27 July 2000

Ž .Physics Letters B 486 2000 172–178www.elsevier.nlrlocaternpe

Brane solutions in strings with broken supersymmetryand dilaton tadpoles

E. Dudas a, J. Mourad b

a LPT 1 , Bat. 210, UniÕ. de Paris-Sud, F-91405 Orsay, Franceˆb LPTM, Site NeuÕille III, UniÕ. de Cergy-Pontoise, NeuÕille sur Oise, F-95031 Cergy-Pontoise, France

Received 5 May 2000; accepted 13 June 2000Editor: L. Alvarez-Gaume

Abstract

The tachyon-free nonsupersymmetric string theories in ten dimensions have dilaton tadpoles which forbid a MinkowskiŽ . Ž . Ž .vacuum. We determine the maximally symmetric backgrounds for the USp 32 Type I string and the SO 16 =SO 16

heterotic string. The static solutions exhibit nine dimensional Poincare symmetry and have finite 9D Planck and Yang–Mills´constants. The low energy geometry is given by a ten dimensional manifold with two boundaries separated by a finitedistance which suggests a spontaneous compactification of the ten dimensional string theory. q 2000 Elsevier Science B.V.All rights reserved.

1. Introduction

Nonsupersymmetric string models are genericallyplagued by divergences which raise the question oftheir quantum consistency. In particular, ten dimen-sional nonsupersymmetric models have all dilatontadpoles and some of them have tachyons in thespectrum. It is however believed that some tadpolesdo not signal an internal inconsistency of the theory

w xbut merely a bakground redefinition 1 . In particu-lar, in orientifolds of Type II theories there should bea difference between tadpoles of Ramond–RamondŽ .RR closed fields and tadpoles of Neveu–Schwarz

Ž .Neveu–Schwarz NS–NS closed fields. While thefirst ones cannot be cured by a background redefini-tion and signal an internal inconsistency of the the-

Ž .E-mail address: dudas@qcd.th.u-psud.fr E. Dudas .1 Ž .Unite mixte de recherche du CNRS UMR 8627 .´

w xory asking therefore always to be cancelled 2 , thelatter ones could in principle be cured by a back-ground redefinition. The NS–NS tadpoles removeflat directions, generate potentials for the correspond-

Ž .ing fields for example dilaton in ten dimensionsand break supersymmetry. The difference betweenRR and NS–NS tadpoles play an important role inŽ .some orientifold models with broken supersymme-

w xtry recently constructed 3–6 .The purpose of this letter is to explicitly find the

background of the nonsupersymmetric tachyon-freestrings in 10D: the type I model in ten dimensionsw x3 , containing 32 D9 antibranes and 32 O9 planes,y

Ž . Ž . w xand the heterotic SO 16 =SO 16 7,8 . For the twotheories, there is no background with maximal

Ž .SO 10 Lorentz symmetry, a result to be expected byvarious considerations. We find explicitly the classi-cal backgrounds with a 9D Poincare symmetry. We´find an unique solution for the Type I model and two

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00734-6

( )E. Dudas, J. MouradrPhysics Letters B 486 2000 172–178 173

independent solutions for the heterotic one. A re-Žmarkable feature in the Type I and one of the two

.heterotic backgrounds of the static solutions is thatthe tenth coordinate is dynamically compactified inthe classical background. Furthemore, the effectivenine-dimensional Planck and Yang–Mills constantsare finite indicating that the low energy physics isnine-dimensional. Another classical background withmaximal symmetry is a cosmological-type solutionwhich we explicitly exhibit for the two theories.They both have big-bang type curvature singularities.

In Section 2 we briefly review the construction ofŽ .the type I USp 32 string. In Section 3, we determine

its classical background with maximal symmetry. InŽ . Ž .Section 4, we consider the SO 16 =SO 16 het-

erotic string and finally we end in Section 5 with adiscussion of the solutions.

( )2. The Type I nonsupersymmetric USp 32 string

The unique supersymmetric Type I model in tenŽ .dimensions is based on the gauge group SO 32 and

Žcontains, by using a modern language see, for ex-w x.ample, 9 32 D9 branes and 32 O9 planes. Thereq

is however another, nonsupersymmetric tachyon-freemodel, with the same closed string spectrum at

Žtree-level, containing 32 D9 branes i.e. branes of.positive tension and negative RR charge and 32

ŽO9 planes i.e. nondynamical objects with positivey.tension and positive RR charge .

w xThe open string partition functions are 3

`1 dt 12KKs V yS ,Ž .H 8 85 6 8X2 t h0 22 4p aŽ .

2`N 1 dt 1

AAs V yS ,Ž .H 8 85 6 8X22 t h08p aŽ .`N 1 dt 1

MMs V qS , 2.1Ž . Ž .H 8 85 6 8X22 t h08p aŽ .where a

X'My2 is the string tension ands

u 4 yu 4 u 4 yu 43 4 2 1

V s , S s , 2.2Ž .8 84 42h 2h

where u are Jacobi functions and h the DedekindiŽ w x. Ž .function see, for example, 9 . In 2.1 the various

modular functions are defined on the double cover-Žing torus of the corresponding Klein, annulus,

.Mobius surface of modular parameter¨

itts2 it Klein , ts annulus ,Ž . Ž .2 2

it 1ts q Mobius , 2.3Ž . Ž .

2 2

where tst q it is the modular parameter of the1 2Ž .torus amplitude and t is the one-loop open string

modulus.As usual, the annulus and Mobius amplitudes¨

have the dual interpretation of one-loop open stringamplitudes and tree-level closed string propagationwith the modulus l, related to the open string chan-nel moduli by

1 2 1KK: ls , AA: ls , MM : ls . 2.4Ž .

2t t 2 t2

From the closed string propagation viewpoint, V8Ždescribe the NS–NS sector more precisely, the dila-

.ton and S the RR sector, corresponding to an8

unphysical 10-form. The tadpole conditions can beŽ .derived from the t™0 l™` limit of the ampli-

tudes above and read

`1 1 2KKqAAqMMs dl Nq32Ž .�H5X22 08p aŽ .

2=1y Ny32 =1 q PPP .Ž . 4

2.5Ž .

It is therefore clear that we can set to zero the RRtappole by choosing Ns32, but we are forced tolive with a dilaton tadpole. The resulting open spec-

Žtrum is nonsupersymmetric the closed spectrum issupersymmetric and given by the Type I supergrav-

.ity and contains the vectors of the gauge groupŽ . ŽUSp 32 and a fermion in the antisymmetric reduci-

.ble representation. However, the spectrum is free ofgauge and gravitational anomalies and therefore themodel should be consistent. It is easy to realize fromŽ .2.1 that the model contains 32 D9 branes and 32O9 planes, such that the total RR charge is zero butyNS–NS tadpoles are present, signaling breaking ofsupersymmetry in the open sector. The effective

Ž .action, identified by writing the amplitudes 2.1 in

( )E. Dudas, J. MouradrPhysics Letters B 486 2000 172–178174

the tree-level closed channel, contains here thebosonic terms

8Ms 210 y2F'Ss d x yG e Rq4 EFŽ .H2

10 yF'yT d x Nq32 yG eŽ .H9

y Ny32 A q PPP , 2.6Ž . Ž .10

where T is the D9 brane tension and we set to zero9

the RR two-form and the gauge fields, which willŽ .play no role in our paper. Notice in 2.6 the peculiar

couplings of the dilaton and the 10-form to an-tibranes and O9 planes, in agreement with theygeneral properties displayed earlier. The RR tadpole

Ž .Ns32 is found in 2.6 simply as the classical fieldequation for the unphysical 10-form A .10

The difference between the supersymmetricŽ . Ž .SO 32 and nonsupersymmetric USp 32 model de-

scribed previously is in the Mobius amplitude de-¨Ž .scribing propagation between anti branes and orien-

tifold planes. Indeed, in the nonsupersymmetric caseŽthere is a sign change in the vector or NS–NS in the

.closed channel character V . Both supersymmetric8

and nonsupersymmetric possibilities are howeverconsistent with the particle interpretation and factor-ization of the amplitudes.

As noticed before, the NS–NS tadpoles generateŽ .scalar potentials for the corresponding closed-string

Ž .fields, in our case the 10D dilaton. The dilatonpotential read

V; Nq32 eyF , 2.7Ž . Ž .

Žand in the Einstein basis is proportional to Nq. Ž . Ž .32 exp 3Fr2 . It has therefore the usual runaway

behaviour towards zero string coupling, a featurewhich is of course true in any perturbative construc-tion. The dilaton tadpole means that the classicalbackground, around which we must consistentlyquantize the string, cannot be the ten dimensionalMinkowski vacuum and solutions with lower sym-metry must be searched for. Once the background iscorrectly identified, there is no NS–NS tadpole any-more, of course.

3. The classical background of the nonsupersym-( )metric USp 32 Type I string

We are searching for classical solutions of theŽ . Ž .effective lagrangian 2.6 of the model 2.1 in the

Einstein frame, which reads

1 2110 'S s d x yG Ry EFŽ .HE 222k3F

E 10 2'yT d x Nq32 yG eŽ .H9

y Ny32 A q PPP , 3.1Ž . Ž .10

where T E indicate that the tension here is in the9

Einstein frame. The maximal possible symmetry ofthe background of the model described in the previ-ous paragraph has a nine dimensional Poincare isom-´etry and is of the following form:

ds2 se2 AŽ y.h dx mdxn qe2 BŽ y.dy2 , FsF y ,Ž .mn

3.2Ž .

where m,ns0 PPP 8 and the antisymmetric tensorfield from the RR sector, the gauge fields and allfermion fields are set to zero. The Einstein and thedilaton field equations with this ansatz are given by

2 2X XX X X X136 A q8 A y8 A B q FŽ . Ž .4

sya e2 Bq3F r2 ,E

2 2X X1 2 Bq3F r236 A y F sya e ,Ž . Ž . E4

FXX q 9AX yBX

FX s3a e2 Bq3F r2 , 3.3Ž . Ž .E

Ž . 2 E 2 Ewhere we defined a s Nq32 k T s64k TE 9 9

and AX'dArdy, etc. The function B can be gauge-

fixed by using the reparametrisation invariance ofthe above equations. It is convenient to choose thecoordinate y where Bsy3Fr4 so that the expo-

Ž .nential factors in the Eqs. 3.3 disappear. In thisŽ .coordinate system, the second equation in 3.3 is

solved in terms of one function2 fX X1A s a sh f , F s2 a ch f . 3.4Ž .( (E E6

2 Ž .It can be checked that the other possible sign choices in 3.4lead to the same solution.

( )E. Dudas, J. MouradrPhysics Letters B 486 2000 172–178 175

The two other field equations become thenX4 2 fa f ch fqa e sya ,( E E E3

X 3 32 f2 a f sh fq a e s a , 3.5Ž .( E E E2 2

and the solution is then3yfe s a yqc , 3.6Ž .( E2

where c is a constant. By a choice of the y originand rescaling of the x m coordinates, the final solu-tion in the Einstein frame reads

3 22 < <Fs a y q ln a y qF ,(E E 04 3

21r92 ya y r8 m nE< <ds s a y e h dx dx(E E mn

2y1 y3F r2 y9a y r8 20 E< <q a y e e dy . 3.7Ž .( E

For physical purposes it is also useful to display thesolution in the string frame, related as usual by a

F

2Weyl rescaling G™e G to the Einstein frame1 1A sAq F , B sBq F . 3.8Ž .s s4 4

In the string frame the solution reads22r3F F 3a y r40 '< <g 'e se a y e ,s

24r92 F r2 a y r4 m n0'< <ds s a y e e h dx dxmn

2y2r3 yF y3a y r4 20'< <q a y e e dy , 3.9Ž .y8 Ž .where as64M T . The solution 3.9 displayss 9

two timelike singularities, one at the origin ys0and one at infinity ysq`, so that the range of they coordinate is 0-y-q`. The dilaton, on theother hand, vanishes at ys0 and diverges at ys

Ž .q`. The brane solution found above 3.9 has astriking feature. Suppose the y coordinate is non-compact 0-y-q`. In curved space however, thereal radius R is given by the integralc

1` ` du 2B yF r2 y y3u r802p R s dy e se a e ,2H Hc 1r3u0 0

3.10Ž .'where us a y. The result is finite, meaning that

despite apparencies the tenth coordinate is actuallycompact. The topology of the solution is thus a tendimensional manifold with two boundaries at ys0and ysq` that is R9 =S1rZ . Moreover, it can2

be argued that gravity and gauge fields of the D9branes are confined to the nine-dimensional noncom-

pact subspace, by computing the nine-dimensionalPlanck mass and gauge couplings, respectively

`7 8 7A qB y2Fs sM sM dy eHP s

0

1 ` du 28 y y3F r4 y3u r40sM a e e ,2 Hs 1r9u0

`16 5 A qB yFs ssM dy eHs2g 0YM

1 ` 26 y yF r4 1r9 yu r20sM a e du u e . 3.11Ž .2 Hs0

Both of them are finite, which indeed suggest thatgravity and gauge fields of the D9 branes are con-fined to the nine-dimensional subspace. The relationsŽ . Ž .3.10 and 3.11 are in sharp contrast with the usualflat space relations obtained by compactifying theten-dimensional theory down to nine-dimensions ona circle of radius Rc

17 y2F 8 yF 60 0M ;e R M , ;e R M . 3.12Ž .P c s c s2gYM

By using exactly the same method we find a cosmo-Ž .logical solution for the USp 32 nonsupersymmetric

Type I model by searching a homogeneous metric ofthe form

ds2 sye2 BŽ t .dt 2 qe2 AŽ t .d dx mdxn , FsF t ,Ž .mn

3.13Ž .

where t is a time coordinate. The solution is easilyŽ .found by following the steps which led to 3.7 and

Ž .3.9 . The result in the string frame is22r3F F y3a t r40 '< <g se se a t e ,s

2y2r32 yF 3a t r4 20'< <ds sy a t e e dt24r9 F r2 ya t r4 m n0'< <q a t e e d dx dx . 3.14Ž .mn

The metric has a spacelike curvature singularities atts0 and tsq`. The laps separating these twosingularities, to be interpreted as the real time param-eter,

2y1r3 yF r2 3a t r80'< <ts dt a t e e , 3.15Ž .His infinite.

( )E. Dudas, J. MouradrPhysics Letters B 486 2000 172–178176

( ) ( )4. Classical background of the SO 16 =SO 16heterotic string

In ten dimensions there is a unique tachyon-freew xnon-supersymmetric heterotic string model 7,8 . It

can be obtained from the two supersymmetric het-erotic strings as a Z orbifold. The resulting bosonic2

spectrum comprises the gravity multiplet: graviton,dilaton, antisymmetric tensor and gauge bosons of

Ž . Ž .the gauge group SO 16 =SO 16 . Compactifica-w xtions of this theory were considered in 10 , and its

strong coupling behavior in nine dimensions wasw x w xexamined in 11 . It has been shown 8 that the

Žcosmological constant the partition function on the.torus at one loop is finite and positive, furthermore

its approximate value is given by

2610LfM =5.67 . 4.1Ž .s 102pŽ .

The effective low energy action for the gravity mul-tiplet is the same as before except for the cosmologi-cal constant term, which now reads

'yL yG 4.2Ž .Hin the string metric. The absence of the dilaton inthis term reflects the one-loop nature of the cosmo-logical constant3. The same ansatz of the Einsteinmetric as in the previous paragraph leads, in theEinstein frame, to the equations

2 2X XX X X X136 A q8 A y8 A B q FŽ . Ž .4

syb e2 Bq5F r2 ,E

2 2X X1 2 Bq5F r236 A y F syb e ,Ž . Ž . E4

FXX q 9AX yBX

FX s5b e2 Bq5F r2 , 4.3Ž . Ž .E

where we defined b sL k 2. The gauge whichE E

eliminates the exponential factors is now BsŽ .y5Fr4. After solving the second equation in 4.3

3 A direct computation shows also that the one-loop dilatontadpole is non-zero and proportional to the cosmological constantL.

X XŽ . Ž .A s b sh h r6, F s2 b ch h , the remaining( (E E

equations give

b y y b y' 'E E1 e qeehe s , 4.4Ž .

b y y b y' 'E E2 e yee

where es"1. An important difference with respectto the type I solution is that here we have two

Žnon-equivalent that is, not related by coordinate.transformations solutions corresponding to es1 or

y1. Let us first consider the es1 case. The solu-tion in the Einstein frame reads

1< <FsF q ln sh b y q2ln ch b y ,( (ž /0 E E2

y11

3122 2< <ds s sh b y ch b y dx( (ž / ž /ž /E E E

y5y54y5F r2 20 < <qe sh b y ch b y dy( (ž / ž /ž /E E

4.5Ž .

and in the string frame

42F 2F 0 < <' 'e se sh b y ch b y ,Ž . Ž .Ž .21

332 F r2 20 < <' 'ds se sh b y ch b y dxŽ . Ž .Ž .y4y1y2F 20 < <' 'qe sh b y ch b y dy ,Ž . Ž .Ž .

4.6Ž .

where here bsLMy8. In both metrics, the solutions

has two timelike singularities at ys0 and ys`.These singularities are separated by a finite distancewhich in the string frame reads

q`y1r2 y1r2 y2yF 02p R s b e du sh u ch u .Ž . Ž . Ž .Hc0

4.7Ž .

The spacetime has therefore the topology of a ninedimensional Minkowski space times an interval. No-tice that the nine dimensional Planck mass

y1r27 8 y5F r40M sM b eŽ .p s

=` y1r3 y11r3du sh u ch u , 4.8Ž . Ž . Ž .H

0

( )E. Dudas, J. MouradrPhysics Letters B 486 2000 172–178 177

as well as the 9D Yang–Mills coupling

1 y1r26 y7F r40sM e bŽ .s2gYM

=` y2r3 y13r3du sh u ch u , 4.9Ž . Ž . Ž .H

0

are finite. This fact together with the finitude of thelength of the tenth coordinate indicate that the lowenergy processes are described by a 9D theory. Thecoupling constant vanishes at 0 and becomes infiniteat large y. We shall comment more on this fact inthe conclusion.

The second solution corresponding to esy1 canŽ Ž . < Ž <' 'be obtained by exchanging ch b y with sh b y

Ž . Ž .in the solutions 4.5 and 4.6 . This solution has twosingularities at 0 and `, however the nine-dimen-sional Planck and Yang–Mills constants as well asthe length of the tenth coordinate are infinite.

The cosmological solution invariant with respectto the nine dimensional Euclidian group can be alsoreadily found and it reads in the string frame

y1 y42 y2F 20 ' 'ds sye sin b t cos b t dtŽ . Ž .1r3 2r3

F r2 20 ' 'qe sin b t cos b t dx ,Ž . Ž .42F 2F 0 ' 'e se sin b t cos b t . 4.10Ž .Ž . Ž .

The variable t in these equations belongs to thew Ž .x'interval 0,pr 2 b . At the boundaries in ts0

Ž .'and tspr 2 b the metric develops curvature sin-gularities. The time separating these two singularitiesis infinite:

y1r2 y2Ž .pr 2 b' ' 'ts dt sin b t cos b t s`.Ž . Ž .H0

4.11Ž .Notice that the solution obtained by exchanging thesine and cosine in the above equations is not a newone since it can be obtained by shifting the time

Ž .'coordinate t™pr 2 b y t.

5. Discussion

Before discussing the solutions we have found weshould mention that there exists an another interest-ing non supersymmetric and tachyon-free model in

w xten dimensions 12 : it is an orientifold of the type

0B string with a projection that removes the tachyonand introduces an open sector with the gauge groupŽ . Ž .U 32 . This model has both a one loop positive

cosmological constant and a disk dilaton tadpole. AtŽ .the lowest disk level , by setting to zero the closed

Žstring RR fields a scalar, a two-from and a four-form.with self-dual field-strength , we find a solution

similar to the one found in Section 3. We do notknow however if this solution is unique in this case.

Ž .Similarly to the U 32 OB orientifold, the Type IŽ .USp 32 model get actually at one-loop a cosmologi-

cal constant, obtained by subtracting from the Mobius¨Ž .amplitude the massless tadpole contribution and

integrating only massive states. If we search there-fore classical solutions at one-loop, in both cases wehave to consider a sum of two terms in the lowenergy effective action

yF' 'yL yG e yL yG , 5.1Ž .H H1 2

in the string metric. The first term is of the type weŽ .encountered in the tree-level USp 32 case and the

Ž .second is analogous to the one we met in the SO 16Ž .=SO 16 case. There is no simple gauge choice for

B that renders the equations as simple as before.However qualitatively the solution should behave as

Ž .the tree-level USp 32 case for small y where thecoupling constant is small and the behavior for large

Ž . Ž .y should resemble that of the SO 16 =SO 16 case.In particular we expect the radius of the tenth dimen-

Žsion to be compactified since the divergence of theŽ . Ž .radius in the second solution of SO 16 =SO 16 is

.due to the behavior at the origin .We have determined the maximally symmetric

solutions to the low energy equations of twotachyon-free non-supersymmetric strings. To whatextent can we consider these solutions as represent-ing the vacuum of these string theories? Perturba-tively, there are two kind of string corrections to thelow energy effective action. The first ones are a

X

corrections involving the string oscillators and thesecond ones are g string loop corrections. A com-s

mon feature of the solutions we found is that the2 A Ž .conformal factor e in nine dimensions as well as

the string coupling vanish at the origin and divergeat ys`. The effective string scale at coordinate y

2Ž . 2 2 A s Xbeing given by M y sM e , we expect as s

( )E. Dudas, J. MouradrPhysics Letters B 486 2000 172–178178

corrections to be important at the origin and the loopcorrections to be dominant at infinity. So strictlyspeaking we cannot trust the classical solution nearthe two singularities where interesting string physicswould occur. Another less ambitious question, is theclassical stability of our solutions. That is, do smallperturbations around the background we found de-stroy the solution? The answer to this question isclosely related to the determination of the Kaluza–

w xKlein excitations 13 .Another common feature of the static solution of

Ž .the Type I model 3.9 and of the first heteroticŽ .solution 4.6 is that the effective Yang–Mills and

Planck constants are finite which means that thegravitational and gauge physics is effectively nine-dimensional. A remarkable feature of the static solu-tions in the low energy approximation is the sponta-neous compactification of one coordinate. Whetherthis feature will survive the string and loop correc-tions is an interesting and open question. The factthat the nine dimensional metric is flat in spite of theten-dimensional cosmological constant is in the spiritof the higher dimensional mechanisms which try to

Ž .explain the vanishing of the effective cosmologicalw xconstant 14 . As in the new approaches to this

w xproblem discussed in 14 , however, a better under-standing of the naked singularities present in oursolutions is needed in order to substantiate this claim.

A notable difference between the type I and het-erotic solutions is that the type I background isunique, whereas there are two classical heteroticbackgrounds with physically very different proper-ties. It is possible that this is due to the low energyapproximation and that this degeneracy will be liftedby string corrections.

All of the nontrivial features of the solutions aredue to the presence of dilaton tadpoles. The latter aregeneric for non-supersymmetric string models. Ac-cording to the Fischler–Susskind mechanism, thequantization around the classical solution should lead

w xto finite string amplitudes 1,9 . It would be interst-ing to confirm this explicitly for the present models.

Acknowledgements

We grateful to C. Angelantonj and A. Sagnotti forilluminating discussions on nonsupersymmetricstrings and to C. Grojean and S. Lavignac for discus-sions concerning naked singularities in relation with

w xthe proposals 14 . E.D. would like to thank theTheory Group at LBNL-Berkeley for warm hospital-ity during the final stage of this work.

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