Volume 185, number 1,2 PHYSICS LETTERS B 12 February 1987

SUPERCONFORMAL INVARIANCE AND DILATON COUPLING IN STRING THEORY

Gerardo ALDAZABAL, Faheem HUSSAIN 1,2 and Ruibin Z H A N G

International Centre for Theoretical Physics, 1-34100 Trieste, Italy

Received 8 September 1986

We consider the propagation of the supersymmetric string in background fields consisting of the metric tensor and the dilaton. Using the operator product expansion method we demonstrate that the equations of these fields are a consequence of the superconformal invariance of the string theory.

To understand the dynamics of strings it is useful to explore the propagation of strings coupled to background fields [1-13]. The consistency requirement which needs to be fulfilled is conforrnal invariance. In this context, the techniques of two-dimensional conformally invariant field theory are proving to be very powerful tools [14,15]. In the case of bosonic strings Banks et al. [11] and Jain et al. [10] have used conformally invariant field theory to check the consistency of the string moving in a general curved background space (including torsion and the dilaton) and have thus obtained the background field equations. Further, Banks et al. [11] have shown that the vanishing of the central charge, in the operator product expansion of the stress tensor, is equivalent to the nilpotence of the BRST charge of the BRST quantised two-dimensional field theory.

Although the operator product expansion technique has been used to study the superstring in a fiat background space [15], it has not previously been applied to the case of a superstring coupled to general background fields. In this letter, we extend the calculation of Banks et al. [11] and Jain [10] to the case of the closed superstring. We use the operator product expansion of the supercurrent for a two-dimensional superconformally invariant field theory [16] to check the consistency of the superstring coupled to a general background field, including the dilaton. We thus obtain the background field equations which have been calculated, previously; using a different technique [3,8]. Our calculations upto order a ' demonstrate that the vanishing of the anomalous terms in the operator product expansion is equivalent to the vanishing of the/3 functions. Just as in the bosonic case, it is important to demonstrate this equivalence. Details of the calculation will be presented in a later publication [17].

The reparametrizafion invariant action for the closed superstring can be written as

s= 1-3-8 f d2x d2OG,j(X; oX'DoXJ, (1)

in the covariant superconformal gauge. This action is the same as for a o-model with N = 1 supersymme- try. Here we are considering a superspace, z M= (x ~, 0~), of two commuting Minkowski coordinates x ~, and two anti-commuting Majorana spinor coordinates 0~. We follow the notation of Alvarez-Gaum6 et al. [18]. The supercovariant derivative is D~ = ~/3/)~- i(~0),~. Xi(zM), i = 1 . . . . . D, is a free superfield mapping the string into a D-dimensional spacetime manifold M. The function Gij can be identified as the

a On leave of absence from the Department of Physics, Quaid-i-Azam University, Islamabad, pakistan. 2 Address from 1 October 1986: Institiit fiir Physik, Postfach 3980, D-6500 Mainz, Fed. Rep. Germany.

0370-2693/87/$03.50 © Elsevier Science Publishers B.V. 89 (North-Holland Physics Publishing Division)

Volume 185, number 1,2 PHYSICS LETTERS B 12 February 1987

background spacetime graviton field in which the string is propagating. Following Fradldn and Tseytlin [2] we now couple the background dilaton field + in the spacetime M, with the following action:

1 = f d2x d20E-1R(2)+(X). (2)

Here E- 1 is the superdeterminant of the vielbein EM A. The superconformal transformations of the theory defined by (1) and (2) are generated by the

supercurrents

J+ 2(z+) = (1/2~ra')(3 +XiD2X:Gi/- a'3 +O2+), (3)

J-1 ( z - ) = (1/2 ~oz')(~) _XiD1XJG,j - e'O_D,+), (4)

where z + = (x+,02), z- = (x-,(1) and x -+ = x ° 4- x 1, ~ + = ½(O 0 _ ~1)- The improved supercurrents are evaluated by varying the action with respect to the vielbein EM A and then going to a conformally fiat gauge. The supercurrents J+2 and J -x generate two independent Virasoro algebras. From now on we only consider the algebra generated by J+2' The same considerations apply to J ~ , leading to (1,1) world sheet supersymmetry.

The requirement of superconformal invariance is that the supercurrent J+2 satisfy the superconformal algebra [16]:

iD 1 ×2 i×2 Dj+2 J+2(z + ) J + 2 ( w + ) = 8~r 2 ~r 777 + - ~ --8+J+2 + 2s ~ ' (5)

k

where w M= (y~,%). We define the coordinate displacements

s+= x+-z y+ + 2ig12#2, s -= x - + y - + 2iglt#l, X , = g ~ - ~ . (6)

The consistency condition we now impose is the requirement that the quantum string theory maintains superconformal invariance. We therefore evaluate the operator product expansion. (5) in perturbation theory. This is most conveniently carried out using the background field expansion method [18]. If ~i denotes normal coordinates in the internal manifold, the background field expansions of various relevant quantities are as follows:

~IxX ira- ~IxX~ "Jr V l , ~ i "Jr- 1 R i l m n ~ l ~ m ~ p . X ~ + 1 D j R i l m n ~ J ~ l ~ m ~ t x X ~

1 i 1 i p j k l m n + (gtOjOk R ,,,, - a~Rjkp R ,m,)~ ~ ~ ~ O~X~ + . . . , (7)

Dax=,i i l_ l,~i ¢:l¢:mn *2"" + 1DjRilmn~J~l~mD~X~

1 i - - 45* ' j kp ~" lrnn " ' '~ + ( ~ O j O k R l m n l~l~i 17,p ) , J , k , l ~ m D a X g q- ( 8 )

+( X) = +( XB) + Die( XB)~i + ½DiDje( X.)l~il~ j + ... . (9)

% ( x ) = c , j ( x . ) - ' R i k j l - - (1/3!)DlRimjk~l~m~ k

+ ( 1 / 5 ! ) ( - 6DkDzRimjn + ~-R~jzPR,,,i,,p)~k~l~m~n + . . . . (10)

where

VI,~ i= 3~*+ Fjk(Xu)~iO~X~, V~5/= D~,~' + Fj~(X,)~JD~X k, (11,12)

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Volume 185, number 1,2 . PHYSICS LETTERS B 12 February 1987

R and F are respectively the Riemann tensor and the Christoffel symbol constructed out of the metric Gij. X~ is the background field satisfying the classical equations of motion. From now on we will drop the subscript B. The terms given in eqs. (7)-(10) are sufficient for our purposes,

Using eqs. (7)-(10) we get

oL, e = (1/8~ra') { 6,jV,~iv,~ j + Rijki~J~kD,~XiD,~X ' + ½D~Rijkz~'n~J~kT)~,XiD,~X j

-k 4 Rijkl~Y~k~a~lo,xxi -I- ½ D mRijkl~m~J~k~a~loax i -k ½ Rijkl~J~k~a~iv~ l

+ @(DiDjRmkb, + 4RPijnRpklm )~i~j~k~l~)aXmDaXn -t- . . . } , (13)

J+2 = (1/2vraO{ GijD2Xi3 + Xj + Gij(D2X'v +~J + V2~ i3 + Xj)

q- Gi jv2~ iv + fJ -1- Rilmn~l~mD2 XiO + X n -t- 1Ri jk lV2~iv + ~l~j~k

q-2Rilmn(V2~i~l~mo+xn "q- V +~i~'~rnD2xn ) - - OL' [D/D/+- (D2Xiv +~J + V2~i3+X j)

+ D i ~ " V +V2~ i + ½(DiDj+ ) • V +V2(~i~J)] + . . . } . (14)

The superfield ~ i (x ) is not quite suitable as a quantum field in diagrammatic calculations because of the presence of Gu(X ) in the kinetic term. We remedy this problem by referring these riemannian vectors to tangent frames on the manifold [18]. We introduce a vielbein e l (X) and define ~ ( X ) = eT(X)~i(X). After this modification the kinetic term becomes V ~ V ~ ~ where

(15) Va~ a = Vez~ a + wabDagi~ b

and where w] b is the spin connection on the manifold. In eqs. (12) and (13) we may replace all the ~ 's by Ei~ ~ and V~ i by i a i a EaV ~ , where E~ is the inverse of e i . We then have the free propagator

GU,w) = f d'p e - ip(~-y) (2er)2 p 2 _ m z

(½DD(p) + rn)32(O ~). (16) =

As pointed out by Alvarez-Gaum6 et al. [18] it is necessary to introduce an infrared cutoff in order to define a meaningful perturbation theory. We have explicitly checked that the infrared divergences cancel to the order of our calculations.

In evaluating diagrams we use supergraph techniques. We first evaluate the contributions to the central charge to order a'. The graphs contributing to this term are given in fig. 1. As usual, we perform all the 0-integration using techniques for partial integrations of D's and ½DD factors until the supergraph is reduced to a multiloop bosonic integral [18]. As we are looking at leading divergences, we only need to consider spinor derivatives acting on the internal quantum lines and covariant derivatives can be replaced

(a) " - J R

(b') (c) (d~ (.e) ( f )

Fig. 1. D i a g r a m s con t r ibu t ing to the centra l charge, x denotes con t r ibu t ions f rom the non-d i l a ton par t of the supercurrent . ® denotes con t r ibu t ions f rom the di la ton. =X denotes b a c k g r o u n d field contr ibut ions . - - - / - - - denotes sp inor der ivat ives on in te rna l lines. Here and in fig. 2 we have no t l is ted the d iagrams re la ted by z ~ w. In (b), (d) and in fig. 2a we have to consider all poss ib le

combina t ions of the spinor der ivat ives on in te rna l lines.

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Volume 185, number 1,2 PHYSICS LETTERS B 12 February 1987

R t/k'~ - -~)¢ ~ ~ ~ - - - " " \ . . . . ~ ~ ~ _ / D~ X c9+ X ~/~'--- \ . Rt~.,

D~X Da X I ua XDaX "D.X

i j x i xj. Fig. 2. Diagrams contributing to D.~XiD1X j, 3+ XiD1X j, 3_ X D 2 X , 2+ 0_

by ordinary flat derivatives. At this stage, ultraviolet divergences are handled by dimensional regularisa- tion. Finally the sum of the graphs in fig. 1 is given by

(i/2~r 2s +3){ ¼D + a ! [ - D 2 ¢ + (D~) 2 + ¼R] }. (17)

The ¼D term cancels against the corresponding contribution from the ghost fields for D = 10. Then, a consistent superconformal invariant string theory requires that the rest of the terms in (17) vanish, leading to the equation

-DZqb + (DO) 2 + ¼R = 0. (lS)

Next we evaluate the contributions to terms involving D1XiD2X j. The graphs contributing to these terms are listed in fig. 2. The sum of these contributions is

- (XIX2/s + 3)D2XiD1XJ(2DiDjq5 + Rij ). (19)

All operators are evaluated at the symmetric point ½(z + w) in order to maintain the symmetry z <--> w. Superconformal invariance requires that terms involving D 1XiD2 X j do not appear in the operator product expansion. Thus we get the equation

2 D i D j O + Ri j = O. (20)

In.fact, the graphs of fig. 2 also lead to contributions tO terms involving O+x i O_x j, O+XiD1X j, 0 XiD2 XL Remarkably, all such terms vanish when we impose eq. (20).

Linear combinations of eqs. (18) and (20) give us the conventional equations of motion for the dilaton and graviton massless states of the string. These equations are equivalent to the vanishing of the fl functions,/3 ~ and/3i~. In the present work, we have not included the effect of torsion. We shall do so in a subsequent publication [17].

We wish to thank S. Randjbar-Daemi, Ashoke Sen and Spenta Wadia for useful discussions and, particularly; J. Helay~l-Neto for helping us to understand supergraph techniques. The authors would like to thank Professor Abdus Salam, the IAEA and UNESCO for hospitality at the ICTP.

References

[1] C. Lovelace, Phys. Lett. B 135 (1984) 75. [2] E.S. Fradkin and A,A. Tseytlin, Phys. Lett. B 158 (1985) 316; Nucl. Phys. B 261 (1985) 1. [3] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 160 (1985) 69. [4] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46. [5] D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett. 54 (1985) 620. [6] S. Jain, R. Shankar and S.R. Wadia, Phys. Rev. D 32 (1985) 2713. [7] A. Sen, Phys. Rev. D 32 (1985) 2102. [8] C.G. Callan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl. Phys. B 262 (1985) 593.

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[9] E. Bergshoeff, S. Randjbar-Daemi, A. Salam, H. Sarmadi and E. Sezgin, Nucl. Phys. B 269 (1986) 77. [10] S. Jain, G. Mandal and S.R. Wadia, Tara Institute preprint No. TIFR/TH/86-25. [11] T. Banks, D. Nemeschansky and A. Sen, SLAC Report No. SLAC-PUB-3885. [12] C.G. Callan and Z. Gan, Princeton University preprint. [13] M.T. Grisaru, A.E.M. van de Ven and D. Zanon, preprints HUTP-86/AD20, BRX-TH-196 (1986); HUTP-86

BRX-TH-198 (1986); HUTP-86/AD27, BRX-TH-199 (1986). [14] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [15] D. Friedan, E. Martinec and S. Shenker, Phys. Lett. B 160 (1985) 55; Nucl. Phys. B 271 (1986) 93. [16] H. Eichenherr, Phys. Lett. B 151 (1985) 26;

M.A. Bershadsky, V.G. Knizhnik and M.G. Teitelman, Phys. Lett. B 151 (1985) 31; D. Friedan, Z. Qni and S. Shertker, Phys. Lett. B 151 (1985) 37.

[17] G. Aldazabal, F. Hussain and R.B. Zhang, in preparation. [18] L. Alvarez-Gaum6, D.Z. Freedman and S. Mukhi, Ann. Phys. 134 (1981) 85.

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