
E
CONTOUR INTEGRATION
Using eqs. (3.15), we rewrite (3.16a) and (3.16b) as
[ ~CwodW
wo
dw'
dw
-
1
1)
-- ~
(W--W') 2
(w+w')
+
(w-w,)Z+
(w+w') z
"--d-'~w ] \ _ _ ~r
_w' _- +w
2
w--7-----+w
IV(1,2,3)),
(E.la)
1V(1,2,3)),
(E.lb)
respectively. In order to calculate (E.1) we expand (2.12) around w = 0,
p(w) -- p0 + 2 w3 + O(wg),
(E.2)
where P0 = p(0) = -~-rri. Then from (3.17) we have w,3 + O ( w , 9 ) = ~ + w ~ + O(wg).
(E.3)
154
K. Itoh et al. / Witten type vertex
We expand w' in terms of 8 as (E.4) n~l
Inserting (E.4) in (E.3) and comparing the coefficients of 8" (n = 1, 2, 3), we obtain
1
-- ~-- 3w 3 -t- O(w9),
(E.5)
f?
- 1 "4-O(W6),
(E.6)
y,
5
f, k
~13 ~---"~ + O(W6).
(Z.V)
From these equations we obtain 1 w;
-
(
)-1
= wEL(w)8 w n>~l
n
/
11
l f2
-~1~
wi?
+0(8)
1 1 = O(w2)a + --w + O(w') + O(8), 1
1
w' + w
2w
(E,8) (E.9)
+ 0(8 / .
For the calculations of the quantities in eq. (E.la)
dw'
(w -t- Wt) 2
dw
]
dw ( w + w , ) 2'
(E.10)
we have the following equations from (E.4) - (E.7),
dw
1
(
dw (w' - w) 2 ~-- 1 + n~>lE- - S h a w
w n>lEfn 8
11 -12 [ dill
= w~lIlJ 82 + - ~
1 2 f2]l w2 fl f 2 8
2 f3 wZ f 3 1
1
2 1
+ o(8)
= O ( w 4 ) ~ -+O(w7)~ + ~ - - - 2 + 0 ( w 4 ) + 0 ( 3 ) , dw' 1 1 = 4w--7 + O(a). dw ( w , + w) 2
(E.11) (E.12)
155
K. Itoh et aL / Witten type vertex
The pole residues of (E.1) are calculated by making use of the formulas, dp(w')
-
1 1 _ 1 ( w2 w5 ) (W'--W) 2 = 9W4 + O 82, 8 ' w2 + O ( 8 ) '
dw'
dp(w') dw'
-1
1
(E.13)
1
(w' + w) 2
-- - -
24w 4
+ O ( w 2) + O ( 8 ) ,
[dp(w)] -11 --1 ( 1w 3 ) + O ( 8 ) dw ] w ' ~ - w = 6w 3 + O 8 '
(d0(W)dw)-l__w,+wl = - - 1 23 w 1 +O(w 3)+O(8).
(E.14)
,
(E.15)
(E.16)
We evaluate (E.1) by performing the contour integration first with 8 kept finite, and hence the 0 ( 8 -2 , 8 -1 ) terms do not contribute. The 0 ( 8 ) terms vanish after taking limit 8 ~ 0. Thus we have
(E.la)=(-2~ri)
=(-i)
22-5d 72~r
i d3C 4 IV(1,2,3)) 3! dw 3 w=o
22 - 5d d3C w=01V(1,2,3)) 216 dw 3
(E.lb) = ( - 2 ~ ' i )
(E.17)
1 1 -d3C w=0I V(1,2, 3)) 2~r 2! dw 3
1 d3C w=OIV(1'2'3)5"
(E.a8)
References
[1] H. Hata, K. Itoh, T. Kugo, I4. Kunitomo and K. Ogawa, Phys. Lett. 172B (1986) 186; A. Neveu and P.C. West, Phys. Lett. 168B (1986) 192; Nucl. Phys. B278 (1986) 601 [2] H. Hata, K. Itoh, T. Kugo, H. Kurtitomo and K. Ogawa, Phys. Rev. D34 (1986) 2360 [3] E. Witten, Nucl. Phys. B268 (1986) 253 [4] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D35 (1987) 1356 [5] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. 172B (1986) 195; Phys. Rev. D35 (1987) 1318
156
K. ltoh et al. / Witten type vertex
[6] S.B. Giddings, Nucl. Phys. B278 (1986) 242; S.B. Giddings and E. Martinec, Nucl. Phys. B278 (1986) 91; S.B. Giddings, E. Martinec and E. Witten, Phys. Lett. 176B (1986) 362; S. Samuel, Phys. Lett. 181B (1986) 249 [7] D.J. Gross and A. Jevicki, Nucl. Phys. B283 (1987) 1 [8] S. Samuel, Phys. Lett. 181B (1986) 255 [9] E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. 179B (1986) 57 [10] N. Ohta, Phys. Rev. D34 (1986) 3785 [11] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Nucl. Phys. B283 (1987) 433 [12] S. Mandelstam, Nucl. Phys. B64 (1973) 205; B69 (1974) 77; B83 (1974) 413 [13] Z. Koba and H.B. Nielsen, Nucl. Phys. B12 (1969) 517 [14] S. Mandelstam, Nucl. Phys. B83 (1974) 413 [15] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D35 (1987) 1318
BRS IN-VARIANCE OF T H E w r I T E N TYPE VERTEX Katsumi ITOH, Kaku OGAWA and KazuhikoSUEHIRO Department of Physics, Kyoto University, Kyoto 606, Japan
Received 10 October 1986
It is proved rigorouslythat the Witten type 3-string vertexis BRS invariant if and only if the space-time dimension is equal to 26. We use the fermionicrepresentation for the reparametrization ghost.
I. Introduction The construction of interacting string field theory amounts to identify the proper interaction vertices between strings. Up to now there are at least two concrete examples of Lorentz covariant and gauge invariant interacting string field theory for open bosonic string [1-3] which are both based on vertices geometric in nature and have some of the algebraic properties in common, but are qualitatively different in many other respects. One approach is a Lorentz covariant extension of light-cone string field theory, whose vertices represent the intuitive joining-splitting type interactions (JS type) between strings [1, 2]. The other is the one introduced by Witten based on the manifestly symmetric decomposition of string surfaces [3]*. The detailed examinations for both of these theories would be desirable for the full understanding of the structures and the possibilities of string field theories. As for the former theory it was proved that the string interaction vertices satisfy the various key algebraic relations, such as the distributive law of the BRS-charge QB and the associative laws [2], governing the gauge properties of the theory. It is also proved that the unphysical freedom of the string-length parameter characteristic to this formulation decouples consistently [4]. Thus a self-consistent unitary field theory of covariant open bosonic string was established and subsequently extended to the closed bosonic string case [5]. On the other hand formal analyses were given by Witten for the latter formulation [3] and more detailed investigations have been developed by various groups * The same type of vertexwas also proposed by T. Goto in Extended pictures of elementaryparticles, lwanami (1978) (in Japanese). 0550-3213/87/$03.50©Elsevier SciencePublishers B.V. (North-Holland Physics Publishing Division)
128
K. ltoh et al. / Witten type vertex
[6-9]. In refs. [7-9] an important advance was made in rewriting the vertex in oscillator modes. (see also ref. [10]). In spite of their efforts, the latter formulation still lacks rigorous proofs for some of the main assumptions, especially for its algebraic axioms (see refs. [3, 7] in this respect). The purpose of the present paper is to give the complete proof of the distributive law of QB with respect to the Witten-type * product: O s ( q ~ , ~ ) = (Osq~), ~/, + (_)letq~, (asko) '
(1.1)
where the * product is defined by [11, 2] ( ~ * '/')[ Z31 = f~tz~l'I'tZE]V[Z2, Z~,23]tdZtdZ21
(1.2)
(see ref. [2] for notation). The distributive law (1.1), which intuitively means the commutativity of QB with the vertex operators, assures directly the following properties at order g level: (i) the closure of the stringy gauge transformation and the nilpotency of the global BRS transformation; (ii) the gauge invariance of the gauge un-fixed action and the BRS invariance of the gauge fixed action. Usually the Faddeev-Popov ghost part of the Witten-type vertex functional V(1, 2, 3) is defined in the bosonized form. However for convenience of the present purpose, we here employ the fermionic representation following Samuel, who also checked (1.1) up to the fourth mass-level [8]. The explicit form of V(1, 2, 3) is given in sect. 2 from the consideration of connection conditions of string coordinates. Then in sect. 3 we prove eq. (1.1) by applying the contour integration method developed in ref. [2], for the case of the JS-type vertex. We thus established the gauge invariance of the Witten-type action at order g level. There remains to be given such a rigorous proof for the O(g 2) gauge invariance as was done in ref. [2] for the JS-type case. Some discussions on this problem and others are given in the final section. The technical details necessary for the proof are given in appendices. 2. Construction of the Witten-type 3-string vertex In this paper we follow the notations of ref. [2] for oscillator expansions of string coordinates and BRS charge and keep the coordinate representations for the zero-mode variables x ~ and ~0: Xt'(o)=
x~ + i £ -(a~-a~-,)cosno
,
n>~l II
1 +~ a"+_(o) ==-P"(o) -7-X'~(o) = ¢ 7 . :_E=~~ e +-'"°' 1
+~ E cn e+-`"°,
1
+¢¢
C+_(°)=-i~r~(°lT-c(°)=-~C-_+(o) = ?(o) -T-ilrc(o ) = v~-
E
e, e -+''°,
(2.1)
K. Itoh et al. / Witten type vertex
1
129
2
Fig. 1. The structure of overlapping 8-functionals in Witten-type 3-string vertex. The strings interact at their mid-points.
and
Q , - - ¼C~ E +
d o C + - A ~ T- 2i--~-o C +I ,
(2.2)
where a ,~, c, and ~, satisfy the following properties*
ag=p"=- - i Ox'--~'
Co = - O?° .
(2.3)
The vacuum state of the oscillators is denoted by 10); (a~, c,, ?,)[0) = 0
(n >I 1).
(2.4)
As was shown in refs. [1, 2], the distributive law (1.1), which assures the gauge invariance or the BRS invariance of the actions at order g, is equivalent to the condition on the 3-string vertex [ V(1, 2, 3)), 3
E Qg')[ V(1,2,3)) = 0.
(2.5)
r=l
The Witten-type 3-string vertex has the following S-functional structure as depicted in fig. 1 [3]: V(1,2, 3) tx
l--I
I-I
~(x(r)(°)-X(r+l)(~-°))
r = 1 , 2 , 3 ~/2~< o~<~r
XC~(c(r)(o) -~ c(r+l)(eff -- 0"))
XS(?(')(o)- ?('+l)(~r- o)), * W e use the metric ~
= d i a g ( - , + , + .... +).
(2.6)
130
K. ltoh et al. / Witten type vertex
Fig. 2. The 3-string scattering process,
where we define r + 3 = r for the 3-string indices. The signs of ghost coordinate connection conditions are chosen such that eq. (2.5) holds at least naively by inspecting the expression (2.2) of BRS charge. We express this &functional rigorously with oscillator modes. In order to construct the vertex I V(1, 2, 3)) using the techniques developed by Mandelstam [12], we must find a conformal mapping from the upper half complex plane into the 3-string diagram (fig. 2). However such a mapping necessarily has a cut singularity, the existence of which causes some complications in the proof in sect. 3. Gross, Jevicki [7] and Samuel [8] avoided this singularity by making use of two copies for each string and considering the conformal mapping from the upper half plane into the 6-string diagram (the p-plane) (fig. 3). By choosing a special gauge for Koba-Nielsen variables [13], this mapping is given by [7, 8]
z(z2- 3) p = ~. aRln(z-- Z R ) - - l n 3 = l n
3-797_1
,
R~I oq = or3 = 0~5 = 1,
0/2 = 0/4 = 0~6 =
z2= , Z4=
-~-3 ,
-1,
z,=o,
Z5-B-- - v @ - ,
Z6=~.
(2.7)
We use capital letters R, S... for the 6-string indices and lower case letters r, s... for the original 3-string indices. In each string strip in the p-plane we introduce a complex coordinates ~R: ~R = ~R + iOR
( ~R < O,
0 <~ 0 R <~¢t ) , R-1
P = aR~R + iflR,
fir = Ir ~
c~s.
(2.8)
S=I
Note that the interaction point z o determined by do ~=~o (z2 + 1)2 dz = (zg-3)(zZ-1)Zo
=0
(2.9)
K. Itoh et al. / Witten type vertex
131
(a)
o1=01 X
I
jo.2=ar
°3== I
IO4=o
a
3
J5
I
t,Az tlo'4=~r
t73=OI
I %=~Itzxt A
%=ol
a
l°e=° 1~6~n
(b) Fig. 3. (a) The 6-string diagram in the p-plane. (b) The connection conditions of the 6-string diagram (a). Each pair of segments labelled X, Y or Z is to be identified. These diagrams should be placed on the top of each other in the p-plane to recover the diagram (a).
is equal to i and Oo - P ( Z o ) = ½rri.
(2.10)
F u r t h e r for later convenience we define a complex w-plane connected to the z-plane via a projective transformation [7] z-i
w=
.
z+i
(2.11)
By this transformation the upper half plane is m a p p e d into the unit disk and the interaction point z o = i to the origin (fig. 4). In terms of w, p can be written as W3+1 p=lnw3_l
6 ½7ri= E ~Rln( w - W R ) - ½ ~ r i ,
w . = w ( Z ~ ) = e-"~"/~
R=I
(R = 1 - e).
(2.a2)
132
K. ltoh et a L /
Witten type vertex
Iw
Fig. 4. The 6-string diagram in the w-plane. The circle around w o = 0 indicates the contour Cwo of the integration (3.6).
Using the Fourier components of the Neumann function (see appendix A), we can write the 6-string vertex corresponding to fig. 3 as l V(6)) = (2qr)d~ a ~=1 f i r ~
expE(l_6)10 )
-"
6
e(1 - 6) =
E
E
R,S=I
n,m~0
~RsC_~a(R,. a% *'rim \2~--n
+
~R~sn~_R2g_~).
(2.13)
Here we add tildes to the 6-string oscillators. This vertex satisfies the following connection conditions
( x,.,( o ), &.,( o ). e,+.,( o ), &.,( o ) )t v,6,> .~- ( X ( R + I ) ( ~
_ O), --~(+R+l)('/7" -- O), -- C(+R +1)('77 _ 0 ) , ~ ( R + I ) ( q 7 -- ( I ) ) 1 V ( 6 ) ) ,
½~r
(2.14)
which can be shown by using the property of Neumann coefficients* -
1
aRs - - e ; , . o . + rl
] ~ -Ns ~s . e - +~. . . .
-
m>~O
8s+l,s
1 -
e;~ ..... + y ' _N, .R.+ I ' S - +¢_i m ° R + I ,
tl
m>>.O
(n>~l),
E - - RN~0cS ^ + i m ° R - - - -{"--i ( ~ R + I ' S -- a R + I ' 6 ) O R + I "~ E
+ i ( ~ R S -- ~ R ' 6 ) O R +
*'mo~R+I'Sp . . . . m>~O
m>~O mN,~,e + -
= --
R+l,Se*i .....
m>~l
1,?ll¥~n
Se+ . . . . . .
,
m>>-I
( n >1 1 ) ,
8ss_sR,6+
~
m N-Rs ~ o e -+ ira°"=-
{8R+I'S--~R+l'6
m>~l % +~ = rr - % ,
+
XyR+t,s~+~ E m'~*mO ~ - . . . . . \] ' m>~l ½~r < % ~< ~r.
* These relations can be derived from eq. (B.17). See also appendix B of ref. [2].
(2.15)
,
K. Itoh et al. /
133
Witten type vertex
In the above expression we define R + 6 = R for 6-string indices. Now we identify the oscillators of the R = rth string with those of the R = ( r + 3)th string and replace them ~-tv(r) ~. ~ s(r) and a~(r+3) . ,
~c(r) ~ ~(r) and ~(~+3) ---(r+ 3) . ¢~-~(r) (.._ c(nr) and c.
(2.16)
Then we obtain 3
E(1,2,3) =
•
Z
• ~rs (r)-(s) ( ½N~Smot(_r)n • ot(_s)m + OtrOtslVFnmnc nC_m) ,
r , s = l n,rn>~O s - - 4r + 3, s _1_/VrmS+3 N Brs n m -~- 21\(" ~' nrm - N nm
N F~" . m = ½ ( N / m - - - r + 3, s __ ~ : ~
_~_Nn m --r+3,s+3),
+ ~ + N/m--r+
~"+~)
(2.17)
With this identification 3 (E 6 = la R?(0R)) vanishes. Samuel omitted this &function and obtained a correct expression [8]
IV(1,2,3))=(2~r)
dSa
Pr expE(1,2,3)10).
(2.18)
By using (2.15) and (2.17), one can easily show that the vertex satisfies the desired connection conditions of Witten-type 3-string vertex expected from (2.6):
(X(r)(o), A(~)(o), C(+f)(o), C~)(o))[ V(1,2, 3)) = (x
o),
-- C(__f+l)(~ - o), C(r+l)(~ _ o))[ V(1,2, 3)),
½7r < o ~<~r. (2.19)
Note that this vertex already has the right ghost number and need not be multiplied by the ghost factor any longer contrary to the JS-type vertex [1,2]. Further the OSp(d/2) symmetry [2] of the vertex (2.13) is violated in (2.17). Although we use this expression (2.18) in the following calculation, we could as well rewrite (2.18) into the c o representation (see appendix B). In this c 0 representation the ghost prefactor as well as the &function of ghost zero modes appears, and the vertex turns out to have manifest OSp(d/2) invariance. Therefore it may be useful in the proof of order g2 invariance. We further discuss this point in sect. 4.
K. ltoh et al. / Witten type vertex
134
3. Proof of the distributive law (2.5)
Since the vertex (2.18) satisfies the connection condition (2.19), 3
Y'. O(sr)l V(1,2,3)) r~l
is naively expected to vanish. However owing to the singularities of A +(o) and C+_(o) near the interaction point on the vertex [1,2], it is not obvious whether it actually vanishes. In order to treat such singularities more systematically, we follow the technique developed in ref. [2] similar to Mandelstam's [14]. The only difference from ref. [2] is that there are now six strings in the p plane. Fig. 5 stands for string diagram fig. 3 plus its complex conjugate. First we divide the complex p plane of fig. 5 into 6-string regions,
o( rth string region) = [ a~( ~r + iO~) + iBr
0~Imo~<~r -rr ~ I m p ~< 0,
[Otr(~r--iOr)--it~ r
o((r+ 3)thstringregion)=(ar+3(~r+ia~)+ifl~+3 ~etr+a(~r--ior)--iflr+3
r = 1,2,3,
~0,
0~
(3.1)
On the r th string region with r = 1, 2, 3, we define (operator-valued) functions by
A(p)=
c(o)
=
arA~)(°r+i~), 'af(f)(Or -- i~) Or +
,
c(P) = { U:)(Or+i r),
(3.2)
where upper (lower) equalities correspond to the region 0
=
c.
where we have divided Q~r) into two halves, one on the to (rth string region) and the other on the o((r + 3)th string region). The contour of integration is depicted in fig. 5.
135
K. Itoh et al. / Witten type vertex
3
1
T
31
2
4
6
2
4
6
(a)
0 < ImP<~
- ~ < ImP
(b) Fig. 5. (a) The 6-string diagram in the p-plane and its complex conjugate. (b) The original contour Co of the integration (3.3) representing E3=lQtar) on the o-plane. Each pair of segments labelled X, Y or Z is to be identified.
[The contribution from the horizontal part of the contour cancels between the two contributions from I m P >< 0.] Since A and C are singular at the interaction point a n d has n o singularity elsewhere, the contour C o of fig. 5 m a y be deformed to one s h o w n in fig. 6. I n order to evaluate (3.3) on the vertex (2.18), we change a variable from P to z c o n n e c t e d via eq. (2.7). Then using the new functions A ( z ) , C ( z ) and C ( z ) , which are regular even at the interaction point, defined as
dz
I (3.4)
136
K. ltoh et aL / Witten type vertex
O< I m P < zr
-zr< ImP
Fig. 6. The contours around Po and pJ which are reduced from the original contour Cp.
we have 8 3 i ~ - - • O~r~lV(1,2,3)) r=l
-1
dC(z) _
=(~C:o+~C:e) dz(dp(z)ldz, ] C(z)[-A(z)2+2----dTC(z)liV(l'2'3))' (3.5) where the contours Cz0 and Cz¢ are depicted in fig. 7. By this change of variables the cuts of figs. 5 and 6 disappear and (3.5) can be evaluated by calculating the residues of the poles at z = z 0 = i and z = z~' = - i. F o r the calculation of the pole residue at z = z 0 = i, it is convenient to m a k e the projective transformation (2.11). By this transformation the contour Czo is taken
lz
Fig. 7. The contours Go and C. d, of the integration (3.5).
K. Itoh et al. / Witten type vertex
137
into CWO,a small circle around the origin w0 = 0 (fig. 4). Since the 3-string vertex (2.18) is invariant under the projective transformation as shown in the appendix C, the integration along CZ, in (3.5) can be rewritten as
$W,Odw(d$+)[
(3.6)
-A(w)2+2d+(w)],V(l,2,3)),
where A(w)
=
c(w) = C(P(W>) 2
C(w) =
(3.7)
By moving the annihilation oscillator part in Qa to the right of exp E&2,3) in 1V(1,2,3)), we obtain the following three contributions in just the same way as in ref. [2]: gWOdw (~)-‘c(w)[-A~iwi+2~~(r)]lV(1.7.3)).
$Qw(s)
(3.8a)
-&42~ k(w), v(1,2,3)),
(3.8b)
In these expressions - stands for taking contraction and the operator 0 (= A, C, c etc.) surviving after the contractions represents the quantity O’-)+
[0, E&2,3)],
(3.9)
where (-) stands for the creation and zero mode part of the operator. First we show (3.8~) vanishes by itself. From (2.12) we obtain
(3.10) and hence -A(w)~+~----
dC(w) _ dw C(w)
I ~(~2~3))
]ri w=o
K. Itohet al. / Wittentypeoertex
138
because of the following properties
A(w = 0)t V(1,2, 3)) = 0,
(3.12a)
C(w = 0)1 V(1,2,3)) = 0,
(3.12b)
dC(w)
)2 =0.
(3.13)
w=O
Indeed, since A(w) and C(w) defined by (3.7) are regular at the interaction point w = 0, eq. (3.12) can be read from the connection condition (2.19)
A(l)(w = 0)1V(1,2, 3)) = -A~2)(w = 0)l V(1,2, 3)5 = AO)(w = 0)l V(1,2, 3)5 = -A¢l)(w = 0)1 V(1,2,3)> = 0,
(3.14)
where A~r)(w= 0) means that w approaches to zero from the rth string region. Eq. (3.12b) follows in quite the same way. These can also be checked directly by using the properties of Neumann coefficients, as is done in appendix D. Eq. (3.13) is trivial since the square of Grassmann number is always zero. For the calculation of (3.8a) and (3.8b) we need the following expressions of contractions which are shown in appendix C
A ~ ( w a ) A ~ ( w 2 )_l ( l = _ (W 1
1) W2)2
')
-wl-w2 + wl+w2 - .
C(wa)&w2)=-
(3.15a)
(W 1-~ W2)2 ~/xv'
(3.15b)
Since eqs. (3.8a) and (3.8b) contain the contractions of operators at the same point, we go back to the expression (3.3) in p variables and shift the coordinate p of one of the contracted operators to p + 2& Then we have, instead of (3.8a) and (3.8b)
~c dw
ida,w,)1( dc,w, ]IV(1,2,3)), dw' C(w) -A(w)fl(w')+ 2 ~ C ( w ' )
wo
(3.16a) -I I
~c d w wo
(d0(w)] dw ]
1
C(w,)2dC(w)c(w)lV(1,2,3)), dw
(3.16b)
K. ltoh et al. / Wittentype vertex
139
where w' and w are related by (3.17)
p(w')=p(w)+23.
The quantities (3.16a) and (3.16b) are calculated in appendix E and there we obtain
(3.16a)=-i
22 - 5d d3C 21~--~ dw 3
1V(1'2'3))
(3.18a)
W=0
d3C3 w=0I V(1,2, 3)), (3.16b) = - i F1 --dw
(3.18b)
respectively. By summing up (3.18), the total contribution to eq. (3.6) from the terms with one contraction is found to be i ( d - 26) 5
d3C w=0I V(1,2, 3)) 216 dw 3
(3.19)
This indeed vanishes when d = 26. The residues of the pole at z = z~' = - i in eq. (3.5) can be calculated in just the same way by making use of the projective transformation z+ i
w=--
z-i'
(3 20)
instead of (2.11) and can be shown that it vanishes when d = 26. Thus we have finished the proof that the vertex IV(l,2, 3)) given by (2.18) actually satisfies the distributive law (2.5). 4. Discussion
We proved that the distributive law of QB (1.1), which is equivalent to (2.5), holds for the Witten-type symmetric 3-string vertex (2.18) if the space-time dimension d is equal to 26. On the vertex functional we have rewritten V3r=lQ~r) into the form of a closed contour integral on the complex z plane. Then the proof of (1.1) was reduced to the simple evaluation of pole residues, the sum of which was found to vanish if and only if d = 26. Alternatively one can prove (1.1) by direct oscillator algebras, in this case one obtains one more condition a(0) (intercept) = 1, which was implicit in the above calculation, from the terms corresponding to the reordering of QBAlong with the line of the arguments given in ref. [2], our next task is to prove the associative law of the • product: ( * * ' / ' ) * A = O* ( # * A ) ,
(4.1)
140
K. Itoh et al. / Witten type vertex
which is conjectured by Witten [3] to hold without introducing 4- and higher-string vertices contrary to the JS type vertex case [2]. Once this axiom is established for the vertex (2.18), one can readily prove that the action [3,11, 2] S = C~ . Q Bcb + Z g cb . ( cb , cb )
(4.2)
is invariant under the stringy local gauge transformation 8 ( A )cb = Q B A + g ( ~ *
A-
A * ~),
(4.3)
with closed algebra [8(A1),6(A2)] = g 6 ( a I * A 2 - A 2 , A a ) .
(4.4)
(In this section we use the notation in ref. [2].) As was done in ref. [2] one can construct a gauge fixed action S simply by deleting the +-component of
(= -~o4, + 4,): g(q~) =S(qb)l¢= 0,
(4.5)
which is invariant under the global (on-shell nilpotent) BRS transformation
= f dl d2 (if(l)[ (dp(2)lW(3)eF(l'2'3)[0>123,
(4.6)
(the proof in sect. 6 of ref. [2] applies also here without any modification), although the precise gauge-fixing procedure is not yet well understood. The associative law (4.1) may naively be expected to hold from the diagramatic considerations. Two diagrams which contribute to (4.1) seem to cancel with each other, independently of the space-time dimension. However the determinants from the contractions of two vertices contain singularities as in the case of the horn diagram contributions in the JS-type string field theory [2]. For the latter case the careful examination lead to the condition d = 26 for the validity of (4.1). Therefore in the present case a more detailed analysis is necessary for a rigorous proof of the associative law. In this paper, we used the vertex with the ghost and the anti-ghost oscillator modes in the fermionic representation, though the original vertex proposed by Witten was written [3] in terms of the bosonized ghost and an oscillator expression of the latter vertex is already given by several authors [7,10]. The precise relation between these two vertices is still to be clarified. The authors would like to express their sincere thanks to H. Hata, T. Kugo and H. Kunitomo for valuable discussions and suggestions. They also thank T. Kugo for
K. Itoh et al. / Witten type vertex
141
careful reading of the manuscript. They are grateful to their colleagues at Kyoto University.
Appendix A PROPERTIES OF THE NEUMANN COEFFICIENTS
The Neumann coefficients ~ f f s corresponding to a 6-string light-cone diagram fig. 3 are defined as the Fourier components of the Neumann function N ( p , tS):
u(o~. &)
= -~
E -e-
~.-~.~cos(,,~)cos(,e~) - 2 m a x ( ~ .
(~
n~>l n
+2
E
~22e"~"+~'c°s('°~/cos(me~) - 2 8 , 6 ~ -
2~6g~
n , r~l ~>0
=lnlz-~l
+lnlz-~*l
(= ½1n((z-~)(z-Z*)(z*-Z)(z*-~*)}),
(a.1) where PR and t5s are assumed to lie in the region of the Rth and the Sth string strip, respectively, and connected to z and ~ via eq. (2.7) PR = p( z ) = an~ R + ifln,
(A.2)
Ps = P(~) = as~s + ifls-
The last two terms in the second expression (A.1) appear since we place Z 6 at the infinity. We can make a projective transformation (2.ll) or equivalently w+l
z = -i--,
(A.3)
w--1
in (A.1). Then the r.h.s, of (A.1) is rewritten as l n [ z - ~l + l n l z - ~'1 = lnl w - r?l + l n l w • * -
11 + 2 ( l n 2 - l n l w - 11 - I n l # - ll) (A.4)
We define another Neumann coefficients N£~s as the Fourier components of the Neumann function in the w-plane:
~(0~.~s) = -8~s
E xe-°'~-~'tcos(,o,)cos(,eA ^
- 2ma~(~.g~)
~
~2.Se"~"+m~scos(.oR)cos(mes)
+2 ~ n,m>~0
=lnlw-
~[ + l n l w ~ * -
11 .
(A.5)
142
K. ltoh et aL/ Wittentypevertex
Then by taking the limits ~--~ W6=1 ( ( s O - ~ ) obtain respectively
or w ~ W 6 ( ~ R ~ - ~ ) ,
we
^
lnlw - 11 = ~R6~R "q- Z N~6e"~"cos(noR), n>~0 ^
ln[~ - 11
=
~$6~S -Jr 2 N6Semg~cos(m6s) •
(A.6)
rn~>0
From eq. (A.4) we obtain a relation between N/,,,--Rsand NAm~Rs
"~R6+ N- -~R S _- N"2-~RS ~ - ~ . o N"6S d ~ - ~moNno
(A.7)
3.oSmoln2.
The general property of the Neumann coefficients can be found in appendix A of ref. [2]. For the case of N f s we can show the following relation in addition,
=
, : 1 . s +1.
( A.8 )
To derive this property, consider a transformation W ~ W' = e-~ri/3w,
(A.9)
which is a rotation around the origin in the w-plane. Thus if w lies in the Rth string region, w' is in the (R + 1)th string region (see fig. 4), and the coordinates ~ in the p plane corresponding to w and w' coincide with each other ~R = {R + 1,
(A.10)
where
p(w) = ~R~R+ i~R, p ( w ' ) = ag+l~g+, + iflR+l. From eq. (A.10) and the fact that Neumann function b~ in eq. (A.5) is invariant under (A.9), we obtain the relation eq. (A.8). Third we show the vertex (2.18) is invariant under the projective transformation (2.11). From eq. (A.7), we can express the coefficients N~Sm and N{~,, defined in eqs. (2.17) in terms of the Neumann coefficients N f s , =
Ars
-
~' = NF~ ^" = , NF~,,
n0t- 68"
"
+
+
°0
oo2,n2,
(A.11)
143
K. ltoh et al. / Witten type vertex
where ^ ^ ^ ^rs l(~rs "~r+3,s /WrmS+ 3 N r + 3 , s + 3 ] _ _ - - r s N_r,s+3 N B n m = 2 ~ " n m q- N ~ m + + - nm ] - - NJnm + nm ,
^ S~rmS+3Ji_Nr+3,s+3)__Nn N^rs F n m = !2( k~ "r ns _m_ ~ r +-3 ,nm - nm
^
m jWrmS + 3 " __ - - r s __
(A.12)
In eq. (A.12), the last equalities hold owing to the relation (A.8). We see from eq. (A.11) the coefficients of the ghost part of the vertex are projective invariant by themselves. As for the orbital part of the vertex, it is easy to see that the momentum conservation assures its invariance under the projective transformation. Thus we obtain an expression of the vertex in terms of the coefficients IVan,, and ]V~m,
IV(1,2,3)) = expE(1,2,3)10)(2~r)a8 a /~(1,2,3) =
E
Pr ,
(1]~SmOL(r)n " ~ ( s ) _~_ ~ ~ ~ r s . , ~ ( r ) ; ( s ) "l _ _ O-r~sZ v Fnmr~t, _ n" - m ] "
(A.13)
n,m~O
r,s=l-3
Finally we point out that one can show the following relations, NB. m - N,~m + rs _ _ - - r s NFnmNnm-
+ 6~o [~*.o Nrms+3
~*.o
]
t~ [ ~ 6 s ]~06mS+ 3 ) + V n O ~ Ore-,
(A.14) (A.15)
which result from an equality Nnm-
Ndm
--~m0(Nn~6- ~R0+3'6),
(A.16)
obtained by using eqs. (A.7) and (A.8). It is easily checked that N ~ , , and N ~ , , in eq. (2.17) can be replaced by the first two terms in eqs. (A.14) and (A.15) respectively.
Appendix B 3-STRING VERTEX IN THE co REPRESENTATION In this appendix, we derive an expression of the vertex in the c o representation, by making a Fourier transformation of (2.18) with respect to the anti-ghost zero
144
K. ltoh et al. / Witten type vertex
mode Co- The resultant vertex 17(1, 2, 3)} is 4~ri
17(1,2,3)) =
-f-~-C'(zo)C'(z~) I V0(1,2,3)),
I go(1,2,3))=(2~r)~M 3
ffS(1,2,3)=
E
(a.1)
(A)(rt ) p. 8
c(or)
ee°'2'3)10 ),
(B.2) (B.3)
N Brs , , ,(l_ot(r) t 2 - , . or(S) -m "4- nc(r)c - , - m(s)s ,]
E
r,s=l n,m>~O
where 1Iio(1 , 2, 3)) is an oscillator expression of (2.6) in the c o representation and thus it satisfies the connection condition (2.19). The operator valued function C(z) in the c o representation is defined as
C(z)= ( dP )
(B.4)
The vertex I V(1, 2, 3)) has the correct ghost number 3 due to the presence of the two ghost prefactors C'(zo) = dC(z)/dzlz=~o and C'(z~) = dC(z)/dzl~=z~. Although I Vo(1, 2, 3)) is projective invariant, the prefactors change under projective transformation. Thus the overall numerical coefficient is fixed when we choose a gauge for the projective transformation. The expression in eq. (B.1) corresponds to the special gauge in eq. (2.7). Now we give a derivation of eq. (B.1). First we rewrite (2.18) as follows, [V(1,2,3)) = I-I (1
--6(o')W('))er(l'z'3)lO>(2~r)dsa
s=l
3 w(S)-~ Z E "~r~s,tZ ~ ~'Mrs ~(r) , FnO~. _n r=l n>.l
(rt)
p~ ,
(B.5)
(B.6)
3 [ F ( 1 , 2 , 3 ) = r,s=lE [ n,m>~OE ~'Bnm2~-nl~]rs L~v(r)'Ol(s)"}--
,~, ~ M rs .o(r)7~(s) ] n,m>.lE t*rt~sZ'Fnm"t'-nt~-m] . (B.7)
The vertex 17(1, 2, 3)) in the c o representation is obtained by the Fourier transformation from eq. (B.5) 17(1,2,3))
-fr~=ld?(o')exp[-s~__?(o~)C(o"}lV(1,2,3))
=
3
~
(A)
H (c o +
Pr
s~l
(B.8)
~
~
=(c(ol)+w(1))(c(o2)+w(2))8
c(o~) r=l
eVO'2'3)10)(2~r)dSd
(A)
p~ . (B.9)
145
K. Itoh et aL / Witten type vertex
In the last equality we use 3 E W(r) = O, r~l
(B.IO)
which results from the following property of the 6-string Neumann coefficients [2], 6
E asN~s=O.
(BAD
S~1
Second we rewrite the ghost part of the exponent F(1, 2, 3) in eq. (B.5) into that of E(1, 2, 3) in eq. (B.3) up to the additive terms which contain c~0~) + w ~). These additive terms can be discarded in eq. (B.9) owing to the presence of factors I-Ir~l{,C " 0(r) + W(r)). For this purpose, we use a relation between the two coefficients N~,,. and N~,,.; 3 m N Brs. , .
+ ~r~snN~Sm
- - - "n m
E
(B.12)
s to , O t r O l t N ~rtn o N B m
t=l
which is derived by using
into
=
-n
+
(B.13)
E - ~TRr~Tsr T ~ I t'tTlYnO lYmO '
and eqs. (A.16) and (B.11). From eq. (B.12) we obtain a desired relation, 3 E E iv ~ M rs la,~(r)7.(s) ~ r ~ s z • F n m , ~ _ n ~. _ m r , s = l n,m>~l
3 E r,s~l
3 E ~ATrBnm s" "'~-"T'(r)~(s)n ~ - m -~- E n,m~O r , s = l n>~l
E nNg~.o(C(o'+W(')e(5).. ( B . 1 4 )
Therefore the IV(1, 2, 3)) is rewritten as
IF(a, 2, 3)) = (C(o1) + w(,)(C(o2)+ w(2))8
C(o" e~(~'2'3)10)(2~)~8~ r=l
= (C(o~ + w (l~)(c(g) + w (2~)1VoO, 2, 3)~
( fl) Pr
(B.aS)
Finally, we show that the ghost prefactor (C(oa) + w(1))(C(o2) + w (2)) in the above expression is equivalent to the operator valued function at the interaction point C'(zo)C'(z~) in front of the vertex 11/o(1,2,3)). Let us calculate C(+r)(~r) on the
146
K. ltoh et al. / Witten type vertex
vertex (B.2), using eqs. (2.1) and (B.14), 1
+o¢
C(+~)(-i~)lVo(1,2,3)) = ~
~
1[
c~)e'~qVo(1,2,3))
3
=~
E C~),.,e-":r--E n>~O
E
"
"
~r~s
,,,,.s
a" Fnm
.~..(~)o":r
Ir~-
m ~"
s = l n,rn>~l
+ ~2 (C(o' ) + w ( ' ) ) E nNg',,oe'~ IVo(1,2,3)), s=l
n>~l
]
(B.16) where we consider only the 1st, 3rd or 5th string which a n = 1 on the p plane. In order to evaluate each term with fr placed near the interaction point in eq. (B.16), we need some relations with the Neumann coefficients which we derive in the following. By picking out the terms analytic in z and ~ from (A.1) with proper redefinition of constant part ~ s , we obtain
kn>~I n
[1
+o((~-~.) Z-e":'<:,)-(~+i~
])
n>~l 1l
-t- E ~[2SenfR+m~s--(~R'6~Rq-~S'6~S)
(B.17)
n,m>~O
(cf. eq. (A.8) of ref. [19]). Taking z in the neighborhood of the interaction point (thus in > (S), we differentiate (B.17) with respect to fn or (s, 0z
__ O~R z -
1
= ~n,S ~
Iz
e n(~s-~R) +
t .>~x -{- E
E
]
1
rl~fRSen~n+m~s-- ~ R , 6 ,
(B.18)
n>~l rn>~O
Off
1
O~s z - ~
.>~1 + E
E mN/se";R+m~-~- ~S,6.
n>~O m>~l
(B.19)
K. Itoh et al. / Witten type oertex
147
In eq. (B.18), we take the limit ~ --+ Z s ( ( s --+ - oo) Oz
+ 8R'6 +
E n N ~ se"~" = - s r ' s n>~l
1
- -
(B.20)
O~R Z -- g S "
We multiply e - " ~ (m >/1) to eq. (B.19) and integrate with respect to (s around the point on the o-plane corresponding to the Sth external string. Then we obtain,
mNRSe"~R = 8RSe-m~r -- mNgS - ~zs d~ _ _ 1 n>~l--" "nm -
2~ri z --
e_m(s,
(m >~ 1).
(B.21)
Using eq. (B.21), we rewrite the first two terms on the r.h.s, of eq. (B.16),
1[
- - . e-< ¢g Y'. ~'~
3
E
-
1[
en]
a/~ sNy.mmc(SL
s = l n,m>~l
n~>O
= ~
E
c'0'~+ w"~ +
E
%~,%
s = l rn~>l
X~
7"-7.
Z7] e 'n~s
_.g
d~
1
1
zs+3 2~ri Zr-- ~.
Z~+3-- ~
where z, and z,+ 3 are related to ~, as follows,
o( z,) = ~,~r + iB,,
(B.23)
P(z~+3) = a,+3~', + ifl~+3,
and thus z,+ 3 = - 1 / z r When ~, takes the value corresponding to the interaction point, z, and z,+ 3 coincides (z~ = z,+ 3 = i) and the terms in the curly bracket in eq. (B.22) vanish. As for the third term of eq. (B.16), we rewrite the summation by using eq. (B.20),
n>~l
"~r
Zr-
Zs
Zr-
ZS+3
zr+3 1 +
0~r ~z,+3
--
1)} Zs +
Zr+ 3 -- Z S + 3
. (B.24)
148
K. Itoh et al. / Witten type vertex
Then we can explicitly calculate the quantity in braces in eq. (B.24) by using eq. (2.7).
( 4__4_e~i/3 a, + 0(1) 17~ z-i { } = ~ 4_4__e_,,i/3 a r lOft-
( s = 1) (B.25)
z_i +0(1)(s=2) ( s = 3).
From eqs. (B.22), (B.24) and (B.25), the singular part of c(+r)(i~,.)lVo(1,2,3)} is given as
2 z Ta,.i [e~,i/,(c(ol)+w(n) 3ff~
C(+")(-i~r)IV°(I'2'3)>
- e - ~ i / 3 ( C ( o 2) +
w(2))] I Vo(1,2,3)>. (B.26)
Similar calculation of Cff)(i~ * )1Vo(1,2, 3)) leads to C!r)(i~*)l V°(1'2'3)) - ~
2
O/r z* + i [e-'~i/3(c(°l) + w(1)) -e'i/3(C(o2)+ w(2))]lVo(1,2,3)}. (B.27)
Therefore from (3.2) and (B.15), we obtain
C(p )C(o* )l go(1,2,3)> = [
4i
1
3~
(z--i)(z*
"}-i) [ il
+0
= -
-z-i"z*+i"
' [ '2
"~[C~)_~.)_W(1)}[C~.)_W(2)}
(1 1)1 (1 1)] z - i' z* + i
+0
z-i'z*+i
I Vo(1, 2, 3)) 1V(1'2'3)) (B.28)
Since eq. (B.28) diverges at the interaction point z o = i, we multiply do/dz: do
3
-dTz = -~i(z- i) 2 + O ( ( z - i)')
(B.29)
K. Itoh et al. /
149
Witten type vertex
and its complex conjugate to eq. (B.28). We then differentiate it with respect to z and z * and take the limit z ~ z o = i and z* ~ z~'. The result is
C'(zo)C'(z~)1V0(1,2, 3)) = -3¢3-_ ~ - 1 V ( 1 , 2 , 3)).
(B.30)
Appendix C CONTRACTION FORMULAS
We show eq. (3.15) by using the expression (A.13) of the vertex. The contraction of two operators O1(0) and 0205) implies the following quantity
Ol(P)b2(p) ~ 0 ( ( - - ~)<0101(P)O2(P)lO>c ~[Ol(p),[O2(/~),E]]:g
-}-
0(~
--
,()
,
where the suffix c denotes the connected part and - ( + ) sign in the r.h.s, should be taken when both of 01 and 0 2 are fermionic (otherwise). For example, from (2.1), (3.2) and (A.13) we have
c(o)c(p)
=
[(O((S-'R)( ~-"en(~R-fsl+l)-O(~R-ffS)n>l y'~ x(~g's+sR+3"s)
+
,v ,~, ..... ~ l O FR,S, , "~ n ~ n ÷ m ~ s /]' z., ~R~'s
n~>0 m~>l
(C.2)
]
where P and t5 are assumed to lie in the regions of the Rth and the Sth string, respectively. Recall that the Rth string and the (R + 3)th string have the same oscillators. We have used the following properties, ^R,S_ ~R+3,S OtROtSNFm n -- OlR+ 3OlslYFm n
~ ^, ~R,S+3 ~ tlRuS+31VFm n
^ 7trR+3, S+3 = OIR+3OIS+31¥Fm n
(c.3)
^RS (R, S = 1 - 6) are defined as where the coefficients NF,,, ^RS
"-~RS "R+3, S
NFn m = Nnm - Nnm
(C.4)
150
K. ltoh et al. / Witten type vertex
(cf. eq. (A.12)). Then (C.2) can be rewritten as
-O(l~,-~s) E e"'gs-~)} 8"s+ E mT~fse"~+"~s n>>.l
n>~O m>~l
aSI{O(~s-liR)( hal
-O(~.-(s) E e"'gs-~') B"+''s+ n>l
E
(c.5)
S e n~a+m( s m N "R+3, ~m
n>~O mNl
The first term of the r.h.s, can be read from eq. (A.5) as
1 2_~Oa~(p,p) = Iras ass I~-pan
1 d~
1
(C.6)
rr d r w - ~ '
since (A.5) can be rewritten as
kn>~l
[1
])
+O(~R-- (S) Z -~n(e-n~"+e-n~Y)(en(s+en~s*)--2~R n>~l -OARS n~R "* +51 E Nnm(e +en~1)(emfs+em~s) •
(C.7)
n, m>>.O
The second term of (C.5) is obtained from (C.6) by the following replacement
P= erROR+ iflR--+ 0/R+3~R+3 -1- iflR+3,
~R+3 = ~R
w--, (e-'i/3)3w= -w.
(c.8)
Thus we have & p ) e {
-
P)
ld#(
~r d~
_
1
1)
_
_
_
w-~
+w+~
"
(C.9)
151
K. ltoh et aL / Wittentype vertex
Eq. (3.15b) is an immediate consequence of (C.9) and (3.7). Derivation of (3.15a) is quite similar.
Appendix D PROOF OF EQ. (3.12) In this appendix we prove (3.12) directly from the properties of Neumann coefficients. First we show (3.12a). By using (2.1) and (A.13) we can write A~+R)(or-- i~r) on the vertex as A~+R)(o, -- i~r) I V(1,2, 3))
1(
=
f~
3
E a(R2e - ' L + E n>~O
S=I
E ,.~~j , R S ~(S).,L I V(1,2,3)), ,Bnm~ rnl.,
(D.1)
n>l rn>~O
where
(D.2)
~ = ~ + ion, n m - - S" n m
- - - "rim
o • ( R__)~ ( R + 3 ) n
----n
(D.3)
:'
(D.4)
•
Since ~ f s has the following properties (see appendix A of ref. [2]),
E
n_n~,se,~R= ( d p ( w ) ] - 1 1
.>~10IR
~ dw
.0
nm
n---~
n ~ -~>,E--NfSe'~R o a r w~
O l s UI=
]
W--
1 W S
8 Rs,
(D.5)
ol R
(D.6)
UtllVnOlVraO~
(dp(w)]-l~ a, m_* dw ] 1 = 1 W0 -- W, a s NS~' (m>/1),
(D.7)
the coefficients of a(_s~[V(1,2,3)) in eq. (D.1) multiplied by d p ( w ) / d w can be
K, ltoh et al. / Witten type vertex
152
evaluated at the interaction point as
dp(w)(
~"nN~Sme'¢') n>~l m
) O~R Z
w'-*wo
dp(w)
/
dw
\
I=1 WO-- W I
°ts
mO
(m>/1)
O/ ~'m0 S+3
18 Rs + 8R,s+3 +
1)
- - + n>~l
/ w-'*Wo
\ Wo-
(D.8)
=0,
W 0 -- W s + 3
Ws
(D.9) where we used the following properties
W S +~ 3-- Ws~
Wo=O~
°ts+3 =
-°ls~
~mS+3,1__~S,I+3 (D.IO) 0
- - S"m0
"
Thus we obtain
A(w = 0)1V(1,2,3)>
(D.11)
= 0,
from the definition in (3.7) and (3.2). Similarly from eqs. (2.1) and (A.13) we can write
c~+R)(o~- i~r)l V(1,2, 3)) =
1(
~R~'S..... r,,,,,'- -., "
n~>l
I V(1,2,3)).
(D.12)
n>~O m>~l S=1~3
In order to prove (3.12b) it is sufficient to show the coefficients of i.e.,
1
( _ 8R,S+3 + 8 R S ) _ _ e - , ~ , _ ~ m ,~0
~'F.,."~RS Ofl~r
(m >~1)
aRasme~ (D.13)
vanish at w = 0. To this end we pick out the pieces analytic in w and ~ from (A.5) with proper redefinition of constant part No~s (see appendix A of ref. [19]),
ln(w-w) =-SRs O(~R--(S) E-e"(~s-~")-~'. n ~ l FI
+
E n, m>~0
N'2mSen~R+m~s-
(D.14)
153
K. Itoh et al. / Witten type vertex
As is explained in appendix A, when ~ lies in the region of the Sth string with the string coordinate ~s, - ~ corresponds to the point in the (S + 3)th string strip with the same string coordinate ~s. Therefore when ~R > ~s we obtain ln(w
_
=
~)-ln(w+ff)
_
f n ~ (8 R s - 3R ' s+3 )1 X - e ( ~ s - ~ R ) - t ' R ] \ .~i n ]
+
~,
(~s_
~f,;s+a)e~R*m?s"
(D.15)
n,m~O
In the above equation when w equals to zero (~R --- ½~ri), the 1.h.s. becomes constant. Thus comparing the coefficients of e mrs (m >~ 1) we obtain ^
--(~RS--sR'S+3) -e-mf"+
m
(m>~ 1),
E N~Sme"~R=0
n~>0 ~'R = ½~ri.
(D.16)
This leads to (3.12b). Appendix
E
CONTOUR INTEGRATION
Using eqs. (3.15), we rewrite (3.16a) and (3.16b) as
[ ~CwodW
wo
dw'
dw
-
1
1)
-- ~
(W--W') 2
(w+w')
+
(w-w,)Z+
(w+w') z
"--d-'~w ] \ _ _ ~r
_w' _- +w
2
w--7-----+w
IV(1,2,3)),
(E.la)
1V(1,2,3)),
(E.lb)
respectively. In order to calculate (E.1) we expand (2.12) around w = 0,
p(w) -- p0 + 2 w3 + O(wg),
(E.2)
where P0 = p(0) = -~-rri. Then from (3.17) we have w,3 + O ( w , 9 ) = ~ + w ~ + O(wg).
(E.3)
154
K. Itoh et al. / Witten type vertex
We expand w' in terms of 8 as (E.4) n~l
Inserting (E.4) in (E.3) and comparing the coefficients of 8" (n = 1, 2, 3), we obtain
1
-- ~-- 3w 3 -t- O(w9),
(E.5)
f?
- 1 "4-O(W6),
(E.6)
y,
5
f, k
~13 ~---"~ + O(W6).
(Z.V)
From these equations we obtain 1 w;
-
(
)-1
= wEL(w)8 w n>~l
n
/
11
l f2
-~1~
wi?
+0(8)
1 1 = O(w2)a + --w + O(w') + O(8), 1
1
w' + w
2w
(E,8) (E.9)
+ 0(8 / .
For the calculations of the quantities in eq. (E.la)
dw'
(w -t- Wt) 2
dw
]
dw ( w + w , ) 2'
(E.10)
we have the following equations from (E.4) - (E.7),
dw
1
(
dw (w' - w) 2 ~-- 1 + n~>lE- - S h a w
w n>lEfn 8
11 -12 [ dill
= w~lIlJ 82 + - ~
1 2 f2]l w2 fl f 2 8
2 f3 wZ f 3 1
1
2 1
+ o(8)
= O ( w 4 ) ~ -+O(w7)~ + ~ - - - 2 + 0 ( w 4 ) + 0 ( 3 ) , dw' 1 1 = 4w--7 + O(a). dw ( w , + w) 2
(E.11) (E.12)
155
K. Itoh et aL / Witten type vertex
The pole residues of (E.1) are calculated by making use of the formulas, dp(w')
-
1 1 _ 1 ( w2 w5 ) (W'--W) 2 = 9W4 + O 82, 8 ' w2 + O ( 8 ) '
dw'
dp(w') dw'
-1
1
(E.13)
1
(w' + w) 2
-- - -
24w 4
+ O ( w 2) + O ( 8 ) ,
[dp(w)] -11 --1 ( 1w 3 ) + O ( 8 ) dw ] w ' ~ - w = 6w 3 + O 8 '
(d0(W)dw)-l__w,+wl = - - 1 23 w 1 +O(w 3)+O(8).
(E.14)
,
(E.15)
(E.16)
We evaluate (E.1) by performing the contour integration first with 8 kept finite, and hence the 0 ( 8 -2 , 8 -1 ) terms do not contribute. The 0 ( 8 ) terms vanish after taking limit 8 ~ 0. Thus we have
(E.la)=(-2~ri)
=(-i)
22-5d 72~r
i d3C 4 IV(1,2,3)) 3! dw 3 w=o
22 - 5d d3C w=01V(1,2,3)) 216 dw 3
(E.lb) = ( - 2 ~ ' i )
(E.17)
1 1 -d3C w=0I V(1,2, 3)) 2~r 2! dw 3
1 d3C w=OIV(1'2'3)5"
(E.a8)
References
[1] H. Hata, K. Itoh, T. Kugo, I4. Kunitomo and K. Ogawa, Phys. Lett. 172B (1986) 186; A. Neveu and P.C. West, Phys. Lett. 168B (1986) 192; Nucl. Phys. B278 (1986) 601 [2] H. Hata, K. Itoh, T. Kugo, H. Kurtitomo and K. Ogawa, Phys. Rev. D34 (1986) 2360 [3] E. Witten, Nucl. Phys. B268 (1986) 253 [4] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D35 (1987) 1356 [5] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. 172B (1986) 195; Phys. Rev. D35 (1987) 1318
156
K. ltoh et al. / Witten type vertex
[6] S.B. Giddings, Nucl. Phys. B278 (1986) 242; S.B. Giddings and E. Martinec, Nucl. Phys. B278 (1986) 91; S.B. Giddings, E. Martinec and E. Witten, Phys. Lett. 176B (1986) 362; S. Samuel, Phys. Lett. 181B (1986) 249 [7] D.J. Gross and A. Jevicki, Nucl. Phys. B283 (1987) 1 [8] S. Samuel, Phys. Lett. 181B (1986) 255 [9] E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. 179B (1986) 57 [10] N. Ohta, Phys. Rev. D34 (1986) 3785 [11] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Nucl. Phys. B283 (1987) 433 [12] S. Mandelstam, Nucl. Phys. B64 (1973) 205; B69 (1974) 77; B83 (1974) 413 [13] Z. Koba and H.B. Nielsen, Nucl. Phys. B12 (1969) 517 [14] S. Mandelstam, Nucl. Phys. B83 (1974) 413 [15] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D35 (1987) 1318