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String ghost number violation in the operator formalism
Volume 2 I 5, number
PHYSICS
2
STRING GHOST NUMBER VIOLATION Th. OHRNDORF
LETTERS
15 December
B
1988
IN THE OPERATOR FORMALISM
’
Institut,fir Theoretische Physik der UniversitZitHeidelberg, Philosophenweg 16, D-6900 Heidelberg, Fed. Rep. Germany Received
25 July 1988
A systematic procedure of incorporating b-c zero modes into an operator formalism on arbitrary Riemann surfaces is presented. It is demonstrated that transition amplitudes on Riemann surfaces must contain certain fermionic &functions to ensure analyticity of the “classical” ghost coordinates. A transition amplitude on a Riemann surface of Euler characteristic x violates ghost number conservation by - 3%units
become functionals
1. Introduction and results
(b(g), One of the major obstacles which prevented the calculation of loop corrections to string amplitudes in the early days was the lack of an understanding of the measure of the string path integral. This has changed since Polyakov [ 1 ] proposed his path integral. By now it is well understood how to incorporate ghost coordinates to obtain a well defined measure. Recent attempts to gain a better insight into “nonperturbative” string phenomenae like covariant string field theory [ 2-6 ] and the diverse operator formalism [ 7- 15 ] are dealing with the construction of various variants of string vertices. These vertices had their forerunners in the so-called reggeon vertices [ 16- 18 1, which were obtained in the old days by factorizing dual amplitudes. In the language of the path integral, the idea of factorization naturally leads one to the path integral on bordered Riemann surfaces. The transition amplitudes (TA) defined by this path integral provide a representation of the string vertices. In Polyakov’s formulation one has to supplement the TA Kx of the string coordinate X by the TA Kg,, of the ghost coordinates. These are the objects we shall deal with in the present paper. To define a TA on a bordered Riemann surface a parametrization (parameter a) of the boundary loops has to be agreed on. In this way the TA Kx and Kgh ’ Supported
280
by DFG.
of the boundary
conditions
X(a),
c(a))
X(a) =X(zY z,
1 ztaM
,
c(a)=-~Im(cZli)I=ta~, b(a)=-i~Im(i’b,,)l..aM,
(1.1)
to be imposed on X, bZ, and cZ along the boundary aM of the world sheet M. The various field theories and operator frameworks differ in the way they slice the underlying Riemann surface into pieces and in their conventions fixing the boundary parametrization. Therefore, TA on bordered Riemann surfaces provide a picture unifying the various field theory and operator approaches. From this point of view the investigation of TA on bordered Riemann surfaces offers several advantages compared with any specific field theory/ operator approach. At first by not choosing any particular slicing and parametrization one does not obscure the essential features by the accidentials of a particular choice. Secondly one has at one’s disposal the possibility of building TA on complicated Riemann surfaces by gluing together TA of much simpler Riemann surfaces. Such “gluing rules” provide an important crosscheck for the general form of the TA. Finally one can employ the knowledge about bordered Riemann surfaces in a definite way to construct TA. In the present paper we shall in particular clarify the way in which the zero modes of the ghost coordi-
0370-26931881s (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
Volume 215, number 2
PHYSICS LETTERS B
nates b=: and c= have to be incorporated into TA. A crucial role therein is played by the R i e m a n n - R o c h theorem on bordered Riemann surfaces. Ghost TA have been investigated recently by the present author. In particular in ref. [ 19 ] it has been found that certain conditions must be imposed on b ( a ) and c(a) in order to ensure that c= and b== are analytic. In this paper we shall demonstrate that these conditions can be naturally incorporated into the definition of Kgh such that Kgh becomes a functional o f b ( a ) and c(a), where b(a) and c(~r) do not have to satisfy any additional constraints. We shall demonstrate that the amount of ghost number non-conservation AN~ of a TA defined on a Riemann surface of Euler characteristic Z is given by - 3Z. This result generalizes an earlier result of ref. [20] for closed surfaces. We would like to point out that due to the definition of Z AN= is an additive quantity, i.e. by gluing M~ and M2 together to form M we have AN= (M) =AN=(M, ) +AN= (M2)
.
(1.2)
Though the property (1.2) appears to be quite natural, a glance at the field theory/operator approaches proposed in the literature reveals that they do in general not satisfy (1.2). For example a basic idea of string field theory is to represent a propagator by the TA on an annulus. It is a common practice [4,21 ] to insert a line integral of the b ghost coordinate into the TA, giving a non-vanishing AN=. However, by gluing together two annuli one gets again an annulus, therefore ( 1.2 ) demands that AN==O for the propagator. Saturating string vertices with physical states and integrating over the moduli produces dual amplitudes. In the language of the path integral the procedure of saturating the vertex with physical states corresponds to filling in the boundary disks of the bordered world sheet, which has originally been associated with the vertex. The boundary disks are filled with operator inserted disks. We shall denote the closed world sheet obtained in this way by M~. In order to make the integrand into a density on moduli space, one has to take into account the discrete determinant of the anti-ghost zero modes obtained by gauge fixing Polyakov's path integral [22,23]. This determinant is defined on M~. It is the treatment of this determinant, which distinguishes the present approach from those in the literature.
15 December 1988
One may view part of this determinant as being left over from the insertions of the b zero modes into the ghost path integral. These insertions are necessary to have the path integral non-vanishing on a compact Riemann surface. The mentioned b line integral insertion into the string field theory propagator represents just such a zero mode insertion. In contrast on a bordered Riemann surface the path integral is nonvanishing without the zero mode insertions. This will become obvious in section 3. Therefore in the process of building physical amplitudes from string vertices one may postpone the insertion of the zero modes just before filling in the last boundary disk. Doing so offers the advantage of having a well defined ghost number and a clear interpretation in terms of the path integral. Moreover, such a procedure offers much more freedom in choosing coordinates on Teichmiiller space. This paper is organized as follows. In the next section we shall list some properties of "classical" ghost coordinates, and state the general form of the ghost TA Kgh, which is justified in section 3. Section 4 (5) is devoted to analyzing the ghost number violation of Kgh for the closed (open) string. In section 6 we shall establish the relation between the coordinate space representation of Kgh used in sections 2-5 and its Fock-space representation.
2. "Classical" ghost coordinates The present section reviews some basic material about ghost fields on bordered Riemann surfaces. Details have been worked out in ref. [ 19 ]. On a bordered Riemann surface quadratic (inverse) differentials h~.(gf) are defined in such a way that they satisfy Im(z2h~:) = I m ( g f / ~ ) = 0 ,
ze0M,
(2.1)
along 0M, where 0M is fixed by z=z(a). An important role is played by those differentials, which are analytic on M. Their number is determined by the Riemann-Roch theorem. If G > 1 (G:=2g+N- 1, g=genus of M, N = n u m b e r of boundary components), then there are ( 3 G - 3 ) finite quadratic and no inverse differentials on M. If M is an annulus (disk) we have 1 (0) quadratic and 1 (3) inverse dif281
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PHYSICS LETTERS B
ferentials. Any differential, not necessarily finite satisfies
Z-P=2u(G-1) ,
(2.2)
where Z(P) counts the number of zeros (poles) an u is the order of the differential. An extremely useful tool to construct ghost coordinates is provided by the so-called variation kernel n(z~, z2) =n-~_-~.[24]. It is defined to have a first order pole along z, = z2
n(z,, z2) = (z~ - z 2 )
- ~+analytic at z~ = z 2 ,
(2.3)
and to fulfill ifz~ ~ z 2 ~ 0 M .
f dac(a2)~2n(z,, z2) .
Due to the additional poles of n (z), z2 ) c =' is not necessarily holomorphic for general c(tT). If c(tT) however satisfies
f dGc(a)Zz2h~:=O
(2.6)
0M
with dtrc(G)h~_.~ 2 ,
(2.10)
provided G > 1. If there are any holomorphic inverse differentials on M, we have to include the associated constraints as well into the product defining 6.
3. Path integral derivation of Kgh We shall demonstrate in this section that (2.9) can be obtained from the path integral representation of the ghost TA Kgh Kgh = j ~ b @c exp ( - Sgh ),
for each holomorphic quadratic differential on M, then c:' as given by (2.5) is holomorphic. If there are any holomorphic inverse differentials on M, we get in the same way a set of constraints to be imposed on b( a), namely
f drrb(G)g~/~=O.
(2.7)
0M
Defining the ghost TA Kg~ by the path integral on a Riemann surface M implies that K ~ [...] = J D b ( a ) D c ( a ) K ~ [...,
b(a),
c(a)] (2.8)
for any decomposition of M into two separate sur-
(3.1)
where it is understood that the b-c configurations to be integrated over satisfy ( 1.1 ). The ghost action S g h is given by [ 19 ]
d2z[b~Osz]+~
Sgh = ~-~ M
c ( a ) .... ] ,
(2.9)
(2.4)
(2.5)
C--I~--- N//2 7~ 0M
282
Kg~=Jdet(P+ pl ) ,/2e x p ( - S gclh ) ,
k=l
For fixed z2 n(z, z2) may be considered as an inverse differential in z,. If G > 1 then n(z~, z2) must have further singularities in z~ besides (2.3) due to (2.2). It can be shown [ 24 ] that the residues of these singularities are given by linear combinations of the holomorphic differential h ~:. A ghost field, which is meromorphic on M and satisfies ( 1.1 ) is given by
XKr~[b(a),
faces MA and MB. The integral D b ( ~ ) D c ( a ) is understood to be taken over all configurations b(a) and c(a) living on the common boundary Of MA and MB. These configurations do not satisfy any constraint. Therefore KM must have the property that it is only non-vanishing provided b(~) and c (tT) satisfy (2.6) and (2.7). We shall argue below that K~, has the following form:
J ' = I-I
Im[(~2/~t)n(zl,z2)]=O
1
15 December 1988
~z:=~-~z.
dg~c~b,z+c.c. OM
(3.2)
A generic configuration contributing to the ~ b integral consists of three pieces el 0 qu . b_~z=b_~z+bz._+b~z
(3.3)
The classical anti-ghost coordinate b§tz is given by the convolution of b(tr) with n(zl, z2) similar to (2.5). Therefore it satisfies ( 1.1 ). As we do not impose any conditions on b ( a ) , b§~ is not necessarily analytic on M. b°. is a linear combination of the analytic quadratic diffferentials hfz with real anti-commuting coefficients, b ~ finally parametrizes the quantum fluctuations. Both b°z and b qf satisfy ( 1.1 ) with the
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left hand side vanishing, c-- has a decomposition similar to (3.3). It is now a simple exercise to verify that (3.1) is indeed equivalent to (2.9). Note that the appearance of the 0-function in (2.9) is an immediate consequence of the presence of the zero mode piece b°-:. We would like to stress that the term of Sgh defined on 0M plays a decisive role in demonstrating that (2.9) can be represented by the path integral (3.1).
4. Ghost number violation of the closed string The b-c system has a global phase invariance, which is reflected by the existence of a current, which is classically conserved, the ghost number current. Integrating this current along a closed curve and taking the real part, one obtains the ghost number N#. At the first quantized level the ghost number is anomalous. It is a well-known fact that on a compact Riemann surface the associated anomaly is given by the intrinsic curvature of the world sheet [ 20 ]. Via the GaussBonnet theorem the ghost number violation is proportional to the Euler characteristic of M. In the present section we would like to show that on a bordered Riemann surface the violation of ghost number is solely determined by O, i.e. it is again proportional to the Euler characteristic of M. The proof of this statement requires to show that there exists a normal ordering prescription of the ghost number current, which is independent of M and has the property that the exponential of the classical action always has vanishing ghost number, no matter what M is. The clasical ghost number current j- is given by j__ := - b = c ~ .
(4.1)
Note that when multiplying Kgh by j~ we can always replace the classical ghost fields appearing in (4.1) by those given by (2.5) and its counterpart for the anti-ghost field, which are not necessarily analytic. If we would have agreed to take only the analytic piece of b_-_- and c-- in (4.1) we would have to introduce some correction terms. In ref. [ 19 ] it has been shown that { ~ 2 } e x p ( - S C ' ) = ~ b¢~ e~'x )p)( - ,( c
.
(4.2)
15 December 1988
~ : an 6 = have the standard expansions
L~=~ z-'-2b., ~-'=Zz-'+%, rl
n
{c,, b m ) = r n _ m •
(4.3)
Though that is a set of oscillators for each boundary component, we do not distinguish between them, as we shall concentrate on a single fixed boundary loop in the following, z is a coordinate on the associated boundary disk. Let us first analyse the case where M is a disk. For the unit disk
1
+lzt 1 2 z-~2 + --'z2
n ° ( z , , z 2 ) - zl-z-~2
(4.4)
We have in general {U ~, b . ~ } = n ( z , , z) .
(4.5)
Therefore employing (4.2) L e x p ( -Scl ) = l i m [ U ' 6 : ~ - - n ° ( z l , z)] e x p ( - S d) Zl ~z
= { :j~ : - 3 / 2 z ) e x p
( - S cz ),
(4.6)
where
Jz
.7,
The normal ordering (4.7) is equivalent to the following oscillator normal ordering: :c,b_,: =c~b_,
n<~l ,
(4.8)
= - b n c n n>~2. If we define
)
N#:= ~ i
dz:j.: +c.c. +1¢,
(4.9)
0M
we have on the disk N=exp ( - S ~ ) = 0 ,
(4.10)
provided we choose x = - 3. Let us now turn to an arbitrary bordered Riemann surface. On an arbitrary surface n°(zt, z) in (4.6) has to be replaced by nM(Zl, Z), the variation kernel of M. Therefore we get 283
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PHYSICS LETTERSB
{ jzexp ( - set ) = ~ :L : - 3/2z
15 December 1988
{cZl, b~=}=Z,[(Z,)-In(z,,z)--(Z,*-*21)] .
Defining the ghost number operator for the open string by
+ l i m [n°(zl, z)--nM(zl, Z)]~ exp( -Scl ) Zl ~Z / (4.11) where the normal ordering has been defined by (4.7). The important point to notice here is that due to (2.4) the difference between n o and n M drops out of Ng, if we integrate along the rth component of 0M. Accordingly
N.=~
i
dz:j~: + x ,
(5.3)
it can be shown by arguments similar to those used for the closed string that
NgUtexp(-SC~)=(l+Xout)exp(-Sc~),
(5.4)
and
N
N~exp( - S c' ) = 0 ,
(4.12)
r=l
an the total ghost number violation of Kgh is due to d. So far we have implicitly assumed that the point z = 0 around which the oscillator expansion (4.3) has been performed lies on M. If the boundaries of M arise by cutting out holes of the complex plane, one uses an expansion like (4.3) where however z = 0 no more lies on M. This implies that we have to let these oscillators act on Kgh from the right. Accordingly we have to distinguish among "in" an "out" operators. So far we have only used "out" operators. N ~ is again given by (4.9) with positive oriented contour of integration and x~,=-Xou~. The violation of ghost number AN~ is now determined by
AN~=~ N~'~+ ~ N~"t'~ • r
(4.13)
r
due to the Riemann-Roch theorem we have AN. = - 3 Z ,
(4.14)
Z=2-2g -N.
exp(-SC~)N~=exp(-S ~) ( - 2 + x i , ) .
(4.15)
Xou,= -x~, = _ 3 .
(bz~)*=b=,
(cZ)*=c e.
(5.7)
As the holomorphic differentials appearing in d must satisfy analogous relations, only a subset of all the holomorphic differentials on M will contribute to d. For the disk (annulus) we have a single inverse (quadratic) differential contributing to d. Therefore (5.4) and (5.5) imply AN.= -3 =0
G=0, G=I.
(5.8)
Assuming that ( 1.2 ) is valid one may show by induction that G>I,
(5.9)
or in general (5.10)
(5.9) implies that there must be 3 g + 2 N - 3 quadratic differentials satisfying the analogue of (5.7) and therefore contributing to d if G > 1.
5. Ghost number violation of the open string The determination of the ghost number violation of the open string is in fact not much more complicated than for the closed string. For the open string we have again (4.3) however with (c,,)*=c~,
(5.6)
Due to (5.1) the ghost fields of the open string satisfy
AN~= - ~ Z .
(b,)*=bn,
(5.5)
We shall choose
A N # = 3 ( G - 1)
with Euler characteristic
(5.1)
in addition. (5.1) implies that (4.5) has to be modified into 284
(5.2)
6. Fock-space representation of transition amplitudes So far we have worked in a representation of Kghin which b(a) and c(a) are diagonal. A Fock-space representation Kgh may be introduced by writing Kgh as a convolution of some integral kernel with coherent
Volume 215, number 2
PHYSICS LETTERS B
state wave functions for the oscillators b, and c,. We shall denote this integral kernel again by Kgh. As most authors prefer to work in the Fock-space representation, we would like to translate our results into this representation. In ref. [25 ] the three string vertex of the covariantized light cone field theory has been constructed. Making use of the technique employed here, it is not too difficult to demonstrate that the general form of the transition amplitude Kgh in the Fock representation is given by Kg h = (01 v g h ~ ,
(6.1)
where the vacuum (01 satisfies (0lc_, =0
n>~-l,
(0[b_, =0
n>~2,
(6.2)
for each of the Nsets of ghost oscillators. N~g is given by
N~=(~:b_,c+,:+c.c.)+3,
(6.3)
therefore (01 carries 3N units of ghost charge. V gh is an exponential function, which is bilinear in b, and c,. Therefore its contributes zero to AN=. V gh is obtained as a solution of transition identities satisfied by Kgh, involving both creation as well as annihilation operators. The function or finally, which should not be confused with 6, which appears in (2.9) arises due to the presence of finite quadratic and/or inverse differentials on the world sheet M~. These differentials give rise to transition identities, which involve only creation operators, an therefore act similarly to fermionic g-functions. For example if g = 0 and N = 3, we have three complex holomorphic inverse differentials on M~, the sphere. The corresponding function or is a product of six fermionic ~-functions involving the anti-ghost oscillators. It has been constructed explicitly in ref. [25]. The presence of this function has previously been found as well in refs. [ 7-9 ]. In general or is a product of as many fermionic 0-functions involving b (c) oscillators, as there are holomorphic inverse (quadratic) differentials on M~. Therefore it violates ghost number conservation by 6 g - 6 units, giving a total ghost number violation of K~h by 6g - 6 + 3N units coinciding with (4.14 ).
15 December 1988
Let us briefly comment on the emergence of the Ofunction in ( 2.9 ). As already mentioned, ( 2.9 ) is obtained from (6.1) by saturating (6.1) with harmonic oscillator wave functions. Technically this may be achieved by employing coherent state techniques. In this way (2.9) is represented by a quadratic integral over coherent state variables. It turns out that the associated quadratic form vanishes for certain linear combinations of coherent state variables, which precisely correspond to the zero modes on M, which satisfy the boundary condition (2.1). Treating these combinations as "collective coordinates", the 0-function in (2.9) factorizes. Let us finally outline how to obtain scattering amplitudes from the string vertex. As alreay pointed out in the introduction, scattering amplitudes of physical states are obtained from the full string vertex by saturating it with such states. In particular BRST invariant states are produced by acting with the product of the ghost coordinate c(z) and one standard vertex operator V(z) on a vacuum [0), which satisfies
c, lO>=O
n~>2,
b,[0> = 0
n>~-l,
(6.4)
and
(010)=+l.
(6.5)
Note that saturating legs with I0) alone, without any vertex operator insertion corresponds to filling in one of the boundary disks. Saturating all external legs with physical states gives a vanishing result. The reason is that the amplitude obtained is represented by a path integral on a Riemann surface without a boundary. To make such a path integral non-vanishing we have to insert enough b-fields to absorb the zero-modes. These b-field insertions allow for an elegant rewriting of the discrete determinant which makes the amplitude into a density on moduli space. Therefore before filling in the last boundary disk we have to act with the operator
--I~II Ld¢iJCd2z'lt (z)b=(z × )1 [c.c.],
I 3g--3+N[-i
M
(6.6) where the /z~--"are the Beltrami differentials associated with the/th mouli. This means that (6.6) should 285
Volume 215, number 2
PHYSICS LETTERS B
be considered as being part of the inner product between two states of the operator formalism Hilbert space, rather than the states themselves.
References [ 1 ] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 207. [2] A. Neveu and P. West, Nucl. Phys. B 278 (1986) 69. [3 ] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 34 (1986) 2360. [4] E. Witten, Nucl. Phys. B 268 (1986) 253. [ 5 ] E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. B 179 (1986) 57. [ 6 ] D. Gross and A. Jevicki, Nucl. Phys. B 283 ( 1987 ) 1; Nucl. Phys. B 287 (1987) 225. [7] P. Di Vecchia, R. Nakayama, I.C. Peterson, J.R. Sidenius and S. Sciuto, Nucl. Phys. B 287 (1987) 621. [ 8 ] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Nucl. Phys. B 298 (1988) 527. [ 9 ] J.L. Petersen and J. Sidenius, Nucl. Phys. B 301 ( 1988 ) 247.
286
15 December 1988
[10IA. Neveu and P. West, Nucl. Phys. B278 (1986) 601; Commun. Math. Phys. 144 (1988) 613. [ 11 ] M.D. Freeman and P. West, Phys. Lett. B 205 (1988) 30. [ 12 ] P. West, Phys. Lett. B 205 ( 1988 ) 38. [13] A. Le Clair, M.E. Peskin and C.R. Preitschopf, preprint SLAC-PUB-4306/4307 (January 1988). [ 14 ] C. Vafa, Phys. Lett. B 190 ( 1987 ) 47; B 199 (1987) 195. [15]L. Alvarez-Gaum6, C. Gomez, G. Moore and C. Vafa, preprint CERN-TH-4883/87. [ 16] S. Sciuto, Lett. Nuovo Cimento 2 (1969) 411. [ 17 ] L. Caneschi, A. Schwimmer and G. Veneziano, Phys. Lett. B 30 (1969) 351. [18] C. Lovelace, Phys. Lett. B 32 (1970) 490. [ 19] Th. Ohrndorf, Nucl. Phys. B 301 (1988) 460. [20] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [21 ] S.B. Giddings and E. Martinec, Nucl. Phys. B 278 (1986) 91. [22] E. D'Hoker and P.H. Phong, Nucl. Phys. B 269 (1986) 205. [ 23 ] G. Moore and P. Nelson, Nucl. Phys. B 266 ( 1986 ) 58. [ 24 ] M. Schiffer and D.C. Spencer, Functionals of finite Riemann surfaces (Princeton U.P., Princeton, 1954). [ 25 ] Th. Ohrndorf, Nucl. Phys. B 310 ( 1988 ) 141.
PHYSICS
2
STRING GHOST NUMBER VIOLATION Th. OHRNDORF
LETTERS
15 December
B
1988
IN THE OPERATOR FORMALISM
’
Institut,fir Theoretische Physik der UniversitZitHeidelberg, Philosophenweg 16, D-6900 Heidelberg, Fed. Rep. Germany Received
25 July 1988
A systematic procedure of incorporating b-c zero modes into an operator formalism on arbitrary Riemann surfaces is presented. It is demonstrated that transition amplitudes on Riemann surfaces must contain certain fermionic &functions to ensure analyticity of the “classical” ghost coordinates. A transition amplitude on a Riemann surface of Euler characteristic x violates ghost number conservation by - 3%units
become functionals
1. Introduction and results
(b(g), One of the major obstacles which prevented the calculation of loop corrections to string amplitudes in the early days was the lack of an understanding of the measure of the string path integral. This has changed since Polyakov [ 1 ] proposed his path integral. By now it is well understood how to incorporate ghost coordinates to obtain a well defined measure. Recent attempts to gain a better insight into “nonperturbative” string phenomenae like covariant string field theory [ 2-6 ] and the diverse operator formalism [ 7- 15 ] are dealing with the construction of various variants of string vertices. These vertices had their forerunners in the so-called reggeon vertices [ 16- 18 1, which were obtained in the old days by factorizing dual amplitudes. In the language of the path integral, the idea of factorization naturally leads one to the path integral on bordered Riemann surfaces. The transition amplitudes (TA) defined by this path integral provide a representation of the string vertices. In Polyakov’s formulation one has to supplement the TA Kx of the string coordinate X by the TA Kg,, of the ghost coordinates. These are the objects we shall deal with in the present paper. To define a TA on a bordered Riemann surface a parametrization (parameter a) of the boundary loops has to be agreed on. In this way the TA Kx and Kgh ’ Supported
280
by DFG.
of the boundary
conditions
X(a),
c(a))
X(a) =X(zY z,
1 ztaM
,
c(a)=-~Im(cZli)I=ta~, b(a)=-i~Im(i’b,,)l..aM,
(1.1)
to be imposed on X, bZ, and cZ along the boundary aM of the world sheet M. The various field theories and operator frameworks differ in the way they slice the underlying Riemann surface into pieces and in their conventions fixing the boundary parametrization. Therefore, TA on bordered Riemann surfaces provide a picture unifying the various field theory and operator approaches. From this point of view the investigation of TA on bordered Riemann surfaces offers several advantages compared with any specific field theory/ operator approach. At first by not choosing any particular slicing and parametrization one does not obscure the essential features by the accidentials of a particular choice. Secondly one has at one’s disposal the possibility of building TA on complicated Riemann surfaces by gluing together TA of much simpler Riemann surfaces. Such “gluing rules” provide an important crosscheck for the general form of the TA. Finally one can employ the knowledge about bordered Riemann surfaces in a definite way to construct TA. In the present paper we shall in particular clarify the way in which the zero modes of the ghost coordi-
0370-26931881s (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
Volume 215, number 2
PHYSICS LETTERS B
nates b=: and c= have to be incorporated into TA. A crucial role therein is played by the R i e m a n n - R o c h theorem on bordered Riemann surfaces. Ghost TA have been investigated recently by the present author. In particular in ref. [ 19 ] it has been found that certain conditions must be imposed on b ( a ) and c(a) in order to ensure that c= and b== are analytic. In this paper we shall demonstrate that these conditions can be naturally incorporated into the definition of Kgh such that Kgh becomes a functional o f b ( a ) and c(a), where b(a) and c(~r) do not have to satisfy any additional constraints. We shall demonstrate that the amount of ghost number non-conservation AN~ of a TA defined on a Riemann surface of Euler characteristic Z is given by - 3Z. This result generalizes an earlier result of ref. [20] for closed surfaces. We would like to point out that due to the definition of Z AN= is an additive quantity, i.e. by gluing M~ and M2 together to form M we have AN= (M) =AN=(M, ) +AN= (M2)
.
(1.2)
Though the property (1.2) appears to be quite natural, a glance at the field theory/operator approaches proposed in the literature reveals that they do in general not satisfy (1.2). For example a basic idea of string field theory is to represent a propagator by the TA on an annulus. It is a common practice [4,21 ] to insert a line integral of the b ghost coordinate into the TA, giving a non-vanishing AN=. However, by gluing together two annuli one gets again an annulus, therefore ( 1.2 ) demands that AN==O for the propagator. Saturating string vertices with physical states and integrating over the moduli produces dual amplitudes. In the language of the path integral the procedure of saturating the vertex with physical states corresponds to filling in the boundary disks of the bordered world sheet, which has originally been associated with the vertex. The boundary disks are filled with operator inserted disks. We shall denote the closed world sheet obtained in this way by M~. In order to make the integrand into a density on moduli space, one has to take into account the discrete determinant of the anti-ghost zero modes obtained by gauge fixing Polyakov's path integral [22,23]. This determinant is defined on M~. It is the treatment of this determinant, which distinguishes the present approach from those in the literature.
15 December 1988
One may view part of this determinant as being left over from the insertions of the b zero modes into the ghost path integral. These insertions are necessary to have the path integral non-vanishing on a compact Riemann surface. The mentioned b line integral insertion into the string field theory propagator represents just such a zero mode insertion. In contrast on a bordered Riemann surface the path integral is nonvanishing without the zero mode insertions. This will become obvious in section 3. Therefore in the process of building physical amplitudes from string vertices one may postpone the insertion of the zero modes just before filling in the last boundary disk. Doing so offers the advantage of having a well defined ghost number and a clear interpretation in terms of the path integral. Moreover, such a procedure offers much more freedom in choosing coordinates on Teichmiiller space. This paper is organized as follows. In the next section we shall list some properties of "classical" ghost coordinates, and state the general form of the ghost TA Kgh, which is justified in section 3. Section 4 (5) is devoted to analyzing the ghost number violation of Kgh for the closed (open) string. In section 6 we shall establish the relation between the coordinate space representation of Kgh used in sections 2-5 and its Fock-space representation.
2. "Classical" ghost coordinates The present section reviews some basic material about ghost fields on bordered Riemann surfaces. Details have been worked out in ref. [ 19 ]. On a bordered Riemann surface quadratic (inverse) differentials h~.(gf) are defined in such a way that they satisfy Im(z2h~:) = I m ( g f / ~ ) = 0 ,
ze0M,
(2.1)
along 0M, where 0M is fixed by z=z(a). An important role is played by those differentials, which are analytic on M. Their number is determined by the Riemann-Roch theorem. If G > 1 (G:=2g+N- 1, g=genus of M, N = n u m b e r of boundary components), then there are ( 3 G - 3 ) finite quadratic and no inverse differentials on M. If M is an annulus (disk) we have 1 (0) quadratic and 1 (3) inverse dif281
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ferentials. Any differential, not necessarily finite satisfies
Z-P=2u(G-1) ,
(2.2)
where Z(P) counts the number of zeros (poles) an u is the order of the differential. An extremely useful tool to construct ghost coordinates is provided by the so-called variation kernel n(z~, z2) =n-~_-~.[24]. It is defined to have a first order pole along z, = z2
n(z,, z2) = (z~ - z 2 )
- ~+analytic at z~ = z 2 ,
(2.3)
and to fulfill ifz~ ~ z 2 ~ 0 M .
f dac(a2)~2n(z,, z2) .
Due to the additional poles of n (z), z2 ) c =' is not necessarily holomorphic for general c(tT). If c(tT) however satisfies
f dGc(a)Zz2h~:=O
(2.6)
0M
with dtrc(G)h~_.~ 2 ,
(2.10)
provided G > 1. If there are any holomorphic inverse differentials on M, we have to include the associated constraints as well into the product defining 6.
3. Path integral derivation of Kgh We shall demonstrate in this section that (2.9) can be obtained from the path integral representation of the ghost TA Kgh Kgh = j ~ b @c exp ( - Sgh ),
for each holomorphic quadratic differential on M, then c:' as given by (2.5) is holomorphic. If there are any holomorphic inverse differentials on M, we get in the same way a set of constraints to be imposed on b( a), namely
f drrb(G)g~/~=O.
(2.7)
0M
Defining the ghost TA Kg~ by the path integral on a Riemann surface M implies that K ~ [...] = J D b ( a ) D c ( a ) K ~ [...,
b(a),
c(a)] (2.8)
for any decomposition of M into two separate sur-
(3.1)
where it is understood that the b-c configurations to be integrated over satisfy ( 1.1 ). The ghost action S g h is given by [ 19 ]
d2z[b~Osz]+~
Sgh = ~-~ M
c ( a ) .... ] ,
(2.9)
(2.4)
(2.5)
C--I~--- N//2 7~ 0M
282
Kg~=Jdet(P+ pl ) ,/2e x p ( - S gclh ) ,
k=l
For fixed z2 n(z, z2) may be considered as an inverse differential in z,. If G > 1 then n(z~, z2) must have further singularities in z~ besides (2.3) due to (2.2). It can be shown [ 24 ] that the residues of these singularities are given by linear combinations of the holomorphic differential h ~:. A ghost field, which is meromorphic on M and satisfies ( 1.1 ) is given by
XKr~[b(a),
faces MA and MB. The integral D b ( ~ ) D c ( a ) is understood to be taken over all configurations b(a) and c(a) living on the common boundary Of MA and MB. These configurations do not satisfy any constraint. Therefore KM must have the property that it is only non-vanishing provided b(~) and c (tT) satisfy (2.6) and (2.7). We shall argue below that K~, has the following form:
J ' = I-I
Im[(~2/~t)n(zl,z2)]=O
1
15 December 1988
~z:=~-~z.
dg~c~b,z+c.c. OM
(3.2)
A generic configuration contributing to the ~ b integral consists of three pieces el 0 qu . b_~z=b_~z+bz._+b~z
(3.3)
The classical anti-ghost coordinate b§tz is given by the convolution of b(tr) with n(zl, z2) similar to (2.5). Therefore it satisfies ( 1.1 ). As we do not impose any conditions on b ( a ) , b§~ is not necessarily analytic on M. b°. is a linear combination of the analytic quadratic diffferentials hfz with real anti-commuting coefficients, b ~ finally parametrizes the quantum fluctuations. Both b°z and b qf satisfy ( 1.1 ) with the
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left hand side vanishing, c-- has a decomposition similar to (3.3). It is now a simple exercise to verify that (3.1) is indeed equivalent to (2.9). Note that the appearance of the 0-function in (2.9) is an immediate consequence of the presence of the zero mode piece b°-:. We would like to stress that the term of Sgh defined on 0M plays a decisive role in demonstrating that (2.9) can be represented by the path integral (3.1).
4. Ghost number violation of the closed string The b-c system has a global phase invariance, which is reflected by the existence of a current, which is classically conserved, the ghost number current. Integrating this current along a closed curve and taking the real part, one obtains the ghost number N#. At the first quantized level the ghost number is anomalous. It is a well-known fact that on a compact Riemann surface the associated anomaly is given by the intrinsic curvature of the world sheet [ 20 ]. Via the GaussBonnet theorem the ghost number violation is proportional to the Euler characteristic of M. In the present section we would like to show that on a bordered Riemann surface the violation of ghost number is solely determined by O, i.e. it is again proportional to the Euler characteristic of M. The proof of this statement requires to show that there exists a normal ordering prescription of the ghost number current, which is independent of M and has the property that the exponential of the classical action always has vanishing ghost number, no matter what M is. The clasical ghost number current j- is given by j__ := - b = c ~ .
(4.1)
Note that when multiplying Kgh by j~ we can always replace the classical ghost fields appearing in (4.1) by those given by (2.5) and its counterpart for the anti-ghost field, which are not necessarily analytic. If we would have agreed to take only the analytic piece of b_-_- and c-- in (4.1) we would have to introduce some correction terms. In ref. [ 19 ] it has been shown that { ~ 2 } e x p ( - S C ' ) = ~ b¢~ e~'x )p)( - ,( c
.
(4.2)
15 December 1988
~ : an 6 = have the standard expansions
L~=~ z-'-2b., ~-'=Zz-'+%, rl
n
{c,, b m ) = r n _ m •
(4.3)
Though that is a set of oscillators for each boundary component, we do not distinguish between them, as we shall concentrate on a single fixed boundary loop in the following, z is a coordinate on the associated boundary disk. Let us first analyse the case where M is a disk. For the unit disk
1
+lzt 1 2 z-~2 + --'z2
n ° ( z , , z 2 ) - zl-z-~2
(4.4)
We have in general {U ~, b . ~ } = n ( z , , z) .
(4.5)
Therefore employing (4.2) L e x p ( -Scl ) = l i m [ U ' 6 : ~ - - n ° ( z l , z)] e x p ( - S d) Zl ~z
= { :j~ : - 3 / 2 z ) e x p
( - S cz ),
(4.6)
where
Jz
.7,
The normal ordering (4.7) is equivalent to the following oscillator normal ordering: :c,b_,: =c~b_,
n<~l ,
(4.8)
= - b n c n n>~2. If we define
)
N#:= ~ i
dz:j.: +c.c. +1¢,
(4.9)
0M
we have on the disk N=exp ( - S ~ ) = 0 ,
(4.10)
provided we choose x = - 3. Let us now turn to an arbitrary bordered Riemann surface. On an arbitrary surface n°(zt, z) in (4.6) has to be replaced by nM(Zl, Z), the variation kernel of M. Therefore we get 283
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{ jzexp ( - set ) = ~ :L : - 3/2z
15 December 1988
{cZl, b~=}=Z,[(Z,)-In(z,,z)--(Z,*-*21)] .
Defining the ghost number operator for the open string by
+ l i m [n°(zl, z)--nM(zl, Z)]~ exp( -Scl ) Zl ~Z / (4.11) where the normal ordering has been defined by (4.7). The important point to notice here is that due to (2.4) the difference between n o and n M drops out of Ng, if we integrate along the rth component of 0M. Accordingly
N.=~
i
dz:j~: + x ,
(5.3)
it can be shown by arguments similar to those used for the closed string that
NgUtexp(-SC~)=(l+Xout)exp(-Sc~),
(5.4)
and
N
N~exp( - S c' ) = 0 ,
(4.12)
r=l
an the total ghost number violation of Kgh is due to d. So far we have implicitly assumed that the point z = 0 around which the oscillator expansion (4.3) has been performed lies on M. If the boundaries of M arise by cutting out holes of the complex plane, one uses an expansion like (4.3) where however z = 0 no more lies on M. This implies that we have to let these oscillators act on Kgh from the right. Accordingly we have to distinguish among "in" an "out" operators. So far we have only used "out" operators. N ~ is again given by (4.9) with positive oriented contour of integration and x~,=-Xou~. The violation of ghost number AN~ is now determined by
AN~=~ N~'~+ ~ N~"t'~ • r
(4.13)
r
due to the Riemann-Roch theorem we have AN. = - 3 Z ,
(4.14)
Z=2-2g -N.
exp(-SC~)N~=exp(-S ~) ( - 2 + x i , ) .
(4.15)
Xou,= -x~, = _ 3 .
(bz~)*=b=,
(cZ)*=c e.
(5.7)
As the holomorphic differentials appearing in d must satisfy analogous relations, only a subset of all the holomorphic differentials on M will contribute to d. For the disk (annulus) we have a single inverse (quadratic) differential contributing to d. Therefore (5.4) and (5.5) imply AN.= -3 =0
G=0, G=I.
(5.8)
Assuming that ( 1.2 ) is valid one may show by induction that G>I,
(5.9)
or in general (5.10)
(5.9) implies that there must be 3 g + 2 N - 3 quadratic differentials satisfying the analogue of (5.7) and therefore contributing to d if G > 1.
5. Ghost number violation of the open string The determination of the ghost number violation of the open string is in fact not much more complicated than for the closed string. For the open string we have again (4.3) however with (c,,)*=c~,
(5.6)
Due to (5.1) the ghost fields of the open string satisfy
AN~= - ~ Z .
(b,)*=bn,
(5.5)
We shall choose
A N # = 3 ( G - 1)
with Euler characteristic
(5.1)
in addition. (5.1) implies that (4.5) has to be modified into 284
(5.2)
6. Fock-space representation of transition amplitudes So far we have worked in a representation of Kghin which b(a) and c(a) are diagonal. A Fock-space representation Kgh may be introduced by writing Kgh as a convolution of some integral kernel with coherent
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state wave functions for the oscillators b, and c,. We shall denote this integral kernel again by Kgh. As most authors prefer to work in the Fock-space representation, we would like to translate our results into this representation. In ref. [25 ] the three string vertex of the covariantized light cone field theory has been constructed. Making use of the technique employed here, it is not too difficult to demonstrate that the general form of the transition amplitude Kgh in the Fock representation is given by Kg h = (01 v g h ~ ,
(6.1)
where the vacuum (01 satisfies (0lc_, =0
n>~-l,
(0[b_, =0
n>~2,
(6.2)
for each of the Nsets of ghost oscillators. N~g is given by
N~=(~:b_,c+,:+c.c.)+3,
(6.3)
therefore (01 carries 3N units of ghost charge. V gh is an exponential function, which is bilinear in b, and c,. Therefore its contributes zero to AN=. V gh is obtained as a solution of transition identities satisfied by Kgh, involving both creation as well as annihilation operators. The function or finally, which should not be confused with 6, which appears in (2.9) arises due to the presence of finite quadratic and/or inverse differentials on the world sheet M~. These differentials give rise to transition identities, which involve only creation operators, an therefore act similarly to fermionic g-functions. For example if g = 0 and N = 3, we have three complex holomorphic inverse differentials on M~, the sphere. The corresponding function or is a product of six fermionic ~-functions involving the anti-ghost oscillators. It has been constructed explicitly in ref. [25]. The presence of this function has previously been found as well in refs. [ 7-9 ]. In general or is a product of as many fermionic 0-functions involving b (c) oscillators, as there are holomorphic inverse (quadratic) differentials on M~. Therefore it violates ghost number conservation by 6 g - 6 units, giving a total ghost number violation of K~h by 6g - 6 + 3N units coinciding with (4.14 ).
15 December 1988
Let us briefly comment on the emergence of the Ofunction in ( 2.9 ). As already mentioned, ( 2.9 ) is obtained from (6.1) by saturating (6.1) with harmonic oscillator wave functions. Technically this may be achieved by employing coherent state techniques. In this way (2.9) is represented by a quadratic integral over coherent state variables. It turns out that the associated quadratic form vanishes for certain linear combinations of coherent state variables, which precisely correspond to the zero modes on M, which satisfy the boundary condition (2.1). Treating these combinations as "collective coordinates", the 0-function in (2.9) factorizes. Let us finally outline how to obtain scattering amplitudes from the string vertex. As alreay pointed out in the introduction, scattering amplitudes of physical states are obtained from the full string vertex by saturating it with such states. In particular BRST invariant states are produced by acting with the product of the ghost coordinate c(z) and one standard vertex operator V(z) on a vacuum [0), which satisfies
c, lO>=O
n~>2,
b,[0> = 0
n>~-l,
(6.4)
and
(010)=+l.
(6.5)
Note that saturating legs with I0) alone, without any vertex operator insertion corresponds to filling in one of the boundary disks. Saturating all external legs with physical states gives a vanishing result. The reason is that the amplitude obtained is represented by a path integral on a Riemann surface without a boundary. To make such a path integral non-vanishing we have to insert enough b-fields to absorb the zero-modes. These b-field insertions allow for an elegant rewriting of the discrete determinant which makes the amplitude into a density on moduli space. Therefore before filling in the last boundary disk we have to act with the operator
--I~II Ld¢iJCd2z'lt (z)b=(z × )1 [c.c.],
I 3g--3+N[-i
M
(6.6) where the /z~--"are the Beltrami differentials associated with the/th mouli. This means that (6.6) should 285
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be considered as being part of the inner product between two states of the operator formalism Hilbert space, rather than the states themselves.
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[10IA. Neveu and P. West, Nucl. Phys. B278 (1986) 601; Commun. Math. Phys. 144 (1988) 613. [ 11 ] M.D. Freeman and P. West, Phys. Lett. B 205 (1988) 30. [ 12 ] P. West, Phys. Lett. B 205 ( 1988 ) 38. [13] A. Le Clair, M.E. Peskin and C.R. Preitschopf, preprint SLAC-PUB-4306/4307 (January 1988). [ 14 ] C. Vafa, Phys. Lett. B 190 ( 1987 ) 47; B 199 (1987) 195. [15]L. Alvarez-Gaum6, C. Gomez, G. Moore and C. Vafa, preprint CERN-TH-4883/87. [ 16] S. Sciuto, Lett. Nuovo Cimento 2 (1969) 411. [ 17 ] L. Caneschi, A. Schwimmer and G. Veneziano, Phys. Lett. B 30 (1969) 351. [18] C. Lovelace, Phys. Lett. B 32 (1970) 490. [ 19] Th. Ohrndorf, Nucl. Phys. B 301 (1988) 460. [20] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [21 ] S.B. Giddings and E. Martinec, Nucl. Phys. B 278 (1986) 91. [22] E. D'Hoker and P.H. Phong, Nucl. Phys. B 269 (1986) 205. [ 23 ] G. Moore and P. Nelson, Nucl. Phys. B 266 ( 1986 ) 58. [ 24 ] M. Schiffer and D.C. Spencer, Functionals of finite Riemann surfaces (Princeton U.P., Princeton, 1954). [ 25 ] Th. Ohrndorf, Nucl. Phys. B 310 ( 1988 ) 141.