- Email: [email protected]
String couplings from meromorphic differentials
Nuclear Physics B310 (1988) 141-162 North-Holland, Amsterdam
STRING COUPLINGS FROM M E R O M O R P H I C DIFFERENTIALS Theophil OHRNDORF* Institut ffir Theoretische Physik der Universitiit Heidelberg, Philosophenweg 16, D -6900 Heidelberg, FR G
Received 5 April 1988 (Revised 27 June 1988)
The "Neumann-function method" which allows one to determine oscillator representation of multiple string verticesis put on a firm basis. The Neumann coefficients are related to the Laurent coefficients of certain meromorphic differentials. A systematic procedure of constructing the differentials is developed. The method is exemplifiedby the "covariantized" light-cone vertex. Our results on the ghost part of this vertex are in disagreementwith various proposals in the literature,
1. Introduction Recent attempts to construct interacting second-quantized string field theories [1-5] have produced a considerable interest in finding explicit oscillator representations of string vertices. At present, several groups have presented such oscillator representations for the various field theories which have been proposed [6-14]. As two-dimensional conformal field theory or equivalently Polyakov's path integral provide a natural framework for the first-quantized string, one expects that these two approaches play, as well, a major role in finding string vertices. However, so far the virtues of these two approaches have not been exploited to their full power in the problems of (i) defining vertices, and (ii) their explicit construction. It is the purpose of this paper to present several technical tools which allow one to calculate arbitrary vertices starting from the path integral. These techniques may be considered as a generalization of those originally developed by Mandelstam [15]. Recently, there was a proposal [16,17] to reformulate Polyakov's approach completely in terms of an operator formalism. The operator formalism proposed in refs. [16,17] is based on the path integral on punctured Riemann surfaces. In principle, such a formalism could be a very useful tool to construct string vertices systematically. Unfortunately, the fact that the authors of refs. [16, 17] base their method on punctured Riemann surfaces makes its immediate application to the problem of constructing string vertices somewhat complicated. * Supported by D.F.G. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
142
T. Ohrndorf / String couplings
An equivalent approach which, however, is technically much simpler makes use of bordered Riemann surfaces [18,19]. In the present paper we shall exploit the fact that string vertices may be considered as transition functions on bordered Riemann surfaces to outline a method which allows the systematic construction of string vertices. We shall work out the oscillator representation of the vertex corresponding to the so-called "covariantized light-cone" field theory [1-4] in detail. Our results for the ghost part of this vertex (see eq. (3.43)) are in disagreement with those that can be found in the literature [2, 4]. This discrepancy is caused by the fact that previous authors do not seem to have incorporated the transformation properties of the reparametrization ghosts under conformal transformations in an adequate manner. We have decided to work out the vertex for the covariantized light-cone approach mainly to give an explicit example. Our methods can easily be generalized to other geometrical configurations of interacting strings, like e.g. Witten's vertex [5]. So far in the literature mainly two methods have been employed to construct string vertices, namely Mandelstam's Neumann-function method and the overlap 6-function method of Cremmer and Gervais [20, 21]. Both methods were developed in the early days of string theory. We find that both methods are unsuitable to allow for a systematic derivation of string vertices in a reliable way, in particular of the ghost sector. The overlap 6-function method is heavily based on manipulating infinite-dimensional matrices. However, the method does not ensure that the sums occurring are convergent. The Neumann-function method does not suffer from this disease, but, however, it is a priori unclear why it is sufficient to work with the Neumann function on the world sheet with infinite strips attached, as Mandelstam does, to get the transition amplitude for finite strips. In particular, there are factors of 2 which have to be adjusted " b y hand" (see commentary preceding formula (4.4a) of ref. [15]). In contrast to these methods our method is solely based on the fact that the derivative OzX of the classical string coordinate X, as well as the classical ghost coordinates c z and bzz are analytic on the world sheet. In this way it is easy to control the region of convergence of the infinite sums involved. By integrating products of suitable differentials with the fields O~X, c z and bzz along the boundary of the world sheet, identities among the various oscillators acting on the vertex are derived, which we shall call transition identities. These identities may be viewed as an infinite set of first-order differential equations, satisfied by the vertex. They may be integrated to yield the vertex explicitly. We would like to emphasize that the virtue of this method lies in the fact that only a minimal set of assumptions is needed as input. This method has first been applied in ref. [2]. The same authors have developed this method further in refs. [22-24]. Technically, it is obviously of vital importance to form products 0, X, c: and bzz with differentials which are chosen in such a way that the solutions of the transition identities become as simple as possible. One way of finding such differentials
7". Ohrndorf / String couplings
143
consists of first filling in the boundary disks of the world sheet which represents the vertex, and then constructing differentials which have a single multiple pole on one such boundary disk. In fact, the Neumann coefficients are just the Laurent coefficients of these differentials. Using a kind of dispersion relation, it is easy to show that these Laurent coefficients are almost uniquely fixed. The rest of the paper is organized as follows. As a warm-up we shall rederive the bosonic vertex in sect. 2. The purpose of this section is to introduce the method with the help of a simple example and to convince the reader that our method does in fact reproduce the known result for the bosonic vertex. For completeness we shall evaluate the Mandelstam coefficients in appendix A. In sect. 3 we shall demonstrate how the method introduced in sect. 2 can be generalized to calculate the ghost vertex. The relevant coefficients will be evaluated explicitly in appendices B and C. We conclude in section 4 with some comments.
2. Bosonic couplings This section has mainly an introductory character. We shall present a systematic way of deriving the form of the three-string vertex, as it has originally been obtained by Mandelstam [15]. Our starting point will be the fact that the three-string vertex is just a Fock-space representation of the path integral defined on a three-strip world sheet, if considered as a bordered Riemann surface. The major tool, which will enable us to write down an explicit expression for the vertex, is simply Cauchy's theorem. In contradiction to Mandelstam's original approach we shall not make use of the Neumann function. Therefore, our approach can be generalized to the string ghost coordinates with only minor modifications. The basic object of interest in this section is the transition amplitude Kx, which describes the interaction of three strings. It may be represented by the path integral on the corresponding world sheet M, which in the present case consists of three rectangular strips glued together as indicated in fig. 1. We shall denote the individual strip regions by M s, their vertical boundaries by Cs, s = 1, 2, 3, and the horizontal boundary of M by (0M)i~. In contrast to a procedure often employed in the literature, we shall not let the strips tend to ~ , rather they may take any finite length. Nevertheless, sometimes it is useful to consider the infinite-strip world sheet, which we shall denote by M~. Working on M, K x is a functional of the boundary value, which we impose on 0,X(o), the normal derivative of X along the vertical component of the boundary of the world sheet and a function of the moduli of M.
]2
i q
I
,
3
I i
Fig. 1. Geometry of the world sheet M.
144
72 Ohrndorf / String couplings
For simplicity we shall not display any arguments of K x explicitly. In order to make our ideas quantitative, we have to introduce coordinates on the world sheet. We shall place the strip diagram displayed in fig. 1 onto the complex plane equipped with a coordinate p in such a way that the bottom horizontal boundary coincides with the real axis. To label the width of the three strips we introduce three real parameters o/,, s = 1,2,3 with %, a 2 > 0; a 3 < 0 such that a, = 0. The width of the s string is then given by ~r[o/~[. To make K x well defined, we have to introduce a parametrization of the boundaries Cs. We shall use the same parametrization as it has already been employed implicitly by Mandelstam. Namely we shall use the angular variable on the half annulus obtained by mapping M, via
0/1
/~1 =
0,
/~2 = -- --iTr,
]~3 = +iqr,
(2.1)
O/2
onto a f, plane. Technically it is convenient to consider the strip region with the strips extending to infinity M~ as being obtained by a conformal mapping from the upper half plane. The so-called Mandelstam mapping o(:)
= % l o g ( z - 1) + o/2 log : ,
(2.2)
achieves precisely this. Note that ( 0 M ) H is the image of the real z-axis. The inverse mapping z(o) has a square-root branch point at O = Py = P(ZT), where ZT, the "turning point", is determined by do dz
=0.
(2.3)
It is given by ZT =
-- 0/2//O/3 •
(2.4)
Though one may derive a formal representation of K x in terms of the Neumann function on M, finding the explicit form of the Neumann function on the three-strip world sheet, and of the functional determinant of the associated laplacian, is highly nontrivial. Fortunately, this problem can be circumvented. As we shall demonstrate below, a Fock-space representation of K x is readily given by solving certain identities, which we shall call transition identities (TI), which exist among the oscillators of the operator associated with OzX. We shall distinguish "transition identities", which hold for oscillators, from "overlap identities" which hold for OnX(o ), b(o), and c(o), which are only defined on OM. These identities are a direct consequence of the fact that OzX is holomorphic on M. Compared with the
T. Ohrndorf / String couplings
145
"overlap identities" [20,21], the transition identities have the advantage of being represented by convergent sums. We shall start from the following assumption, which can easily be justified by employing the path-integral representation of Kx, as it has been given e.g. in ref. [18]. For each boundary component C~., s = 1, 2, 3 of M, there exists an operator 0~](', which has an expansion 0~s 2 " =
(2.5)
-½i .V .~ -. n .- lo~s . tt
with
[~,", ~2"]
=8"sn"'8,,
_m
.
(2.6)
0f J(" has the property that it reproduces the classical field Of X by acting on K x. The oscillators a s are represented as differential operators in terms of the coefficients of the normal-mode expansion of O,,X(o) along C~.. Instead of working with K x as being defined by the path integral on a bordered surface, it is technically much simpler to work with a Fock-space representation. This may easily be obtained by writing K x as a convolution of some integral kernel, with coherent-state wave functions for the oscillators a,~'~, n =~ 0, and taking the Fourier transform with respect to the momentum pr. We shall denote this integral kernel as well by the symbol K x. An explicit example of how to go from the X( o )-representation to the Fock-space representation may be found in ref. [25]. Within this formalism it is readily understood that the path integral on the disk represents just the commonly adopted vacuum state. Denoting the standard coordinate on the disk by w, and expanding as in eq. (2.5), all the oscillators with n > 0 must annihilate Kx, otherwise 3wX would have a singularity at the origin. In the present context the world sheet is represented topologically by a three-holed sphere. Accordingly, a single string state would be represented by the path integral on the complement of the disk. In other words, the vacuum had to be annihilated by all the oscillators a ,~,r, n ~< 0. In order to be in accordance with the standard Fock-space notation, we shall therefore always let 0~)( act to the left, i.e.
KxO~SU= Kxa~ X,
(2.7)
if p is near the boundary curve Cs. Note that the right-hand side of (2.7) is independent of s. However, by acting with O~Xs on K x we obtain O~X represented by a particular series. This series need only to converge in a small neighborhood around C,. Nevertheless, the function which is represented by such a series has a holomorph continuation to all of M. It is now a simple matter to derive the general form of the TI. Suppose a function F(p) is given, which is analytic in the coordinate z on M if eq. (2.2) is employed,
T. Ohrndorf / String couplings
146
and has the additional property that it is real if p ~ ( 3 M ) H
lm( F(p ) )lo~(OM). = O.
(2.8)
Then, C a u c h y ' s theorem implies that 3
~, f
(2.9)
]
as Im(lSOoX ) = 0 along (¢gM)H. Making use of eq. (2.7), this statement m a y be turned into the operator statement 3
Kx ~, fcIm(d'~,F(o)o;f(s)=o.
(2.10)
,';=1
Let us next turn to the construction of functions F ( p ) , potentially to be used in eq. (2.10). A p a r t from the constants which result into the conservation of energy m o m e n t u m , if inserted into eq. (2.10), there are no other functions which are analytic on Moo. We shall therefore postulate that F(p) has a m-fold pole at ~r = 0. In this way we expect to obtain a set of functions Fr(O), labelled by m > + 1 and r = 1,2, 3, normalized like lim F £ ( p ) z~z
= ~'7 m + analytic,
(2.11)
r
where z 1 = 1, z 2 = 0 , and z 3 = + ~ . The requirement that F,~(p) has only a single pole allows one to determine F~(p) almost completely from eq. (2.11). To this end we start from Cauchy's integral formula
1 F,(,(p') = ~, dz z - z' Fm~(P) '
(2.12)
where the c o n t o u r of integration runs around z'. We shall always absorb a factor 2~ri into 9~. T h o u g h F,~(z) has only been defined on the upper half plane, it can be analytically continued to the lower half plane with the help of the Schwarz reflection principle, as Fr(z) is real along the real axis. N o w we deform the contour of integration in eq. (2.12) in such a way that it encircles z = oo and z = z r if r 4: 3. If r 4= 3 F,~(p) has to go to a constant if z ~ oo. Therefore the contour around ov produces just this constant. N e a r z = z r the analytic piece of eq. (2.11) can be dropped, only if r = 3 a constant term contributes. In this way we get
Fr(p ) = -
z~
N o t e that apart from the constant c,~,,
dz
1 Z --
Z i
fr m + £ r ~n "
(2.13)
F,(,(O) is uniquely determined by eq. (2.11).
T. O h r n d o r f /
147
S t r i n g couplings
Writing down a Taylor expansion for the analytic piece of F,~(p), let us say in powers of ~,, we cannot expect that such an expansion converges on all of M. A generic function which is analytic in z has a branch cut if considered as a function of p, and therefore of ~s as well. Therefore Fr(O) has different Taylor expansions on the various 34, r ~ ,r( p )
__
--~rs~r
m
--
rs s+n , mN~,~
p~M,,
n>~O,
m>~ + 1 .
(2.14)
In eq. (2.14) summation over n, but no summation over s is understood. The coefficients N ~ are determined from eq. (2.13) to be
mNr~ =
dz .
~
[
fPs
n
]
dz~r
m
Zr
~n,oC~, " Z
--
Z~
(2.15)
It m a y be checked that the coefficients N ~ are indeed real, and the requirement (2.8) is satisfied. Eq. (2.15) will be evaluated explicitly in appendix A. Before we solve eq. (2.10) with eq. (2.14) we would like to emphasize one point which has been ignored so far. In deriving eq. (2.9) we have to be careful that any contribution originating from going around the turning point z T is absent. This will obviously be the case as we may perform the integral around the turning point in the z-plane, and F~ is non-singular at z T. One may be tempted to replace F,~(p) by its derivative d F £ ( o ) / d p to get new TI. However, as d z / d p - ( p - p T ) -1/2 eq. (2.9) ceases to hold as we get a nonvanishing contribution from going around the turning point. Let us finally turn to the solution of eq. (2.10) with eq. (2.14). To write the solution in a compact way we shall agree on the following summation convention to be used in the remainder of this paper. If not otherwise stated, we shall always sum over repeated indices with r, s = 1, 2, 3 and m, n . . . . running as indicated. Inserting eq. (2.14) into eq. (2.10) gives
Kx(a~_m - mN,~,a~) = 0,
n >~ 0,
m >/1.
(2.16)
The solution of eq. (2.16) is conveniently expressed with the help of the operator
V x = exp( ~amlVm,a,+amN,~oao}, 1 r ~rrs s r rs s
m,n>~l.
(2.17)
Due to N ~ = N;,~,
n, m >~ + 1,
(2.18)
we have
vX(a~m - mN,~a~) = a~mV x,
m >~ + 1 ,
n >/0.
(2.19)
Therefore, the general solution of eq. (2.16) is
K x = 8(p~ +P2 + P3)(P[ v X f ( P , ) ,
(2.20)
148
T. Ohrndorf / String couplings
where 3
(P[ := I-I (0, Prl.
(2.21)
r=l
f ( p , ) is a function depending on p, and the moduli of M. In the present case one may take the lengths of the strips M,, or equivalently the radii R, of the image of the Cs under ~s(P) as parameters. To make the R s dependence of f explicit, we exploit the fact that changes of R s are generated by L 0 (see ref. [180, i.e.
0 Rs--Kx OR,
= KxL'o,
(2.22)
where it is understood that K x is in the X( o )-representation. For the n ~ 0 modes one may solve eq. (2.22) simply by rescaling the arguments of the coherent-state wave functions, so that in the Fock-space representation only 0 R.~--Kx=
2Ps12Kx '
ORs
(2.23)
is left. Therefore f ( p , ) ~ exp(½)2 p2 log R s ) .
(2.24)
Letting the lengths of the strips go to zero, we may express R s in terms of the % R, = exp
Re(o-r) ,
(2.25)
with Re(0T) = c q l o g ( - - ~ ) +
%log(- ~).
(2.26)
3. Ghost couplings In the present section we shall generalize the method which has been exemplified in the preceding section to the couplings of the ghost oscillators. We shall proceed in a manner completely parallel to the determination of the bosonic couplings. The main difference being that the function F,~(O ) has to be replaced by either an inverse differential or by a quadratic differential. In ref. [19] it has been shown how to construct ghost operators in terms of bS(o) and cS(o) being defined on C,. The oscillators being defined by the usual expansion t5~3, = E~'s n 2bS '
(bn)* = b,,,
(3.1)
(c',,)* = c,,
(3.2)
n
6,~ = EfS"+'c~;, tl
T. Ohrndorf / String couplings
149
satisfy the anti-commutation relations (3.3) The ghost transition function K~h is represented by Kgh = Be- Sc~.
(3.4)
In the c ( o ) - b ( o ) representation there is an additional determinant, which we shall, however, suppress. The explicit form of ScL has been given in ref. [19]. 8 is defined by
8 = FlSk,
(3.5)
k
with 8 k := foMdO Im(h~z~2C(O)),
(3.6)
where h~z is a holomorphic quadratic differential on the Riemann surface obtained by taking the union of M with its mirror image with respect to real axis. Note that only those differentials contribute which satisfy (h~z) k , = h~, k as we are dealing with the open string. Therefore, we have
Im(eZh z)l
0M = 0.
(3.7)
For the closed string the product (3.5) includes all the holomorphic quadratic differentials on M. It will be shown in ref. [26] that the 8 k must be present in Kgh to ensure that the classical ghost field c z is analytic on M, and that it is possible to define a ghost-number operator in such a way that the ghost-number violation is solely determined by 8. We would like to make clear that the &function (3.5) has to appear in the c ( a ) - b(o) representation. It has to be distinguished from those &function-like contributions previously identified in ref. [13] which we shall call G k (cf. eqs. (3.17) and (3.43)) which appear in the Fock representation. The ghost transition amplitude Kg h s a t i s f i e s e - Sc@s~s;8 = Kghb~,; , Kgh@ = Kghc~, ,
(3.8) (3.9)
if p is near Cs. b~, and c ~, have to be considered as classical ghost coordinates. 3.1 ANTI-GHOST TI
Let us start with a systematic derivation of TI for the anti-ghost operator b~. As b~ is a quadratic differential, we have to replace the function F used in the
T. Ohrndorf / String couplings
150
foregoing section by an inverse differential, which we shall call G ~, so that dfb;;G ~ is invariant. Along (OM)H the classical ghost field b~f satisfies open string boundary conditions
Im(f}2boo) o~{o.) = 0 .
(3.10)
In order to have a vanishing contribution from the contour along ( 3 M ) H after applying Cauchy's theorem to the product GOboo, we have to impose (3.11)
Im( G°/ts )lo~(oM} ~ = O. The general form of the anti-ghost TI 3
K~h ~ ;Im(d~'sG~s/;~,~) = 0,
(3.12)
s=l
is easily derived with the help of Cauchy's theorem. As we shall construct G ~, by using Cauchy's theorem in the z-plane, it is necessary to know the behavior of a holomorphic inverse differential at z = oo. To study a differential in the neighborhood of m it is useful to introduce a coordinate Z = 1 / z , which vanishes at z = m. At ~ = 0 the transformed differential G ~ has an ordinary Taylor expansion in powers of L Therefore G z has the following expansion around z = 1
GZ=a 2 z 2 + a _ t z + a o + a + t - + Z
....
(3.13)
Note that the first three terms are just a linear combination of the three holomorphic differentials on the sphere. If inserted into eq. (3.12) these differentials yield TI which have the peculiar property of involving only annihilation operators. In the light-cone parametrization a useful basis of these differentials is provided by
g;=z(z-
1),
g~ = z,
g~ = z - 1.
(3.14)
k = 1, 2,3,
(3.15)
Expanding these differentials for O ~ M, like =
n
- 1,
(no sum over s) we get the following TI to be satisfied by Kgh
KghG k = O,
(3.16)
where G k := G~''b,~,
n >~ - 1.
The coefficients G,*'" will be calculated explicitly in appendix C.
(3.17)
151
T. Ohrndorf / String couplings
In order to obtain TI involving creation operators as well, G t must be singular somewhere on the z-plane. We shall employ differentials Gm "'t, which have an ( m - 1)-fold pole at f, = 0, i.e.
ar'z=
dz ] m+ ~rr )~r 1 + analytic.
(3.18)
To reconstruct the analytic piece in eq. (3.18) we shall employ Cauchy's theorem in the same manner as in sect. 2. Because of eq. (3.11) we can again employ the Schwarz reflection principle to continue G,~'z to the lower half z-plane. Deforming the contour of integration across oc we obtain =
1
zr
Z--Z'
~r
+
a-az'2)"
(3.19)
F o r r 4= 3 the piece in parenthesis is left over from pulling the contour of integration across oo. The coefficients a0, a 1 and a 2 are left undetermined by eq. (3.18). Their appearance is a consequence of the fact that we may add any linear combination of the differentials g~ to eq. (3.18) without changing the singularity. We shall expand Gr't, according to
p ~ Ms
G r ' f ' = ~ r s ~ -rn+l -- vrs~ n+l ,
(3.20)
where no sum over s is understood. The coefficients are only nonvanishing for m >/ + 2, n >/ - 1 . They will be calculated explicitly in appendix B. Note that the coefficients V ~ are only unique up to m-dependent linear combinations of G~". We find that these coefficients do not coincide with the N e u m a n n coefficients of sect. 2 as the results of refs. [2,4] would suggest. The coefficients turn out to be real. Therefore the condition (3.11) is indeed satisfied and the neglection of ( 3 M ) H in deriving eq. (3.12)justified. Inserting eq. (3.20) into eq. (3.12) we obtain the following T I for the anti-ghost coordinate bz: Kgh(br_m - V ~ b S ) = 0 ,
m>~ + 2 ,
n>~ - 1 .
(3.21)
3.2 GHOST TI Before we can solve eqs. (3.16) and (3.21) we have still to determine the TI satisfied by the ghost field G" cp is an inverse differential, therefore we have to substitute F~ by a quadratic differential, which we shall call Hpp. As c p satisfies
Im('-cl ] o~(oM).
322) '
T. Ohrndorf / String couplings
152
we have to require
Im({~2Hoo) p~(an) = 0 .
(3.23)
The general TI now reads 3
Kgh ~ f c I m ( d ~ . H ~ , ~ ' ) = 0 . s=l
(3.24)
s
A quadratic differential which is analytic at oo has to vanish at least like z - 4 for z ---, ~ . If we try to reconstruct H = from its singularities by applying Cauchy's theorem in the same way as we have done previously for F and G", we find that it is singular at ~ . In order to construct a differential H = , which vanishes fast enough at oo, we have to perform suitable subtractions. If r ~ 3, a suitable ansatz to start with is provided by
Hz, ~,=
,dz
{ a -
Z -- Z t
+ z'
1 2 z_zr)k} -- Z r
(3.25)
H'z"
If r = 3, we do not need any subtractions. In the present case there is no left-over from pulling the contour of integration across oo. It is clear from eq. (3.25) that //~,,, will always have at least a second order pole at z = G. To write the sums appearing in the remainder of this section in an economical way, we shall extend our summation convention by agreeing that the index a always runs from - 1 to + 1. With this convention the expansion of H{;, m in powers of ~s reads H ~ ,r m
=
~rs(~rm--2__
V~rs r a ~ r a--2 )
"rs n - 2 , -V,~n~s
p~M,.,
n>2,
m>2,
(3.26)
where no summation over s is understood. If r ~ 3, the curly bracket in eq. (3.25) vanishes for I z - Zrl < [Z'--Zrl at least with the third power of z - z r. Therefore, the coefficients l?m~s are completely determined by the singular piece ~r '~-2 of H[~, ,,. Inserting eq. (3.26) into eq. (3.24) we obtain the TI -- rs
S
Kgh(Cr_m - V r n c n ) = 0 ,
m > 2,
n> -1.
(3.27)
F r o m eq. (3.27) it appears that there are no quadratic differentials with poles of order less than four. However, this is only true if we insist on having only a single pole. In fact there are quadratic differentials with more than one pole located at ~r = 0 such that the order of all the poles does not exceed 3. As these differentials are linearly independent from those given by eq. (3.26), we have to take them into a c c o u n t as well, to be able to find a unique solution for Kg h. In general these
T. Ohrndorf / Stringcouplings
153
differentials are of the following form (no sum over s) H~=H2'S~
-2 ,
n
> -1,
O ~M~.
(3.28)
We can again employ Cauchy's theorem to show that the coefficients of the analytic part of the series of eq. (3.28) H2 's, n > + 2, are completely determined by the coefficients of the singular part H 2 " , n = - 1 , 0, + 1. In other words, there appear to be nine linear independent differentials of the form (3.28). However, due to z j
k= 1,2,3,
foMIm( dzg, H~z ) = 0
(3.29)
we have three constraints so that in fact there are only six independent differentials, and the coefficients H ~ " satisfy j,s _ aok , s H_o - 0.
(3.30)
F r o m eq. (3.28) we get a further set of TI
KghH k = O,
(3.31)
where H k := H2"c,',
n >i - 1.
(3.32)
3.3. CONSISTENCY OF THE TI So far we have obtained four homogeneous relations (3.16), (3.21), (3.27), and (3.31), which have to be imposed on K~h. In order for these relations to be compatible with one another, the multiple anti-commutators among the operators annihilating Kgh must form a closed algebra. We would like to demonstrate in the sequel that this is indeed the case. We can derive " T I " for the differentials H[~,,,, G~; ~, g~, and H~ themselves in exactly the same way as the TI for the ghost coordinates have been obtained previously. They imply
amP,r__ak,rprr
=0,
m>/ + 2 ,
-,sm - 1/"" V~t,,+ V,~ . . . ~. " _ a = O ,
m,n> +2,
(3.34)
m >~ + 2.
(3.35)
--a
'
m,
--a
k=1,2,3,
(3.33)
and Hi,
r __
rs j, s V,~aH_ a = 0,
T. Ohrndorf / String couplings
154
These three relations may be supplemented by eq. (3.30). Note that eq. (3.33) guarantees that V,~t, satisfies eq. (3.34) after adding any linear combination of G~". It is now easy to show that all the anti-commutators among cr m -- VrnCn, ~ . . . br_m. . . V,~,bn ,
H k and
Gk,
in fact vanish identically. 3.4 S O L U T I O N OF T H E TI
Let us finally solve the constraints (3.16), (3.21), (3.27) and (3.31). The transition amplitude Kgh is usually represented by letting a Fock-space operator act on a 3-string vacuum state. We shall construct Kgh in a similar way. The Fock-space operator which we will employ is given by V ~ := e-C;'vL"b~;,
m >~ + 2,
n >~ --1.
(3.36)
It has the property that V~bL,~ (vgh) -1
=br-m+
{ 0, V,~S~b~, n > / - 1 ,
Vg%~-m(Vgh)-l=c~-m-
0,
csV;,~,,
n>~2,
if m ~< 1, if m > / 2 ,
(3.37)
if m ~< - 2,
(3.38)
ifm>~-l,
VghHk( vgh ) - i = r~a.,k,s sca•
(3.39)
It follows from eq. (3.34) that vgh(cr
m -- V m S c S ) ( V g h )
l=cr_m--
~lrSacS ,
m>~ + 2 ,
n>~ --1.
(3.40)
Therefore by letting V gh act on a vacuum (01 satisfying (0[c~, = 0,
n>~ - 1 ,
(01b~_n = 0,
n >/ + 2 ,
(3.41)
we obtain a solution of eqs. (3.21) and (3.27). All that remains is to incorporate eqs. (3.16) and (3.31). V gh commutes with G k vghak( vgh ) -1 = ak "
(3.42)
155
T. Ohrndorf / String couplings
Therefore Kgh = <01V ~h
Gk ,
(3.43)
represents a solution of eqs. (3.16), (3.21), and (3.27). Note that due to vghHk(v~h)
1 = H ka ..... Ca
(3.44)
,
eq. (3.43) satisfies eq. (3.31) as well. Eq. (3.43) is unique modulo its normalization. Another way of solving the constraints may be based on the operator 17"gh := e x p ( - b m Vr r~, rcsn ) s,
m >1 + 2 ,
n >~ - 1 ,
(3.45)
which commutes with H k. With the help of eq. (3.45) Kgh can be written as
where the vacuum ~01 satisfies
(OlCr.=0,
n>~ + 2 ,
(13]b~, = 0,
n >~ - 1 ,
(3.47)
The equivalence of eq. (3.46) with eq. (3.43) is demonstrated by noting that s ~sr V ~ = exp( bmV:oC or )exp( --CmV.;.aOo)VX.'.gh ,
m >1 + 2 ,
(0, (kI~__lGk'sb~)- (0, ( kI~__ 1, , ' s C ~ ) .
(3.48) (3.49)
Let us finally clarify the Rs-dependence of Kgh. This can be done in exactly the same way as it has been carried out for K x by replacing L~ by L0gh. As Logh does not act on the zero modes co and b0, all the Rs-dependence can again be absorbed into the coherent-state wave function. The only freedom which is left is to multiply Kg h by a function depending on R . and the width a s.
4. Comments The aim of the present paper is twofold. At first we have demonstrated that the Neumann coefficients which determine the couplings among arbitrary string states are nothing else than the Laurent coefficients of some particular meromorphic
156
T. Ohrndorf / String couplings
differentials, where the coordinate system in which the expansion has to be performed coincides with the one used in the oscillator expansion of the string coordinates. Secondly, we have given a method to calculate these Laurent coefficients. In particular we have shown that, apart from some ambiguities due to zero modes, these Laurent coefficients are uniquely determined. Whereas for the bosonic sector it is sufficient to consider only functions which have a single multiple pole, we have found that for the ghosts one has to consider, as well, differentials with several poles. However, the order of these poles may not exceed three. Though we have presented our method only for the case of the three-string vertex of the covariantized light-cone theory, the generalization to couplings of more than three strings or to other field-theory approaches, like e.g. Witten's theory, should not be too difficult. The major change is induced by the necessity to replace the Mandelstam mapping O(z) used for the covariantized light-cone vertex by another appropriate conformal mapping. If one interprets a vertex as a transition amplitude on a bordered Riemann surface M, the functions ~r(Z) are conformal mappings which map the region inside the boundary loops, which have been removed from M, onto regular unit disks in the ~'r plane. Agreeing to parametrize the boundaries of these disks by the angular variable, one induces a parametrization of OM. Therefore, the functions ~'r(z) may be understood as a choice of boundary parametrization. As there is no conceptual difference between field-theory vertices and reggeon vertices, the present technique may, as well, be used to calculate the latter. In this case one has to choose for the mappings ~'r(z) appropriate projective transformations. This allows for example a derivation of the overlap conditions which have been postulated in refs. [22-24] in the context of the so called "group-theoretic" approach. We would like to emphasize that the present approach is more general than the one pursued in refs. [22-24], as the mappings ~r(Z) must not necessarily be projective. The technique presented in this paper can easily be adjusted to the case where the world sheet M has genus different from zero. In that case one must replace the kernel 1 / ( z - z') in eq. (2.12) by an abelian differential of the third kind. In contradiction to the claims of refs. [2, 4] we have not been able to verify that the ghost Neumann coefficients coincide with the X Neumann coefficients. We believe that this discrepancy is due to the fact that in refs. [2, 4] the transformation properties of the ghost coordinate have not been incorporated appropriately. These findings at least exclude that the vertices presented in refs. [2, 4] have an interpretation in terms of transition functions on bordered Riemann surfaces. However, more seriously, we find it hard to believe that any vertex which does not share this interpretation is BRST invariant. We have used our technique to construct, as well, a BRST completion of the Caneschi-Schwimmer Veneziano vertex. In this case our results are in agreement with those of ref. [2], besides the contributions of the zero modes.
I57
T. Ohrndorf / String couplings
The authors of refs. [8, 9], who have constructed the oscillator representation of Witten's vertex in the ghost sector, claim as well that X Neumann coefficients and ghost N e u m a n n coefficients essentially coincide. In the light of the results of the present paper we feel that this claim deserves reconsideration. The author wishes to express his thanks to A. Neveu for an instructive discussion.
Appendix A
EVALUATION OF N~ In the present appendix we shall calculate the Neumann coefficients Nm% explicitly, to convince the reader that our formalism leads exactly to the known results for the couplings of the bosonic degrees of freedom. The point of departure is the relation (2.15). For convenience we shall set the constants c~, to zero throughout. Technically a direct evaluation of the coefficients from eq. (2.15) is quite involved. If we, however, take suitable combinations of F r and its derivatives, such that the m-fold singularity at ~ = 0 cancels, the evaluation of the coefficients simplifies enormously. A suitable combination is provided by the following expression
do do'l-1 |\ 1 a , m ~ f z f 7 m 1 - dz j z-z'
. (A.I)
Note that the integrand on the right-hand side of eq. ( A . 1 ) 1s non-singular at z = z'. This is immediately obvious, as due to
dp dz
z -- Z T a 3 z ( z - 1) '
(A.2)
we have
1
d0(d0'l dz~dz']
=(z--zt)[(Z--ZT)(Z'--ZT)+(1--ZT)ZT]
X[Z(Z--I)(z'--ZT)]
(A.3)
-1
Inserting eq. (A.3) into eq. (A.1) the right-hand side factorizes completely
( a~;S'--d;. a + a,m ) G ( P
)=
1TlOClOC20¢rOlsOL33( Z' -- ZT)
,z ~r
Z
-1ZT
(A.4)
T. O h r n d o r f /
158
S t r i n g couplings
The coefficients N~s can now immediately be read off eq. (A.4) by noting that
--
- Z -- ZT
nf,(Ts)~ +" -- --~sl
1 ]
n=1
0
d
-1- - - ~ s , 2 '
p E Ms,
~2
l
(A.5)
where
L(vs)-= r(n+ l)r(n(vs-1) +1) "}ts =
O/s+ 1 -- - - , O/s
0/4=0/1
(A.6)
'
(A.7)
•
In this way we get exactly Mandelstam's result alaza3fm (7~)f. (7~)
rs __
Nrn -
-ran
OtrOts(tlar_t_mOts )
,
+1,
(A.8)
rn >/ + 1 .
(A.9)
m,n>~
1 N r s o = "~ - - f m ( ' ~ r ) ( O L l ~ s , 2 - Ot2(~s,1),
o/r
The coefficients N% may be brought into a symmetric form by making use of the freedom to add the constants c~, [2]. Appendix B
EVALUATION OF Vmn r' In order to evaluate the coefficients Vrs explicitly which have been defined by eq. (3.20) it is useful to start from the following expression 1 G~, '~: = -
% ~:(~ d z
a s
Jzr
Z -- Z ;~r m
% s ~-Tz,]
dz
z
_z,~r m
dp
do'
+ ....
(B.1)
where the dots signalize the freedom to adjust the coefficients ak, k = 0,1, 2 in eq. (3.19). The first term of eq. (B.1) is just Fr multiplied by a factor of ( a J a s ) ~ ;. It has been evaluated in appendix A. By choosing the undetermined coefficients in eq. (3.19) in a particular way, the second term in eq. (B.1) factorizes ~ ' F ~' "
G,~ '~; %
) a3as
1 dz
~
dz~'~"r
(B.2) Z -- ZT
159
T. Ohrndorf / String couplings
Therefore, the coefficients V~ are given by O/r
O~lOg20~r V s
V,~= --mN~
+ - -
OLs
r ,
(B.3)
, Vr
O130L 2
where
if
(B.4)
We have V.*= J.*(-2, +1),
V~ = Jm~(0,- 1),
(B.5)
where J~( p, l):= ~. dz ~7" Zr
[z(z -
1)] P ( Z
(B.6)
-- Z T ) I .
The integral (B.6) can be represented in terms of hypergeometric functions by: J~=0
if m - p ~ < 0 ,
whereas, if l>10and m - p > / l + l o r l < ~ 0 a n d 1_ J~ - F ( m
p)r(mv,
-
xr(-l,p-m+
j2=0
if
m - p > ~ +1,
r ( m Y l + p + 1) -
m
(B.Va)
(1 - V l ) - '
+ 2(p + 1))
l;myl-m+
Z(p+ l);1-Vx);
m-p~<0,
whereas, if l>~0and m - p > / l + l
j2 = ( _ 1)P+Iy2I
×F(-I,
(B.Vb) (B.8a)
orl~<0and m - p >
+1,
F ( m y 2 - l - 1 - 2p) _ m - 1-p)F(m - p)
r(mv2
p - m +
1; my2 --
m - l + p;
1 - 72);
(B.8b)
and finally J ~3 -_ 0
if
whereas, if l>~0and m + 2 p + l > t 0 o r l ~ < 0 a n d j~ = ( _ 1),+ 1
(B.9a)
m+2p+2+l<~O,
m+2+l>~-l,
F(mv3 + p + 1) 2)F(my3- m-p-
F ( m + 2 p + 1+
×r(-l,-1-m-Zp-l;my3-m-p-l;1-73).
l) (B.9b)
T. Ohrndorf / String couplings
160
The coefficients V~,, V.~ are given by
F(nTs- 1)n V2 = C(n + 2 ) F ( n ( v , - 1) + 2) (1 + 71)(n (71 - 1) - 1)(r/(~/1 (']/2 2)(nY2 - 1)(n72 + 1), (2V3 - 1)(n + 1)(n - 1),
×
-
-
1) + 1),
-
-
s=l, s:2, s:3,
n>~ - 1 ,
n>~ - 1 , n>2,
(B.IO)
for n ~ 0 . If n = 0 --
__m
v°~= / ( v 2 - 2)/v2 ' ~0,
s = +1, (B.11)
s = +2, s = +3,
F(myr- 1)m2yr v2 =/'(m + 2)F(m(rr-- 1) + 2)
)(
71(1 - 7 1 ) m F ( 1 , - m + 1; m ( 7 1 - 1) + 2,1 - 71),
r=l,
-'Y2(m ('g2- 1) + 1)F(1, - m + 1; m(72 - 1) + 1;1 - T2),
r=2,
73(m(~,3- 1) + 1)F(1, - m ; m ( 7 3 - 1) + 1;1 - 73),
r=3, (B.12)
if m>~2. Appendix C EVALUATION
O F G,~"s
This appendix is devoted to the explicit determination of the coefficients G,,k," of the expansion (3.15) of the holomorphic inverse differentials in powers of ~',. They
TABLE
k 1 2 3
s=l +1 1 - 271 +1
1
s=2 -1 + 1 2-/3 3
s=3 273 1 + 1 +1
161
T. Ohrndorf / String coupling~ TABLE2
a,f" ( . ~ o) k
s=l
s=2
s=3
/'(n?' l + 1)(1 - ?'l) F(n + 1)F(n (?'1 - i) + 2)
F(n?'2 1)(?'2 1)?'2 F(n)F(n(?'2 - 1) + 1)
m
n)+l
n ) +1
/'(n?' 3 + i)(i - ?'3) C(,, + 2)F(n (?'~ - 1) + 1) n > -1
F(,,?', + 1)(1 - ?'1) F(n+2)F(n(?'l 1)+1) n) -1
r(n?'2)(1 - ?'2)?'2 V ( n ) r ( n ( ? ' 2 - 1) + 2) n ) +1
r(n?"3 1)?'3(?'3 - 1) r ( n ) r ( ~ ( ? ' 3 - 1) + 1) n ) +1
r(n?'l - 1)?'1(1 - Yl) r ( , , ) r ( n ( ? ' , - 1) + 1) n ) +1
r(n?'2)n(?" 2 - 1)?'2 r ( n + 2)r(n(?'2 1 ) + 1 ) n) 1
F(n?'3) ?'3(?'3 - 1) F(n)F(n(y 3 1)+2) n ) +1
are given by 2
= a~
dz~;"
[ z ( z - 1)] 2 g k "
(C.1)
C h o o s i n g for t h e h o l o m o r p h i c i n v e r s e d i f f e r e n t i a l s the basis (3.14) has the a d v a n t a g e t h a t t h e c o e f f i c i e n t s G,k's c a n b e w r i t t e n in a c o m p a c t f o r m . W e h a v e listed t h e m in t a b l e s 1 a n d 2.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A. Neveu and P. West, Phys. Lett. B168 (1986) 192 A. Neveu and P. West, Nucl. Phys. B278 (1986) 601 H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B172 (1986) 186 H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D34 (1986) 2360 E. Witten, Nucl. Phys. B268 (1986) 353 E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. B179 (1986) 57 S. Samuel, Phys. Lett. B181 (1986) 249 S. Samuel, Phys. Lett. B181 (1986) 255 D. Gross and A. Jevicki, Nucl. Phys. B283 (1987) 1; 287 (1987) 225 A. Eastaugh and J.G. McCarthy, Nucl. Phys. B294 (1987) 845 G.T. Horowitz and S.P. Martin, Nucl. Phys. B296 (1988) 220 P. Di Vecchia, R. Nakayama, J.L. Petersen, J. Sidenius and S.Sciuto, Phys. Lett. B182 (1986) 164 P. Di Vecchia, R. Nakayama, JL. Petersen, J. R. Sidenius and S. Sciuto, Nucl. Phys. B287 (1987) 621 [14] U. Carow-Watamura and S. Watamura, Nucl. Phys. B288 (1987) 500 [15] S. Mandelstam, Nucl. Phys. B64 (1973) 205 [16] C. Vafa, Phys. Lett. B199 (1987) 195
T. Ohrndorf / String couplings"
162 [i7] [18] [19] [20] [21] [22] [23] [24] [25] [26]
L. T. T. E. E. A. A. A. T. T.
Avarez Gaum& C. Gomez, G. Moore and C. Vafa, Nucl. Phys. B303 (1988) 455 Ohrndorf, Nucl. Phys. B293 (1987) 709 Ohrndorf, Nucl. Phys. B301 (1988) 460 Cremmer and J.L. Gervais, Nucl. Phys. B76 (1976) 1100 Cremmer and J.L. Gervais, Nucl. Phys. B90 (1975) 410 Neveu and P. West, CERN-TH 4757/87 Neveu and P. West, Phys. Lett. B193 (1987) 187 Neveu and P. West, Phys. Lett. B194 (1987) 200 Ohrndorf, Phys. Lett. B194 (1987) 490 Ohrndorf, in preparation
STRING COUPLINGS FROM M E R O M O R P H I C DIFFERENTIALS Theophil OHRNDORF* Institut ffir Theoretische Physik der Universitiit Heidelberg, Philosophenweg 16, D -6900 Heidelberg, FR G
Received 5 April 1988 (Revised 27 June 1988)
The "Neumann-function method" which allows one to determine oscillator representation of multiple string verticesis put on a firm basis. The Neumann coefficients are related to the Laurent coefficients of certain meromorphic differentials. A systematic procedure of constructing the differentials is developed. The method is exemplifiedby the "covariantized" light-cone vertex. Our results on the ghost part of this vertex are in disagreementwith various proposals in the literature,
1. Introduction Recent attempts to construct interacting second-quantized string field theories [1-5] have produced a considerable interest in finding explicit oscillator representations of string vertices. At present, several groups have presented such oscillator representations for the various field theories which have been proposed [6-14]. As two-dimensional conformal field theory or equivalently Polyakov's path integral provide a natural framework for the first-quantized string, one expects that these two approaches play, as well, a major role in finding string vertices. However, so far the virtues of these two approaches have not been exploited to their full power in the problems of (i) defining vertices, and (ii) their explicit construction. It is the purpose of this paper to present several technical tools which allow one to calculate arbitrary vertices starting from the path integral. These techniques may be considered as a generalization of those originally developed by Mandelstam [15]. Recently, there was a proposal [16,17] to reformulate Polyakov's approach completely in terms of an operator formalism. The operator formalism proposed in refs. [16,17] is based on the path integral on punctured Riemann surfaces. In principle, such a formalism could be a very useful tool to construct string vertices systematically. Unfortunately, the fact that the authors of refs. [16, 17] base their method on punctured Riemann surfaces makes its immediate application to the problem of constructing string vertices somewhat complicated. * Supported by D.F.G. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
142
T. Ohrndorf / String couplings
An equivalent approach which, however, is technically much simpler makes use of bordered Riemann surfaces [18,19]. In the present paper we shall exploit the fact that string vertices may be considered as transition functions on bordered Riemann surfaces to outline a method which allows the systematic construction of string vertices. We shall work out the oscillator representation of the vertex corresponding to the so-called "covariantized light-cone" field theory [1-4] in detail. Our results for the ghost part of this vertex (see eq. (3.43)) are in disagreement with those that can be found in the literature [2, 4]. This discrepancy is caused by the fact that previous authors do not seem to have incorporated the transformation properties of the reparametrization ghosts under conformal transformations in an adequate manner. We have decided to work out the vertex for the covariantized light-cone approach mainly to give an explicit example. Our methods can easily be generalized to other geometrical configurations of interacting strings, like e.g. Witten's vertex [5]. So far in the literature mainly two methods have been employed to construct string vertices, namely Mandelstam's Neumann-function method and the overlap 6-function method of Cremmer and Gervais [20, 21]. Both methods were developed in the early days of string theory. We find that both methods are unsuitable to allow for a systematic derivation of string vertices in a reliable way, in particular of the ghost sector. The overlap 6-function method is heavily based on manipulating infinite-dimensional matrices. However, the method does not ensure that the sums occurring are convergent. The Neumann-function method does not suffer from this disease, but, however, it is a priori unclear why it is sufficient to work with the Neumann function on the world sheet with infinite strips attached, as Mandelstam does, to get the transition amplitude for finite strips. In particular, there are factors of 2 which have to be adjusted " b y hand" (see commentary preceding formula (4.4a) of ref. [15]). In contrast to these methods our method is solely based on the fact that the derivative OzX of the classical string coordinate X, as well as the classical ghost coordinates c z and bzz are analytic on the world sheet. In this way it is easy to control the region of convergence of the infinite sums involved. By integrating products of suitable differentials with the fields O~X, c z and bzz along the boundary of the world sheet, identities among the various oscillators acting on the vertex are derived, which we shall call transition identities. These identities may be viewed as an infinite set of first-order differential equations, satisfied by the vertex. They may be integrated to yield the vertex explicitly. We would like to emphasize that the virtue of this method lies in the fact that only a minimal set of assumptions is needed as input. This method has first been applied in ref. [2]. The same authors have developed this method further in refs. [22-24]. Technically, it is obviously of vital importance to form products 0, X, c: and bzz with differentials which are chosen in such a way that the solutions of the transition identities become as simple as possible. One way of finding such differentials
7". Ohrndorf / String couplings
143
consists of first filling in the boundary disks of the world sheet which represents the vertex, and then constructing differentials which have a single multiple pole on one such boundary disk. In fact, the Neumann coefficients are just the Laurent coefficients of these differentials. Using a kind of dispersion relation, it is easy to show that these Laurent coefficients are almost uniquely fixed. The rest of the paper is organized as follows. As a warm-up we shall rederive the bosonic vertex in sect. 2. The purpose of this section is to introduce the method with the help of a simple example and to convince the reader that our method does in fact reproduce the known result for the bosonic vertex. For completeness we shall evaluate the Mandelstam coefficients in appendix A. In sect. 3 we shall demonstrate how the method introduced in sect. 2 can be generalized to calculate the ghost vertex. The relevant coefficients will be evaluated explicitly in appendices B and C. We conclude in section 4 with some comments.
2. Bosonic couplings This section has mainly an introductory character. We shall present a systematic way of deriving the form of the three-string vertex, as it has originally been obtained by Mandelstam [15]. Our starting point will be the fact that the three-string vertex is just a Fock-space representation of the path integral defined on a three-strip world sheet, if considered as a bordered Riemann surface. The major tool, which will enable us to write down an explicit expression for the vertex, is simply Cauchy's theorem. In contradiction to Mandelstam's original approach we shall not make use of the Neumann function. Therefore, our approach can be generalized to the string ghost coordinates with only minor modifications. The basic object of interest in this section is the transition amplitude Kx, which describes the interaction of three strings. It may be represented by the path integral on the corresponding world sheet M, which in the present case consists of three rectangular strips glued together as indicated in fig. 1. We shall denote the individual strip regions by M s, their vertical boundaries by Cs, s = 1, 2, 3, and the horizontal boundary of M by (0M)i~. In contrast to a procedure often employed in the literature, we shall not let the strips tend to ~ , rather they may take any finite length. Nevertheless, sometimes it is useful to consider the infinite-strip world sheet, which we shall denote by M~. Working on M, K x is a functional of the boundary value, which we impose on 0,X(o), the normal derivative of X along the vertical component of the boundary of the world sheet and a function of the moduli of M.
]2
i q
I
,
3
I i
Fig. 1. Geometry of the world sheet M.
144
72 Ohrndorf / String couplings
For simplicity we shall not display any arguments of K x explicitly. In order to make our ideas quantitative, we have to introduce coordinates on the world sheet. We shall place the strip diagram displayed in fig. 1 onto the complex plane equipped with a coordinate p in such a way that the bottom horizontal boundary coincides with the real axis. To label the width of the three strips we introduce three real parameters o/,, s = 1,2,3 with %, a 2 > 0; a 3 < 0 such that a, = 0. The width of the s string is then given by ~r[o/~[. To make K x well defined, we have to introduce a parametrization of the boundaries Cs. We shall use the same parametrization as it has already been employed implicitly by Mandelstam. Namely we shall use the angular variable on the half annulus obtained by mapping M, via
0/1
/~1 =
0,
/~2 = -- --iTr,
]~3 = +iqr,
(2.1)
O/2
onto a f, plane. Technically it is convenient to consider the strip region with the strips extending to infinity M~ as being obtained by a conformal mapping from the upper half plane. The so-called Mandelstam mapping o(:)
= % l o g ( z - 1) + o/2 log : ,
(2.2)
achieves precisely this. Note that ( 0 M ) H is the image of the real z-axis. The inverse mapping z(o) has a square-root branch point at O = Py = P(ZT), where ZT, the "turning point", is determined by do dz
=0.
(2.3)
It is given by ZT =
-- 0/2//O/3 •
(2.4)
Though one may derive a formal representation of K x in terms of the Neumann function on M, finding the explicit form of the Neumann function on the three-strip world sheet, and of the functional determinant of the associated laplacian, is highly nontrivial. Fortunately, this problem can be circumvented. As we shall demonstrate below, a Fock-space representation of K x is readily given by solving certain identities, which we shall call transition identities (TI), which exist among the oscillators of the operator associated with OzX. We shall distinguish "transition identities", which hold for oscillators, from "overlap identities" which hold for OnX(o ), b(o), and c(o), which are only defined on OM. These identities are a direct consequence of the fact that OzX is holomorphic on M. Compared with the
T. Ohrndorf / String couplings
145
"overlap identities" [20,21], the transition identities have the advantage of being represented by convergent sums. We shall start from the following assumption, which can easily be justified by employing the path-integral representation of Kx, as it has been given e.g. in ref. [18]. For each boundary component C~., s = 1, 2, 3 of M, there exists an operator 0~](', which has an expansion 0~s 2 " =
(2.5)
-½i .V .~ -. n .- lo~s . tt
with
[~,", ~2"]
=8"sn"'8,,
_m
.
(2.6)
0f J(" has the property that it reproduces the classical field Of X by acting on K x. The oscillators a s are represented as differential operators in terms of the coefficients of the normal-mode expansion of O,,X(o) along C~.. Instead of working with K x as being defined by the path integral on a bordered surface, it is technically much simpler to work with a Fock-space representation. This may easily be obtained by writing K x as a convolution of some integral kernel, with coherent-state wave functions for the oscillators a,~'~, n =~ 0, and taking the Fourier transform with respect to the momentum pr. We shall denote this integral kernel as well by the symbol K x. An explicit example of how to go from the X( o )-representation to the Fock-space representation may be found in ref. [25]. Within this formalism it is readily understood that the path integral on the disk represents just the commonly adopted vacuum state. Denoting the standard coordinate on the disk by w, and expanding as in eq. (2.5), all the oscillators with n > 0 must annihilate Kx, otherwise 3wX would have a singularity at the origin. In the present context the world sheet is represented topologically by a three-holed sphere. Accordingly, a single string state would be represented by the path integral on the complement of the disk. In other words, the vacuum had to be annihilated by all the oscillators a ,~,r, n ~< 0. In order to be in accordance with the standard Fock-space notation, we shall therefore always let 0~)( act to the left, i.e.
KxO~SU= Kxa~ X,
(2.7)
if p is near the boundary curve Cs. Note that the right-hand side of (2.7) is independent of s. However, by acting with O~Xs on K x we obtain O~X represented by a particular series. This series need only to converge in a small neighborhood around C,. Nevertheless, the function which is represented by such a series has a holomorph continuation to all of M. It is now a simple matter to derive the general form of the TI. Suppose a function F(p) is given, which is analytic in the coordinate z on M if eq. (2.2) is employed,
T. Ohrndorf / String couplings
146
and has the additional property that it is real if p ~ ( 3 M ) H
lm( F(p ) )lo~(OM). = O.
(2.8)
Then, C a u c h y ' s theorem implies that 3
~, f
(2.9)
]
as Im(lSOoX ) = 0 along (¢gM)H. Making use of eq. (2.7), this statement m a y be turned into the operator statement 3
Kx ~, fcIm(d'~,F(o)o;f(s)=o.
(2.10)
,';=1
Let us next turn to the construction of functions F ( p ) , potentially to be used in eq. (2.10). A p a r t from the constants which result into the conservation of energy m o m e n t u m , if inserted into eq. (2.10), there are no other functions which are analytic on Moo. We shall therefore postulate that F(p) has a m-fold pole at ~r = 0. In this way we expect to obtain a set of functions Fr(O), labelled by m > + 1 and r = 1,2, 3, normalized like lim F £ ( p ) z~z
= ~'7 m + analytic,
(2.11)
r
where z 1 = 1, z 2 = 0 , and z 3 = + ~ . The requirement that F,~(p) has only a single pole allows one to determine F~(p) almost completely from eq. (2.11). To this end we start from Cauchy's integral formula
1 F,(,(p') = ~, dz z - z' Fm~(P) '
(2.12)
where the c o n t o u r of integration runs around z'. We shall always absorb a factor 2~ri into 9~. T h o u g h F,~(z) has only been defined on the upper half plane, it can be analytically continued to the lower half plane with the help of the Schwarz reflection principle, as Fr(z) is real along the real axis. N o w we deform the contour of integration in eq. (2.12) in such a way that it encircles z = oo and z = z r if r 4: 3. If r 4= 3 F,~(p) has to go to a constant if z ~ oo. Therefore the contour around ov produces just this constant. N e a r z = z r the analytic piece of eq. (2.11) can be dropped, only if r = 3 a constant term contributes. In this way we get
Fr(p ) = -
z~
N o t e that apart from the constant c,~,,
dz
1 Z --
Z i
fr m + £ r ~n "
(2.13)
F,(,(O) is uniquely determined by eq. (2.11).
T. O h r n d o r f /
147
S t r i n g couplings
Writing down a Taylor expansion for the analytic piece of F,~(p), let us say in powers of ~,, we cannot expect that such an expansion converges on all of M. A generic function which is analytic in z has a branch cut if considered as a function of p, and therefore of ~s as well. Therefore Fr(O) has different Taylor expansions on the various 34, r ~ ,r( p )
__
--~rs~r
m
--
rs s+n , mN~,~
p~M,,
n>~O,
m>~ + 1 .
(2.14)
In eq. (2.14) summation over n, but no summation over s is understood. The coefficients N ~ are determined from eq. (2.13) to be
mNr~ =
dz .
~
[
fPs
n
]
dz~r
m
Zr
~n,oC~, " Z
--
Z~
(2.15)
It m a y be checked that the coefficients N ~ are indeed real, and the requirement (2.8) is satisfied. Eq. (2.15) will be evaluated explicitly in appendix A. Before we solve eq. (2.10) with eq. (2.14) we would like to emphasize one point which has been ignored so far. In deriving eq. (2.9) we have to be careful that any contribution originating from going around the turning point z T is absent. This will obviously be the case as we may perform the integral around the turning point in the z-plane, and F~ is non-singular at z T. One may be tempted to replace F,~(p) by its derivative d F £ ( o ) / d p to get new TI. However, as d z / d p - ( p - p T ) -1/2 eq. (2.9) ceases to hold as we get a nonvanishing contribution from going around the turning point. Let us finally turn to the solution of eq. (2.10) with eq. (2.14). To write the solution in a compact way we shall agree on the following summation convention to be used in the remainder of this paper. If not otherwise stated, we shall always sum over repeated indices with r, s = 1, 2, 3 and m, n . . . . running as indicated. Inserting eq. (2.14) into eq. (2.10) gives
Kx(a~_m - mN,~,a~) = 0,
n >~ 0,
m >/1.
(2.16)
The solution of eq. (2.16) is conveniently expressed with the help of the operator
V x = exp( ~amlVm,a,+amN,~oao}, 1 r ~rrs s r rs s
m,n>~l.
(2.17)
Due to N ~ = N;,~,
n, m >~ + 1,
(2.18)
we have
vX(a~m - mN,~a~) = a~mV x,
m >~ + 1 ,
n >/0.
(2.19)
Therefore, the general solution of eq. (2.16) is
K x = 8(p~ +P2 + P3)(P[ v X f ( P , ) ,
(2.20)
148
T. Ohrndorf / String couplings
where 3
(P[ := I-I (0, Prl.
(2.21)
r=l
f ( p , ) is a function depending on p, and the moduli of M. In the present case one may take the lengths of the strips M,, or equivalently the radii R, of the image of the Cs under ~s(P) as parameters. To make the R s dependence of f explicit, we exploit the fact that changes of R s are generated by L 0 (see ref. [180, i.e.
0 Rs--Kx OR,
= KxL'o,
(2.22)
where it is understood that K x is in the X( o )-representation. For the n ~ 0 modes one may solve eq. (2.22) simply by rescaling the arguments of the coherent-state wave functions, so that in the Fock-space representation only 0 R.~--Kx=
2Ps12Kx '
ORs
(2.23)
is left. Therefore f ( p , ) ~ exp(½)2 p2 log R s ) .
(2.24)
Letting the lengths of the strips go to zero, we may express R s in terms of the % R, = exp
Re(o-r) ,
(2.25)
with Re(0T) = c q l o g ( - - ~ ) +
%log(- ~).
(2.26)
3. Ghost couplings In the present section we shall generalize the method which has been exemplified in the preceding section to the couplings of the ghost oscillators. We shall proceed in a manner completely parallel to the determination of the bosonic couplings. The main difference being that the function F,~(O ) has to be replaced by either an inverse differential or by a quadratic differential. In ref. [19] it has been shown how to construct ghost operators in terms of bS(o) and cS(o) being defined on C,. The oscillators being defined by the usual expansion t5~3, = E~'s n 2bS '
(bn)* = b,,,
(3.1)
(c',,)* = c,,
(3.2)
n
6,~ = EfS"+'c~;, tl
T. Ohrndorf / String couplings
149
satisfy the anti-commutation relations (3.3) The ghost transition function K~h is represented by Kgh = Be- Sc~.
(3.4)
In the c ( o ) - b ( o ) representation there is an additional determinant, which we shall, however, suppress. The explicit form of ScL has been given in ref. [19]. 8 is defined by
8 = FlSk,
(3.5)
k
with 8 k := foMdO Im(h~z~2C(O)),
(3.6)
where h~z is a holomorphic quadratic differential on the Riemann surface obtained by taking the union of M with its mirror image with respect to real axis. Note that only those differentials contribute which satisfy (h~z) k , = h~, k as we are dealing with the open string. Therefore, we have
Im(eZh z)l
0M = 0.
(3.7)
For the closed string the product (3.5) includes all the holomorphic quadratic differentials on M. It will be shown in ref. [26] that the 8 k must be present in Kgh to ensure that the classical ghost field c z is analytic on M, and that it is possible to define a ghost-number operator in such a way that the ghost-number violation is solely determined by 8. We would like to make clear that the &function (3.5) has to appear in the c ( a ) - b(o) representation. It has to be distinguished from those &function-like contributions previously identified in ref. [13] which we shall call G k (cf. eqs. (3.17) and (3.43)) which appear in the Fock representation. The ghost transition amplitude Kg h s a t i s f i e s e - Sc@s~s;8 = Kghb~,; , Kgh@ = Kghc~, ,
(3.8) (3.9)
if p is near Cs. b~, and c ~, have to be considered as classical ghost coordinates. 3.1 ANTI-GHOST TI
Let us start with a systematic derivation of TI for the anti-ghost operator b~. As b~ is a quadratic differential, we have to replace the function F used in the
T. Ohrndorf / String couplings
150
foregoing section by an inverse differential, which we shall call G ~, so that dfb;;G ~ is invariant. Along (OM)H the classical ghost field b~f satisfies open string boundary conditions
Im(f}2boo) o~{o.) = 0 .
(3.10)
In order to have a vanishing contribution from the contour along ( 3 M ) H after applying Cauchy's theorem to the product GOboo, we have to impose (3.11)
Im( G°/ts )lo~(oM} ~ = O. The general form of the anti-ghost TI 3
K~h ~ ;Im(d~'sG~s/;~,~) = 0,
(3.12)
s=l
is easily derived with the help of Cauchy's theorem. As we shall construct G ~, by using Cauchy's theorem in the z-plane, it is necessary to know the behavior of a holomorphic inverse differential at z = oo. To study a differential in the neighborhood of m it is useful to introduce a coordinate Z = 1 / z , which vanishes at z = m. At ~ = 0 the transformed differential G ~ has an ordinary Taylor expansion in powers of L Therefore G z has the following expansion around z = 1
GZ=a 2 z 2 + a _ t z + a o + a + t - + Z
....
(3.13)
Note that the first three terms are just a linear combination of the three holomorphic differentials on the sphere. If inserted into eq. (3.12) these differentials yield TI which have the peculiar property of involving only annihilation operators. In the light-cone parametrization a useful basis of these differentials is provided by
g;=z(z-
1),
g~ = z,
g~ = z - 1.
(3.14)
k = 1, 2,3,
(3.15)
Expanding these differentials for O ~ M, like =
n
- 1,
(no sum over s) we get the following TI to be satisfied by Kgh
KghG k = O,
(3.16)
where G k := G~''b,~,
n >~ - 1.
The coefficients G,*'" will be calculated explicitly in appendix C.
(3.17)
151
T. Ohrndorf / String couplings
In order to obtain TI involving creation operators as well, G t must be singular somewhere on the z-plane. We shall employ differentials Gm "'t, which have an ( m - 1)-fold pole at f, = 0, i.e.
ar'z=
dz ] m+ ~rr )~r 1 + analytic.
(3.18)
To reconstruct the analytic piece in eq. (3.18) we shall employ Cauchy's theorem in the same manner as in sect. 2. Because of eq. (3.11) we can again employ the Schwarz reflection principle to continue G,~'z to the lower half z-plane. Deforming the contour of integration across oc we obtain =
1
zr
Z--Z'
~r
+
a-az'2)"
(3.19)
F o r r 4= 3 the piece in parenthesis is left over from pulling the contour of integration across oo. The coefficients a0, a 1 and a 2 are left undetermined by eq. (3.18). Their appearance is a consequence of the fact that we may add any linear combination of the differentials g~ to eq. (3.18) without changing the singularity. We shall expand Gr't, according to
p ~ Ms
G r ' f ' = ~ r s ~ -rn+l -- vrs~ n+l ,
(3.20)
where no sum over s is understood. The coefficients are only nonvanishing for m >/ + 2, n >/ - 1 . They will be calculated explicitly in appendix B. Note that the coefficients V ~ are only unique up to m-dependent linear combinations of G~". We find that these coefficients do not coincide with the N e u m a n n coefficients of sect. 2 as the results of refs. [2,4] would suggest. The coefficients turn out to be real. Therefore the condition (3.11) is indeed satisfied and the neglection of ( 3 M ) H in deriving eq. (3.12)justified. Inserting eq. (3.20) into eq. (3.12) we obtain the following T I for the anti-ghost coordinate bz: Kgh(br_m - V ~ b S ) = 0 ,
m>~ + 2 ,
n>~ - 1 .
(3.21)
3.2 GHOST TI Before we can solve eqs. (3.16) and (3.21) we have still to determine the TI satisfied by the ghost field G" cp is an inverse differential, therefore we have to substitute F~ by a quadratic differential, which we shall call Hpp. As c p satisfies
Im('-cl ] o~(oM).
322) '
T. Ohrndorf / String couplings
152
we have to require
Im({~2Hoo) p~(an) = 0 .
(3.23)
The general TI now reads 3
Kgh ~ f c I m ( d ~ . H ~ , ~ ' ) = 0 . s=l
(3.24)
s
A quadratic differential which is analytic at oo has to vanish at least like z - 4 for z ---, ~ . If we try to reconstruct H = from its singularities by applying Cauchy's theorem in the same way as we have done previously for F and G", we find that it is singular at ~ . In order to construct a differential H = , which vanishes fast enough at oo, we have to perform suitable subtractions. If r ~ 3, a suitable ansatz to start with is provided by
Hz, ~,=
,dz
{ a -
Z -- Z t
+ z'
1 2 z_zr)k} -- Z r
(3.25)
H'z"
If r = 3, we do not need any subtractions. In the present case there is no left-over from pulling the contour of integration across oo. It is clear from eq. (3.25) that //~,,, will always have at least a second order pole at z = G. To write the sums appearing in the remainder of this section in an economical way, we shall extend our summation convention by agreeing that the index a always runs from - 1 to + 1. With this convention the expansion of H{;, m in powers of ~s reads H ~ ,r m
=
~rs(~rm--2__
V~rs r a ~ r a--2 )
"rs n - 2 , -V,~n~s
p~M,.,
n>2,
m>2,
(3.26)
where no summation over s is understood. If r ~ 3, the curly bracket in eq. (3.25) vanishes for I z - Zrl < [Z'--Zrl at least with the third power of z - z r. Therefore, the coefficients l?m~s are completely determined by the singular piece ~r '~-2 of H[~, ,,. Inserting eq. (3.26) into eq. (3.24) we obtain the TI -- rs
S
Kgh(Cr_m - V r n c n ) = 0 ,
m > 2,
n> -1.
(3.27)
F r o m eq. (3.27) it appears that there are no quadratic differentials with poles of order less than four. However, this is only true if we insist on having only a single pole. In fact there are quadratic differentials with more than one pole located at ~r = 0 such that the order of all the poles does not exceed 3. As these differentials are linearly independent from those given by eq. (3.26), we have to take them into a c c o u n t as well, to be able to find a unique solution for Kg h. In general these
T. Ohrndorf / Stringcouplings
153
differentials are of the following form (no sum over s) H~=H2'S~
-2 ,
n
> -1,
O ~M~.
(3.28)
We can again employ Cauchy's theorem to show that the coefficients of the analytic part of the series of eq. (3.28) H2 's, n > + 2, are completely determined by the coefficients of the singular part H 2 " , n = - 1 , 0, + 1. In other words, there appear to be nine linear independent differentials of the form (3.28). However, due to z j
k= 1,2,3,
foMIm( dzg, H~z ) = 0
(3.29)
we have three constraints so that in fact there are only six independent differentials, and the coefficients H ~ " satisfy j,s _ aok , s H_o - 0.
(3.30)
F r o m eq. (3.28) we get a further set of TI
KghH k = O,
(3.31)
where H k := H2"c,',
n >i - 1.
(3.32)
3.3. CONSISTENCY OF THE TI So far we have obtained four homogeneous relations (3.16), (3.21), (3.27), and (3.31), which have to be imposed on K~h. In order for these relations to be compatible with one another, the multiple anti-commutators among the operators annihilating Kgh must form a closed algebra. We would like to demonstrate in the sequel that this is indeed the case. We can derive " T I " for the differentials H[~,,,, G~; ~, g~, and H~ themselves in exactly the same way as the TI for the ghost coordinates have been obtained previously. They imply
amP,r__ak,rprr
=0,
m>/ + 2 ,
-,sm - 1/"" V~t,,+ V,~ . . . ~. " _ a = O ,
m,n> +2,
(3.34)
m >~ + 2.
(3.35)
--a
'
m,
--a
k=1,2,3,
(3.33)
and Hi,
r __
rs j, s V,~aH_ a = 0,
T. Ohrndorf / String couplings
154
These three relations may be supplemented by eq. (3.30). Note that eq. (3.33) guarantees that V,~t, satisfies eq. (3.34) after adding any linear combination of G~". It is now easy to show that all the anti-commutators among cr m -- VrnCn, ~ . . . br_m. . . V,~,bn ,
H k and
Gk,
in fact vanish identically. 3.4 S O L U T I O N OF T H E TI
Let us finally solve the constraints (3.16), (3.21), (3.27) and (3.31). The transition amplitude Kgh is usually represented by letting a Fock-space operator act on a 3-string vacuum state. We shall construct Kgh in a similar way. The Fock-space operator which we will employ is given by V ~ := e-C;'vL"b~;,
m >~ + 2,
n >~ --1.
(3.36)
It has the property that V~bL,~ (vgh) -1
=br-m+
{ 0, V,~S~b~, n > / - 1 ,
Vg%~-m(Vgh)-l=c~-m-
0,
csV;,~,,
n>~2,
if m ~< 1, if m > / 2 ,
(3.37)
if m ~< - 2,
(3.38)
ifm>~-l,
VghHk( vgh ) - i = r~a.,k,s sca•
(3.39)
It follows from eq. (3.34) that vgh(cr
m -- V m S c S ) ( V g h )
l=cr_m--
~lrSacS ,
m>~ + 2 ,
n>~ --1.
(3.40)
Therefore by letting V gh act on a vacuum (01 satisfying (0[c~, = 0,
n>~ - 1 ,
(01b~_n = 0,
n >/ + 2 ,
(3.41)
we obtain a solution of eqs. (3.21) and (3.27). All that remains is to incorporate eqs. (3.16) and (3.31). V gh commutes with G k vghak( vgh ) -1 = ak "
(3.42)
155
T. Ohrndorf / String couplings
Therefore Kgh = <01V ~h
Gk ,
(3.43)
represents a solution of eqs. (3.16), (3.21), and (3.27). Note that due to vghHk(v~h)
1 = H ka ..... Ca
(3.44)
,
eq. (3.43) satisfies eq. (3.31) as well. Eq. (3.43) is unique modulo its normalization. Another way of solving the constraints may be based on the operator 17"gh := e x p ( - b m Vr r~, rcsn ) s,
m >1 + 2 ,
n >~ - 1 ,
(3.45)
which commutes with H k. With the help of eq. (3.45) Kgh can be written as
where the vacuum ~01 satisfies
(OlCr.=0,
n>~ + 2 ,
(13]b~, = 0,
n >~ - 1 ,
(3.47)
The equivalence of eq. (3.46) with eq. (3.43) is demonstrated by noting that s ~sr V ~ = exp( bmV:oC or )exp( --CmV.;.aOo)VX.'.gh ,
m >1 + 2 ,
(0, (kI~__lGk'sb~)- (0, ( kI~__ 1, , ' s C ~ ) .
(3.48) (3.49)
Let us finally clarify the Rs-dependence of Kgh. This can be done in exactly the same way as it has been carried out for K x by replacing L~ by L0gh. As Logh does not act on the zero modes co and b0, all the Rs-dependence can again be absorbed into the coherent-state wave function. The only freedom which is left is to multiply Kg h by a function depending on R . and the width a s.
4. Comments The aim of the present paper is twofold. At first we have demonstrated that the Neumann coefficients which determine the couplings among arbitrary string states are nothing else than the Laurent coefficients of some particular meromorphic
156
T. Ohrndorf / String couplings
differentials, where the coordinate system in which the expansion has to be performed coincides with the one used in the oscillator expansion of the string coordinates. Secondly, we have given a method to calculate these Laurent coefficients. In particular we have shown that, apart from some ambiguities due to zero modes, these Laurent coefficients are uniquely determined. Whereas for the bosonic sector it is sufficient to consider only functions which have a single multiple pole, we have found that for the ghosts one has to consider, as well, differentials with several poles. However, the order of these poles may not exceed three. Though we have presented our method only for the case of the three-string vertex of the covariantized light-cone theory, the generalization to couplings of more than three strings or to other field-theory approaches, like e.g. Witten's theory, should not be too difficult. The major change is induced by the necessity to replace the Mandelstam mapping O(z) used for the covariantized light-cone vertex by another appropriate conformal mapping. If one interprets a vertex as a transition amplitude on a bordered Riemann surface M, the functions ~r(Z) are conformal mappings which map the region inside the boundary loops, which have been removed from M, onto regular unit disks in the ~'r plane. Agreeing to parametrize the boundaries of these disks by the angular variable, one induces a parametrization of OM. Therefore, the functions ~'r(z) may be understood as a choice of boundary parametrization. As there is no conceptual difference between field-theory vertices and reggeon vertices, the present technique may, as well, be used to calculate the latter. In this case one has to choose for the mappings ~'r(z) appropriate projective transformations. This allows for example a derivation of the overlap conditions which have been postulated in refs. [22-24] in the context of the so called "group-theoretic" approach. We would like to emphasize that the present approach is more general than the one pursued in refs. [22-24], as the mappings ~r(Z) must not necessarily be projective. The technique presented in this paper can easily be adjusted to the case where the world sheet M has genus different from zero. In that case one must replace the kernel 1 / ( z - z') in eq. (2.12) by an abelian differential of the third kind. In contradiction to the claims of refs. [2, 4] we have not been able to verify that the ghost Neumann coefficients coincide with the X Neumann coefficients. We believe that this discrepancy is due to the fact that in refs. [2, 4] the transformation properties of the ghost coordinate have not been incorporated appropriately. These findings at least exclude that the vertices presented in refs. [2, 4] have an interpretation in terms of transition functions on bordered Riemann surfaces. However, more seriously, we find it hard to believe that any vertex which does not share this interpretation is BRST invariant. We have used our technique to construct, as well, a BRST completion of the Caneschi-Schwimmer Veneziano vertex. In this case our results are in agreement with those of ref. [2], besides the contributions of the zero modes.
I57
T. Ohrndorf / String couplings
The authors of refs. [8, 9], who have constructed the oscillator representation of Witten's vertex in the ghost sector, claim as well that X Neumann coefficients and ghost N e u m a n n coefficients essentially coincide. In the light of the results of the present paper we feel that this claim deserves reconsideration. The author wishes to express his thanks to A. Neveu for an instructive discussion.
Appendix A
EVALUATION OF N~ In the present appendix we shall calculate the Neumann coefficients Nm% explicitly, to convince the reader that our formalism leads exactly to the known results for the couplings of the bosonic degrees of freedom. The point of departure is the relation (2.15). For convenience we shall set the constants c~, to zero throughout. Technically a direct evaluation of the coefficients from eq. (2.15) is quite involved. If we, however, take suitable combinations of F r and its derivatives, such that the m-fold singularity at ~ = 0 cancels, the evaluation of the coefficients simplifies enormously. A suitable combination is provided by the following expression
do do'l-1 |\ 1 a , m ~ f z f 7 m 1 - dz j z-z'
. (A.I)
Note that the integrand on the right-hand side of eq. ( A . 1 ) 1s non-singular at z = z'. This is immediately obvious, as due to
dp dz
z -- Z T a 3 z ( z - 1) '
(A.2)
we have
1
d0(d0'l dz~dz']
=(z--zt)[(Z--ZT)(Z'--ZT)+(1--ZT)ZT]
X[Z(Z--I)(z'--ZT)]
(A.3)
-1
Inserting eq. (A.3) into eq. (A.1) the right-hand side factorizes completely
( a~;S'--d;. a + a,m ) G ( P
)=
1TlOClOC20¢rOlsOL33( Z' -- ZT)
,z ~r
Z
-1ZT
(A.4)
T. O h r n d o r f /
158
S t r i n g couplings
The coefficients N~s can now immediately be read off eq. (A.4) by noting that
--
- Z -- ZT
nf,(Ts)~ +" -- --~sl
1 ]
n=1
0
d
-1- - - ~ s , 2 '
p E Ms,
~2
l
(A.5)
where
L(vs)-= r(n+ l)r(n(vs-1) +1) "}ts =
O/s+ 1 -- - - , O/s
0/4=0/1
(A.6)
'
(A.7)
•
In this way we get exactly Mandelstam's result alaza3fm (7~)f. (7~)
rs __
Nrn -
-ran
OtrOts(tlar_t_mOts )
,
+1,
(A.8)
rn >/ + 1 .
(A.9)
m,n>~
1 N r s o = "~ - - f m ( ' ~ r ) ( O L l ~ s , 2 - Ot2(~s,1),
o/r
The coefficients N% may be brought into a symmetric form by making use of the freedom to add the constants c~, [2]. Appendix B
EVALUATION OF Vmn r' In order to evaluate the coefficients Vrs explicitly which have been defined by eq. (3.20) it is useful to start from the following expression 1 G~, '~: = -
% ~:(~ d z
a s
Jzr
Z -- Z ;~r m
% s ~-Tz,]
dz
z
_z,~r m
dp
do'
+ ....
(B.1)
where the dots signalize the freedom to adjust the coefficients ak, k = 0,1, 2 in eq. (3.19). The first term of eq. (B.1) is just Fr multiplied by a factor of ( a J a s ) ~ ;. It has been evaluated in appendix A. By choosing the undetermined coefficients in eq. (3.19) in a particular way, the second term in eq. (B.1) factorizes ~ ' F ~' "
G,~ '~; %
) a3as
1 dz
~
dz~'~"r
(B.2) Z -- ZT
159
T. Ohrndorf / String couplings
Therefore, the coefficients V~ are given by O/r
O~lOg20~r V s
V,~= --mN~
+ - -
OLs
r ,
(B.3)
, Vr
O130L 2
where
if
(B.4)
We have V.*= J.*(-2, +1),
V~ = Jm~(0,- 1),
(B.5)
where J~( p, l):= ~. dz ~7" Zr
[z(z -
1)] P ( Z
(B.6)
-- Z T ) I .
The integral (B.6) can be represented in terms of hypergeometric functions by: J~=0
if m - p ~ < 0 ,
whereas, if l>10and m - p > / l + l o r l < ~ 0 a n d 1_ J~ - F ( m
p)r(mv,
-
xr(-l,p-m+
j2=0
if
m - p > ~ +1,
r ( m Y l + p + 1) -
m
(B.Va)
(1 - V l ) - '
+ 2(p + 1))
l;myl-m+
Z(p+ l);1-Vx);
m-p~<0,
whereas, if l>~0and m - p > / l + l
j2 = ( _ 1)P+Iy2I
×F(-I,
(B.Vb) (B.8a)
orl~<0and m - p >
+1,
F ( m y 2 - l - 1 - 2p) _ m - 1-p)F(m - p)
r(mv2
p - m +
1; my2 --
m - l + p;
1 - 72);
(B.8b)
and finally J ~3 -_ 0
if
whereas, if l>~0and m + 2 p + l > t 0 o r l ~ < 0 a n d j~ = ( _ 1),+ 1
(B.9a)
m+2p+2+l<~O,
m+2+l>~-l,
F(mv3 + p + 1) 2)F(my3- m-p-
F ( m + 2 p + 1+
×r(-l,-1-m-Zp-l;my3-m-p-l;1-73).
l) (B.9b)
T. Ohrndorf / String couplings
160
The coefficients V~,, V.~ are given by
F(nTs- 1)n V2 = C(n + 2 ) F ( n ( v , - 1) + 2) (1 + 71)(n (71 - 1) - 1)(r/(~/1 (']/2 2)(nY2 - 1)(n72 + 1), (2V3 - 1)(n + 1)(n - 1),
×
-
-
1) + 1),
-
-
s=l, s:2, s:3,
n>~ - 1 ,
n>~ - 1 , n>2,
(B.IO)
for n ~ 0 . If n = 0 --
__m
v°~= / ( v 2 - 2)/v2 ' ~0,
s = +1, (B.11)
s = +2, s = +3,
F(myr- 1)m2yr v2 =/'(m + 2)F(m(rr-- 1) + 2)
)(
71(1 - 7 1 ) m F ( 1 , - m + 1; m ( 7 1 - 1) + 2,1 - 71),
r=l,
-'Y2(m ('g2- 1) + 1)F(1, - m + 1; m(72 - 1) + 1;1 - T2),
r=2,
73(m(~,3- 1) + 1)F(1, - m ; m ( 7 3 - 1) + 1;1 - 73),
r=3, (B.12)
if m>~2. Appendix C EVALUATION
O F G,~"s
This appendix is devoted to the explicit determination of the coefficients G,,k," of the expansion (3.15) of the holomorphic inverse differentials in powers of ~',. They
TABLE
k 1 2 3
s=l +1 1 - 271 +1
1
s=2 -1 + 1 2-/3 3
s=3 273 1 + 1 +1
161
T. Ohrndorf / String coupling~ TABLE2
a,f" ( . ~ o) k
s=l
s=2
s=3
/'(n?' l + 1)(1 - ?'l) F(n + 1)F(n (?'1 - i) + 2)
F(n?'2 1)(?'2 1)?'2 F(n)F(n(?'2 - 1) + 1)
m
n)+l
n ) +1
/'(n?' 3 + i)(i - ?'3) C(,, + 2)F(n (?'~ - 1) + 1) n > -1
F(,,?', + 1)(1 - ?'1) F(n+2)F(n(?'l 1)+1) n) -1
r(n?'2)(1 - ?'2)?'2 V ( n ) r ( n ( ? ' 2 - 1) + 2) n ) +1
r(n?"3 1)?'3(?'3 - 1) r ( n ) r ( ~ ( ? ' 3 - 1) + 1) n ) +1
r(n?'l - 1)?'1(1 - Yl) r ( , , ) r ( n ( ? ' , - 1) + 1) n ) +1
r(n?'2)n(?" 2 - 1)?'2 r ( n + 2)r(n(?'2 1 ) + 1 ) n) 1
F(n?'3) ?'3(?'3 - 1) F(n)F(n(y 3 1)+2) n ) +1
are given by 2
= a~
dz~;"
[ z ( z - 1)] 2 g k "
(C.1)
C h o o s i n g for t h e h o l o m o r p h i c i n v e r s e d i f f e r e n t i a l s the basis (3.14) has the a d v a n t a g e t h a t t h e c o e f f i c i e n t s G,k's c a n b e w r i t t e n in a c o m p a c t f o r m . W e h a v e listed t h e m in t a b l e s 1 a n d 2.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A. Neveu and P. West, Phys. Lett. B168 (1986) 192 A. Neveu and P. West, Nucl. Phys. B278 (1986) 601 H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B172 (1986) 186 H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D34 (1986) 2360 E. Witten, Nucl. Phys. B268 (1986) 353 E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. B179 (1986) 57 S. Samuel, Phys. Lett. B181 (1986) 249 S. Samuel, Phys. Lett. B181 (1986) 255 D. Gross and A. Jevicki, Nucl. Phys. B283 (1987) 1; 287 (1987) 225 A. Eastaugh and J.G. McCarthy, Nucl. Phys. B294 (1987) 845 G.T. Horowitz and S.P. Martin, Nucl. Phys. B296 (1988) 220 P. Di Vecchia, R. Nakayama, J.L. Petersen, J. Sidenius and S.Sciuto, Phys. Lett. B182 (1986) 164 P. Di Vecchia, R. Nakayama, JL. Petersen, J. R. Sidenius and S. Sciuto, Nucl. Phys. B287 (1987) 621 [14] U. Carow-Watamura and S. Watamura, Nucl. Phys. B288 (1987) 500 [15] S. Mandelstam, Nucl. Phys. B64 (1973) 205 [16] C. Vafa, Phys. Lett. B199 (1987) 195
T. Ohrndorf / String couplings"
162 [i7] [18] [19] [20] [21] [22] [23] [24] [25] [26]
L. T. T. E. E. A. A. A. T. T.
Avarez Gaum& C. Gomez, G. Moore and C. Vafa, Nucl. Phys. B303 (1988) 455 Ohrndorf, Nucl. Phys. B293 (1987) 709 Ohrndorf, Nucl. Phys. B301 (1988) 460 Cremmer and J.L. Gervais, Nucl. Phys. B76 (1976) 1100 Cremmer and J.L. Gervais, Nucl. Phys. B90 (1975) 410 Neveu and P. West, CERN-TH 4757/87 Neveu and P. West, Phys. Lett. B193 (1987) 187 Neveu and P. West, Phys. Lett. B194 (1987) 200 Ohrndorf, Phys. Lett. B194 (1987) 490 Ohrndorf, in preparation