String field theory in curved space

Volume 197, number 4 PHYSICS LETTERS B 5 November 1987

STRING FIELD THEORY IN CURVED SPACE

Keiji KIKKAWA, Masahiro MAENO and Shiro SAWADA Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan

Received 1 July 1987

The purely cubic action in the string field theory is shown to provide a set of equations of motion for background fields which agree to those obtained by the vanishing condition of fl-functions in the non-linear sigma model. Using the sigma model as an auxiliary tool, a systematic method for solving the string field theory in curved space is proposed.

The purely cubic action in the string field theory, which has been recently proposed by Hata, Itoh, Kugo, Kunitomo and Ogawa (HIKKO) [ 1 ] and also by Horowitz, Lykken, Rohm and Strominger [2], is an inter- esting model that provides the string field theory in curved space. The familiar form [ 3,4] of the string action can arise by expansion around some classical solution to the equation of motion of the former. In their papers they looked for general conditions that the solution should satisfy to generate the ordinary form of field theory. Except for some delicate regularization problems about operator products, they found that the crucial point is the existence of BRST operators associated with some background fields. As a special case they demonstrated that a solution constructed with the ordinary BRST charge for a fiat Minkowski space does generate the well- known string field theory.

In this paper we ask ourselves what sort of background fields would be allowed as solutions to the equation derived from the cubic action. We find a set of equations of motion for background fields which agree to those previously obtained in the context of the conformal invariant sigma model [ 5-7 ].

In the sigma-model approach the conditions over background fields were derived either by requiring the fl- functions for background field couplings being zero [ 6 ], or by requiring the nilpotency of BRST charge QB associated with the background field [ 7 ]. If our work is viewed from the standpoint of the sigma model, the derivation is different from those two and the result follows from a linear equation in QB.

In our arguments we adopt HIKKO's closed string model [ 1,3], since it is the only available covariant theory for the closed string. Although HIKKO's formalism contains unphysical width parameters and makes some troubles in loop amplitudes, the effect is factored out and no problems occur in tree amplitudes [ 3 ]. Since we are interested in a neighborhood around the string vertex on the world sheet, our results are not affected by the width parameter being involved in the formalism. However, a remark with respect to this point will be given later.

Let us first review the relevant results of HIKKO. The purely cubic action in the bosonic closed string theory is given by ~t

S= (2/3g 2) ~v. ( ~ , ~ ) , (1)

which is invariant under the local gauge transformation 8 ~v = 2 ~v ,A, where the ,-product is defined by using the three-string vertex operator I V) as follows:

J( t/-Jt * t/~2) [ 3] ) = | ( t/ll (1)1 (g"2(2) I I V(1, 2, 3 ) ) dl d2 . (2) d

~ The notations are the same as those used in ref. [ 1 ].

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As pointed out by Friedan [ 8 ], (1) is formally background geometry independent if the string field T transforms as a one-half-density under the transformation of string coordinate X u.

Suppose a classical solution 71o satisfies the equation of motion

~ o * ~ o = 0 , (3)

which is derived from (1) by variation with respect to ~ . I f one represents the full field by ~v = ~Uo+gq~ and substitutes it back into (1), the action reduces to the familiar form

S = ~ . Q ~ + ~g~.( ~ ,dp) , (4)

where Q is a linear operator defined by

~ o , ~ = ½ a ~ (5)

for an arbitrary field q~. I f the operator Q satisfies all the properties that the BRST operator does, i.e., the nilpotency

a~ = 0 (6)

and the distribution law

Q( cb, ~ ) = Q ~ , q~ + ( - )'~ ~ , Q q ~ , (7)

the action (4) defines a new string action for certain background fields whose information is supposed to be contained in Q.

HIKKO show that ~o is given by

~v o = - ½QF, (8)

provided that a field F obeys the equations ~2

F * ~ = (Nvr, + 1 - a l l a l - a OlOa)~ (9)

and

FvpF = - 2 F (10)

for arbitrary ~ , where Nvp is the Faddeev-Popov ghost number operator and a the width parameter of ~ . It is rather easy to show that (8) satisfies (3) and (5):

~ o * 71o = ¼ Q ( F • Q F ) = ~ Q ( N F p + 1 - a l l a l - a OlOa) QF oc Q2F = O ,

where the distribution law (7) and the nilpotency (6) have been used, and

~Uo,~ = - ½ [ Q ( 1 - ' , ~ ) - I - ' , Q 1 - ' ] = ½ [NFp, Q]~=½Q4~,

where [Nvp, Q] =Q and (7) have been used. The existence and the uniqueness of QF were shown by HIKKO. An explicit form of F given by them is

independent of background fields. Summing up the above arguments, one can conclude that, if any non-trival operator Q satisfying both the

nilpotency and the distribution law is found, ~Uo formed by (8) defines a string field theory associated with the geometry specified by Q.

In the following we first prepare a test operator Q[G~,,, Bu,, ~ ] as a function of the unspecified metric field

~: Eq. (9) is different from the one given by HIKKO in which a/la I is missing. One can prove that F obtained by HIKKO satisfies (9) and also the consistency condition (1 + a O/Oa)F = 0, which is needed if ~ = F in (9).

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G~,,, the antisymmetric tensor B~, and the dilaton field q~. Then, instead of solving eq. (3), we solve the nilpotency (6) and the distribution relation (7), which is more explicitly represented as

3

Z Q°)[G,B,~]IV(1,2,3)) =0, (11) r = l

where the superscript r refers to the channels 1, 2 and 3. The vertex operator I V) is given from the outset by the overlapping condition, hence (11 ) requires conditions over Q. The relation (11 ) means that the vertex should transform as a conformal tensor under the operation Q.

As far as the nilpotency condition (6) is concerned, Banks, Nemeschansky and Sen [ 7] studied its impli- cation in the context of the non-linear sigma model and obtained those equations for background fields that Callan, Friedan, Martinec and Perry [ 6 ] found.

Here we address ourselves to the condition (11 ), which has a straightforward connection with the cubic theory. In choosing the test operator Q we take advantage of the non-linear sigma model as an auxiliary tool:

S = ~ 1 4~za' f d2a [x/ggabGu"(X)OaX~' ObX" +iEahBuv(X)OaX~'ObXV +a'x/gR~2)q)(X)] ' (12)

where g,t, and R (2) represent the metric tensor and the curvature scalar of the two-dimensional world sheet. In what follows we assume the slope parameter a ' is small and make the normal coordinate expansion of X~'= x/' + ~ ~" around a constant classical solution x ~ (the center of mass). The metric in (12) is euclid- ean. The renormalized action which follows from (12), up to O ( a ' ) is given by

S= So + S,., + Sc, + Sgh , (13)

So= ½ J d ' a (6at 'Oa~MOb~M-[-m2~M~M) ,

+ f d'o" x~g~b[ - }go~'RMxNz.~K~LOa~MOb~N l

+ f d ' a

+ f d ' a

+f

x/g R (') [( 2 x / ~ / 4 r t ) V gq~ M + l a'V MV NCI~M~ N]

[ i l ~ SLmN~ab~LOaCMOb~ N exp (eO)l

d ' a [i½na'V KSLMN~ab~K~LOa~MObCN exp (e0)] ,

1 D Id.ax/~R(m+fd.axf~g~b[_(1/12e)oZ,RgNOa~MOb~U] So,- 2e 24~

+ f d'a . ~ [ +(1/12e)OdRMNm2~M~N]+ I d'a [+(1/4e)a'SMIjSN zg xfggaOOa~MOb~ N]

+ f d'a [i(1]8e)Ol'V xSKMNeabOa~MOb~N exp ( e ~ ) ] ,

sgh = I d2a ½ [ ~(~)0~ c (a ) + C(a)ooc(e)l,

where ~g=e~,g(x)~U with eu M being a vielbein, Su, p is the antisymmetric field strength of B~, and n =2 +2 e . Now our test operator Q for the BRST charge is defined by

Q(")[G,B,~] f daj(a) f daIC(a)(TX+++½T~+)+C(e)(r x_ + _ ~ h , = = 2 __j] , (14) o o

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o F i

J

CO

i p = r + i a

- ~ v = 0 + ~

Fig. 1. A half of the world-sheet diagram for the three closed- string vertex. The other half, a mirror image of the figure with respect to the bottom line, is implicit.

where C (C) stands for the (anti-)ghost field and Tab is the two-dimensional stress tensor defined by

T~,,( a) = (Z/x/g) 8S/~g~b( a) .

The superscripts X and gh of T in (14) represent the string coordinate and ghost parts of T, respectively. In the hamiltonian-operator formalism, the test operator Q should be written in terms of the normal coor-

dinate ~x(a) and its canonical momentum pN(O') -=~S/~N(0") together with ghost fields. Formula (1 1 ) implies that Q should be operated at r = 0 on each channel of the vertex shown in fig. I. In order to make the operation definite, we operate the charge density j(p = v + ia) along the contour Co in fig. 1, and then take the limit fi ~ 0. The effect of this operation can be equivalently estimated by the use of the lagrangian formalism ~3, i.e., by calculating

dpj(a) J V) : l i m J [ ~ X ~ C ~C' J j dpj(p)exp(--SD)I V ) , (15) 6 4 0

CO

where So stands for the action (12) defined over the strip domain D in fig. 1. Readers might wonder that the shift of the j-operation by r = + ~ with the background dependent action So

could not be legitimate in the cubic theory because all information about background fields should be contained i n j only. This is, however, allowed. As one sees in the following calculation, the corrections due to So appears only along the contour Co. The role of e x p ( - S o ) , therefore, can be considered to take care of background corrections to the canonical momenta, which are no more ~, as it should be.

The calculation of (15) is performed on the complex z-plane instead of the p-plane after making the Man- delstam mapping

3

p(r, a) = Z ar I n ( z - z ) . (16) r = [

It is important to note the role of the vertex operator I V). As is well known, ] V) is determined from the overlapping condition and is best defined in terms of oscillators with Neumann function coefficients. In par- ticular in the zeroth approximation where the space is flat, the following relation holds in the operator formalism:

2n

,.2.3 (0117 ~(Pi) lVo) = Z f i U(zr,, zr2), (17) i = 1 r = I

where J Vo) is J V) in which ghost parts are factored out, N(Zr,, Zr2) is the Neumann function on the z-plane, and the summation on the right-hand side covers all possible partitions of 2n indices into n pairs of (rl, r2).

The relation (I 7) seems to keep the conformal invariance under the transformation p-,z. If, however, some

~ The equivalence is known as Matthew's theorem. Nice proofs can be found in ref. [9].

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of the p coincide, the conformal factor dependence remains non-vanishing in the right-hand side when a certain regularization is made. (We have used the dimensional regularization.) In our calculation, j is a composite field with some derivatives, and it transforms no more as a conformal tensor under (16) and some conditions are required to recover the covariance. This is the essential feature in our calculation.

Let us look at a typical matrix element o f (15) when Bu~ is disregarded:

3

j,2.3(01 ~ a(r) lV(1,2, 3)> r - - I

- - I

- - I

- 2 ( C( z)C( z) ) O-_C(z)C( zo) + 2 ( C( z) C( zo) > C( z)OzC(z) ]

-~4x/~ [D-26-3a'(R+4VuV~,q)-4V~q)V~,¢)][(1/a)C"(zo)V(zo)-(b/a2)V'(zo)V(zo)] , (18)

where a and b are certain constants and

( TX+ + ( z ) ) = ( l / 24~r ) [D- 2 6 - 3oe'(R+4VuVuq) -4Vuq)Vucb)](OzO~¢- ½0~¢0~0),

(C( z )C(z ) > = - (1/4~) ½0:0, (O~C(z)C(z) > = (1 /4~)~(0~0~0- ½0.OOJp),

( C(z)C(zo) ) =(1/4n)(Z-Zo)- ' , ¢=lnldp/dz[ 2 (19)

In passing the first equality in (18) we transformed the variable p to z, hence the integration contour Co to C6 in fig. 2. It should be noted that the perturbation corrections due to S~,t occur only on the line Co in the p-plane or C; in the z-plane. There are potential sources o f non-local corrections, for instance, coming from the diagram shown in fig. 3a. In the amplitude, however, some of the propagators which link L~.t(w) with T ~ + (z) and ¢(u) collapse into delta functions due to derivatives on the propagator, leaving a local correction to T~)+ (z). A similar situation also happens in fig. 3b which occurs in the calculation of (21) below. Careful inspection o f the integrand shows that no singularities appear except at the image of interaction point Po, so that the integration can shrink to a small circle C; around Zo. Singularities at the image Zo of the vertex point appear f rom two sources, one from (dp /dz ) - 1 in (18) which comes from the jacobian and the transformation factor of j, and the other from the conformal factor in g~b=e°6ab in the z-plane (g,b=Oab in the p-plane) ~4 Collecting those singular terms and performing the Cauchy integral, we obtain the final equality in (18).

I f the BRST charge should eliminate the vertex operators as is required by (11 ), the last expression in (18) should vanish. Taking account o f the contributions from Bu, we have obtain eventually ~5

D - 2 6 - -32 a ' (R + 4V ~'V ~, ~ - 4V ~' ~ V u q~ - ~S~,~pS u"p) = 0 . (20)

From another matrix element of (1 1 ), namely

3

1.2.3(0[~,(1)~u(2) ~ Q(r)[v(1,2,3)> , (21) r = l

we have obtained

R,,, + 2V vV , ¢-S,~ooS, P° = 0 , (22)

~4 The ghost fields C and C' are those defined in ref. [ 10 ], where C(p) = C~°"e(z ) with C ¢°"f being the conformal field. To see the relation to the ordinary eonformal ghost fields, see ref. [ 11 ].

,5 Details of the calculation will be published in a separate paper.

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c;

Fig. 2. Image of the three-string vertex on the complex z-plane.

Sint=f dnw ~int{w)

1

b ,~c~

2 Fig. 3. Examples of diagrams which give rise to corrections to the BRST charge densityj.

VpS, ," - 2S,,pVPq~ = 0 . (23)

All matrix elements other than (18) and (21 ) are easily shown to be trivially zero. One can conclude, therefore, that the background fields Gu~ , Bu, and q) should obey eqs. (20), (22) and (23) up to the field redefinition X---.X+F(X) [6] . Ours agree to those obtained in refs. [6,7]. Banks et al. studied the nilpotency constraints in O ( a ' ) and found that those equations for G,v, Bu, and q~ are necessary. It is worthwhile to emphasize that the same equations have been deduced from a new requirement which is linear in Q ~6, and that the equations are sufficient to define a field theory o f the type (4).

The method developed above can be extended to higher orders in a ' . Using the sigma model as an auxiliary tool one constructs a test operator Q[ Gu,, B~,~, q~], then derives the equations for background fields to any order in a ' from the conditions o f the distribution law (11 ) and the nilpotency (6). Substituting a set o f solutions that obey the equations one determines a BRST operator

Q=Q¢O~ + a , Q ~ , ) + (o¢,)2Q~2) + .... (24)

The string action (4) associated with (24) then provides the S-matrix theory in the curved space. A practical method for small oe' is to develop a perturbation theory in (4) where q~Q~O)q~ is treated as an unperturbed term. Note that the theory is self-contained because it generates all conformal independent background fields as solutions of (3). The cubic theory, therefore, can be qualified as a background-independent field theory o f strings.

Finally, we remark on a renormalization correction to the background-field equations due to string-loop amplitudes. As was discussed by Fischler and Susskind [ 13 ] and many others [ 14 ], the divergences in ampli-

~6 The conformal covariance of the tachyon emission vertex was studied in ref. [ 12] and they obtained the same equations for background fields. In the context of the sigma model, ours can be regarded as a generalization of theirs to the three-string vertex.

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tudes having non-tr ivia l topology require new counter terms in (13) and the equat ions for background fields are to be modi f i ed thereby. In so far as we stick to H I K K O ' s formal i sm for the closed string, our method should not be appl ied s t raightforwardly to this problem, because the unphysical width pa ramete r seems to violate the modu la r invar iance in loop ampl i tudes . There may be some way out of this t rouble by modif ica t ions as proposed by Neveu and West [ 1 ] and Taylor [ 16 ].

To pract ice our program in the f ramework o f Wi t t en ' s theory is o f extreme interest. As has been s tudied by Horowitz and St rominger [ 17 ], however, special care must be pa id about the middle poin t o f the string when the operators QL and QR are mult ipl ied. Moreover , the closed string state seems not to be included in the phys- ical Hi lber t space which is annih i la ted by the BRST charge for the open string [ 18 ]. On these problems we hope to report elsewhere in the near future.

The authors thank Professor T. Kugo for informing us on the detai ls o f calculating F and also thank Dr. T. Kubota and Dr. R. Endo for useful discussions. One o f the authors (K.K. ) thanks Professor R. Schriefer, Pro- fessor J.H. Schwarz, and Professor M.B. Green for k ind hospi ta l i ty at the Inst i tute for Theoret ical Physics, Santa Barbara, where this work was in i t ia ted in the fall of 1986. This research was suppor ted in par t by the Nat ional Science Founda t ion under grant No. Phys. 82-1785 3, supp lemented by funds f rom the Nat iona l Aer- onautics and Space Admin i s t r a t ion in the USA, and also in par t by the Gran t - in -Aid No, 61540202 for Sci- entific Research f rom the Minis t ry of Educat ion, Science and Culture in Japan.

Note added. After this manuscr ip t was comple ted the authors received a prepr in t by St rominger [ 19 ], in which it was shown that, in the f ramework o f Wi t t en ' s formalism, each solut ion of the cubic theory represents every conformal invar iant s igma model .

References

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String field theory in curved space