colliding string waves and duality

18 October 2001

Physics Letters B 518 (2001) 306–314www.elsevier.com/locate/npe

Colliding string waves and duality

Ashok Dasa, J. Maharanab,c, A. Melikyana

a Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171, USAb Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany

c Institute of Physics, Bhubaneswar 751005, India

Received 27 July 2001; accepted 27 August 2001Editor: G.F. Giudice

Abstract

The collision of plane waves corresponding to massless states of closed string is considered inD-dimensional space–time.The reduced tree level effective action is known to be manifestlyO(d,d) invariant,d being the number of transverse spatialdimensions in the collision process. We adopt a coset space reformulation of the effective two-dimensional theory and discussthe relation of this process with classical integrable systems in two dimensions in the presence of gravity. We show how it ispossible to generate new backgrounds for the scattering process, from known background solutions to the equations of motion,in the coset reformulation. We present explicit calculations for the case of four space–time dimensions as an illustrative example. 2001 Elsevier Science B.V. All rights reserved.

Recently, there have been attempts to study collisionof plane waves in the frame work of string theory [1,2].A scenario is envisaged where infinite plane-frontedwaves undergo a head on collision. One of the goals ofthe investigation of this problem is to gain insight intothe mechanism for the creation of the Universe in thepre-big-bang (PBB) hypothesis [3,4]. In this scenario,it is proposed that the Universe, in its infancy, is in theweak coupling regime with low curvature, where wecan trust the low energy string effective action. In thisphase, we can admit the Minkowski vacuum order byorder in the perturbative frame work. The Universe, inthis early epoch, can be described as a superpositionof massless waves, travelling in all directions, whoseenergies lie below the string scale. These waves collidein the process of propagation, constructively interfereand eventually gravitational collapse takes place, when

E-mail address: [email protected] (J. Maharana).

over-dense regions are formed. It has been argued byBuonanno, Damour and Veneziano [5] that the interiorof an adequately large and collapsing region may givebirth to a Universe which possesses attributes of ourown Universe. In other words, collisions of planewaves could result in nontrivial geometries in theinteraction region of the waves, leading to the creationof a Universe as proposed in the PBB scenario. Thesingularity, arising from the interaction process of theplane waves, finds an interpretation as a cosmologicalsingularity from the perspective of the PBB proposal.In the study of the scattering process, when one solvesthe underlying equations in the interaction region, theresulting metric assumes Kasner’s form. Moreover,the exponents fulfill the requirements of the PBBhypothesis in the space of admissible parameters.Furthermore, it has been argued that initial statesof the PBB cosmology could be thought of as theplane waves associated with the massless spectrumof strings [6]. It is important to note that Kasner

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01067-X

A. Das et al. / Physics Letters B 518 (2001) 306–314 307

solutions have attracted considerable attention fromvarious perspectives [7,8].

It is ideal to consider configurations of plane wavesof infinite front undergoing collisions and several in-teresting results have already emerged in recent stud-ies [1,2]. In the work of FKV [1], the collision processwas studied in four-dimensional (D = 4) space–timefor graviton and dilatonic waves. Subsequently, Bozzaand Veneziano [2] examined the case of arbitraryD-dimensional space–time (withD−2 = d transversedimensions) and incorporated the presence of the an-tisymmetric tensor background through implementa-tion of theO(d,d) symmetry, since the effective two-dimensional tree level action is invariant under this tar-get space symmetry. We mention in passing that colli-sion of plane waves has been studied in detail in gen-eral theory of relativity [9]; however, interest, in this,has been revived since there is an interesting connec-tion between this process and the issue of the cre-ation of the Universe in the PBB scenario. Since theproblem under consideration is described by a two-dimensional effective action, it is natural to expect anintimate connection with integrable systems [10]. In-deed, there has been a considerable amount of workin investigating a class of two-dimensional field theo-ries in curved space–time and super-gravity theories tostudy their integrability properties [11].

It is expected that the techniques adopted in thestudy of integrable systems will unravel other fasci-nating features of the problem under consideration.In the past, the symmetries of two-dimensional stringeffective action have been investigated in some de-tail by several authors [12–15]. Moreover, if we adoptthe string frame metric in expressing the reduced ac-tion, then the manifestO(d,d) symmetry can be ex-ploited to generate new background configurationsfrom known solutions of the equations of motion.

The purpose of this note is two fold. First, we shallargue how the effective action, studied from a differ-ent perspective, can be shown to have a direct connec-tion with integrable systems. This is achieved ratherelegantly by adopting the coset space reformulation ofthe action as was derived by Schwarz and one of theauthors [16]. Thus the action can be expressed in acurrent–current form involving the scalar fields (mod-uli). Our second result is the special case of collisionof waves in four space–time dimensions (i.e., trans-verse dimensionsd = 2). In this case, we start with

a background where the antisymmetric tensor field isvanishing along transverse directions while the metricand the dilaton are nonzero. We generate a nontrivialB-field along the transverse directions by exploitinga property of the coset reformulation, so that the newbackground configuration can be extracted rather eas-ily. This technique is applicable to the case with arbi-trary d transverse dimensions and is a property of thereformulation alluded to above.

TheD-dimensional tree level string effective actionis

SD =∫dDx

√−G e−φ

(1)×(RG + (∂φ)2 − 1

12HµνρH

µνρ

).

Here Gµν is theD-dimensional metric in the stringframe,G is its determinant andRG is the curvaturescalar derived from this metric. Note thatφ is the dila-ton andHµνρ = ∂µBνρ + ∂ρBµν + ∂νBρµ is the fieldstrength associated with the antisymmetric fieldBµν .When one studies the problem of head on collision oftwo plane-front waves of infinite extension, it is pos-sible to define a coordinate system where the wavestravel alongx1 direction. If we assume that there istranslational symmetry in the transverse directions,then there will be Abelian isometries along these,D − 2 = d , directions. Thus the process is describedby a two-dimensional reduced effective effective ac-tion, which can be derived following the prescriptionof Ref. [16]

S =∫dx0dx1√−g e−φ

(2)×(R + (∂φ)2 + 1

8Tr

(∂αM

−1∂αM)).

In general, there will be additional terms in thereduced action corresponding tod Abelian gaugefields from the original metric and another set ofdgauge fields from the antisymmetric tensorBµν asa result of dimensional reduction [16]. Furthermore,there would have been the field strength of the two-dimensional tensor fieldBαβ whereα,β = 0,1 arespace–time indices. Since we are effectively in twospace–time dimensions, we have dropped the gaugefield terms and, in the same spirit, have not kept fieldstrengths ofBαβ , which can be removed if it dependsonly on coordinatesx0 and x1. The other terms

308 A. Das et al. / Physics Letters B 518 (2001) 306–314

appearing in the above action are defined as follows:gαβ with α,β = 0,1 is the two-dimensional space–time metric, andR is the corresponding curvaturescalar. The shifted dilaton isφ = φ − 1

2 logdetG,with G the metric corresponding to the transversecoordinatesxi , i = 2,3, . . . ,D − 1. M is a 2d × 2dsymmetric matrix

(3)M =(G−1 −G−1B

BG−1 G−BG−1B

),

where B represents the moduli coming from thereduction of theB-field inD space–time dimensions.The symmetric nature ofM is evident sinceG issymmetric andB is antisymmetric.

The reduced effective action is invariant underglobalO(d,d) transformations

(4)gαβ → gαβ, φ→ φ,

(5)M →ΩTMΩ,

whereΩ ∈O(d,d) is the transformation matrix. TheO(d,d) metric

(6)η=(

0 11 0

)remains invariant under the non-compact global trans-formations,1 being thed-dimensional unit matrix.Furthermore, it is easy to check that the matrixM be-longs toO(d,d).

The relevant field equations are

R + 2gαβDαDβφ − gαβ∂αφ∂βφ(7)+ 1

8gαβ Tr

(∂αM

−1∂βM) = 0,

(8)Rαβ +DαDβφ + 1

8Tr

(∂αM

−1∂βM) = 0,

which result from the variation of the effective actionwith respect to the shifted dilaton and the metricgαβ ,respectively. The equation

(9)DαDαe−φ = 0

follows from the above two equations in a straightfor-ward manner. The equation of motion forMij is

(10)∂α(e−φ

√−g gαβM−1∂βM) = 0

which is obtained by a constrained variation of the ef-fective action, taking account of the fact thatM ∈ O(d,d). Eqs. (7)–(10) can be simplified further

in the light cone coordinates, if we recall that the two-dimensional metric can be written as the flat metrictimes a conformal factor, i.e.,gαβ = eFηαβ . In thiscase, the relevant equations take a rather simple formas has been shown in [2]

(11)∂u∂ve−φ = 0,

which is also equivalent to∂u∂vφ = ∂uφ∂vφ, and

(12)∂u(e−φM−1∂vM

) + ∂v(e−φM−1∂uM

) = 0,

∂2uφ − ∂uF∂uφ + 1

8Tr

(∂uM

−1∂uM) = 0,

(13)another equation withu→ v,

(14)∂u∂vφ − ∂u∂vF + 1

8Tr

(∂uM

−1∂vM) = 0,

whereu= 1√2(x0 − x1) andv = 1√

2(x0 + x1).

Let us recapitulate very briefly how solutions areobtained for the scattering of the plane waves. Thewaves collide atu= 0 andv = 0 which corresponds tox0 = x1 = 0. The fronts of the two waves are definedto be atu = 0 and v = 0, respectively. In order todescribe the collision process, it is convenient to dividespace–time into four regions. First, one considers aregion which is in front of the plane waves, calledregion I such thatu,v < 0 and this is before thecollision occurs. The metric is Minkowskian here andthe string coupling is small, so that we are in theperturbative regime. Here, for the dilaton we haveφ = φ0, with eφ0 1 and we can fixφ = 0; and takeB = 0. The line element is

(15)ds2I = −2dudv+ dxi dxiin this region and we adopt the convention of summa-tion over repeated indices throughout. Before the col-lision occurs, a wave is coming from the left, calledregion II, whereu > 0, v < 0 and the backgroundsGIIij (u), B

IIij (u) andφII (u) are onlyu-dependent and

the superscript specifies that they are in region II. Thefield equations enable us to choose the conformal fac-tor, F = 0 in region II and the line element has thefollowing form

(16)ds2II = −2dudv+GIIij (u) dx

i dxj .

The other wave front is coming from the right, i.e.,v > 0 andu < 0, which is denoted by III. Here the


backgrounds depend only onv and they are designatedwith the superscript III. We shall not display the lineelement explicitly; it has the same form as that for II,except thatGII

ij (u) is replaced byGIIIij (v). Finally,

we have region IV, where interaction takes place andthe conformal factor,F(u, v), makes its appearancein equations and the backgroundsGIV

ij ,BIVij andφIV

depend on bothu andv so that the line element takesthe form

(17)ds2IV = −2eF(u,v) dudv+GIVij (u, v) dx

i dxj .

The solutions to background equations of motion forthis system in the string theory have been discussed in[1,2], especially the solution for the diagonal form ofG has been dealt with in detail. Our goal, in the firststep, is to show the connection between this problemand integrable systems [10]. In order to achieve this,first we have to cast the action in a different form.

We may recall that the reduced effective action ad-mits a coset space reformulation as was demonstratedin Ref. [16]. One introduces a triangular matrix formforM, such thatM = VV T and

(18)V =(E−1 0BE−1 ET

),

whereB is the antisymmetric tensorBij andE is avielbein such that(ET E)ij = Gij . Also it is easy toshow thatV , V T ∈ O(d,d). It was shown that onecan introduce an arbitrary matrixV which belongsto O(d,d) (not the special form given above) andwrite an action which is invariant under globalO(d,d)transformation as well as under local gauge transfor-mations of its maximal compact subgroupO(d) ×O(d). Then, one can make a gauge choice to obtainthe form of the reduced action that appears in standard-dimensional reduction with manifestO(d,d) symme-try. Of course, for the problem at hand, we shall adoptthe results of [16] accordingly. First, we introducea transformation to go over to the diagonal form ofO(d,d) metric,

σ =(

1 00 −1

),

1 being thed × d unit matrix, through the relation

(19)LT ηL= σ, L= 1√2

(1 −11 1

).

TheM and theV matrices transform as follows:

(20)M → M = LTML, V → V = LT VL.Let us next define a current

(21)V−1∂αV = Pα +Qα,where V −1∂αV ∈ O(d,d) andQα ∈ O(d) × O(d),the maximal compact subgroup. It is straightforwardto show that

(22)Tr(∂αM

−1∂βM) = −4 TrPαPβ.

It follows from the symmetric space automorphismproperty of the coset

O(d,d)

O(d)×O(d)thatPTα = Pα , which belongs to the complement andQTα = −Qα asQα ∈ O(d) × O(d). Thus, we canwrite

Pα = 1

2

[V −1∂αV + (

V−1∂αV)T ]

and then (22) follows in a straightforward manner. Wemention another important point in passing, namely,that, under anO(d) × O(d) gauge transformationV → V h(x), h(x) ∈ O(d) × O(d), Pα and Qαtransform as follows

Pα → h(x)−1Pαh(x),

(23)Qα → h−1(x)Qαh(x)+ h−1∂αh(x)

and this transformation leaves (22) invariant. Noticethe form of (22): in flat space–time, this describes aσ -model in two dimensions. Furthermore, it is possi-ble to introduce a constant spectral parameter and azero curvature condition by taking a suitable combi-nation ofPα andQα to show the integrability proper-ties of the classical theory following well known pro-cedures [10]. For the case at hand, the full action con-sists of kinetic energy terms for the conformal factorF and the shifted dilatonφ in addition to the piecegiven by (22). One can scale the metricgαβ → eφgαβto get rid of the kinetic term ofφ in action (2). Then,for the modified action, one can follow the proceduredescribed in [11] to establish relation of this processwith classical integrable systems. Let us consider thespecial case, as an illustrative example, when the met-ricGij is diagonal as was the case in [1,2] andBij = 0.


The correspondingV -matrix is symmetric and so isV .Thus it follows from (21) thatQα = 0 and one needsto suitably reformulate integrability conditions for thiscase. When one studies integrability properties of two-dimensional models in curved space–time, the spectralparameter assumes space–time dependence. Further-more,V (x0, x1)→ V (x0, x1, t) such that

V−1∂vV = 1− t1+ t Pv,

(24)V −1∂uV = 1+ t1− t Pu.

The equations of motion and integrability condition

∂α(V −1∂βV

) − ∂β(V−1∂αV

)(25)+ [

V−1∂αV , V−1∂β V

] = 0

are compatible provided the spectral parameter satis-fies

(26)∂αt = −1

2εαβ∂

β

(e−φ

(t + 1

t

)).

The solution to the above first order equation is

(27)1− t1+ t =

√w− e−φ(u)w+ e−φ(v) ,

where w is an integration constant. We shall seethat, in region IV, e−φ is expressed as a sum oftwo solutions, one depending onu and the otheron v. The next step is to solve for the monodromymatrix. It is well known that the linear system, definedthrough (24), is invariant under the generalization ofsymmetric space automorphism. One defines

(28)τ∞V (t)= (V T

)−1(

1

t

).

In terms of Lie algebra elements, this impliesPα →−Pα , t → 1/t . Furthermore, the currentV −1∂αV isinvariant underτ∞ transformation. The monodromymatrix

(29)M = V τ∞(V )−1 = V (x, t)V T(x,

1

t

)plays a fundamental role in the reconstruction ofsolutions as well as in the study of integrabilityproperties from this point of view. We shall presentthe explicit form ofM for the four-dimensional case,when we discuss solutions.

Another aspect of this reformulation is that onecan generate new background configurations fromknown solutions as follows. First solve for equationsof motion for a given background. Then implement anO(d,d) transformation onV . In general, the resultingV obtained from transformedV will not maintain itstriangular form, which is quite essential to get thedesired form of theM-matrix. At this stage, one canfurther make a transformationh ∈ O(d) × O(d) tobring the transformedV , call it V ′, to a triangularform.

Let us consider, to be specific, the scattering ofplane fronted waves inD = 4 for which the dualitysymmetry isO(2,2). ThusGij and Bij are 2× 2matrices. If we choose a background configurationwhereGij is diagonal andBij = 0, then

(30)V =(E−1 0

0 ET

).

Here,E is the corresponding zweibein with a choice

E =(E1 00 E2

)and thatG = ETE. We recall that under a globalO(2,2) transformation,V →ΩT V withΩ ∈O(2,2).Let us choose

(31)Ω =(a1 bε

−cε d1

),

where1 is unit 2×2 matrix andε is the antisymmetric2× 2 matrix. Note thata, b, c andd are real numberssatisfyingad − bc = 1, in order that the 4× 4 matrixΩ ∈O(2,2). Under such a transformation,

(32)V →ΩT V =(aE−1 cεET

−bεE−1 dET

).

Thus, as mentioned, the transformedV is not in thedesired triangular form, although we can read off thenew transformed backgrounds to be

(33)E′ = E

a, B ′ = −b

cε.

Note that the resultingB ′ is just a constant.However, let us recall that we also have the free-

dom of making a localO(2) × O(2) transformationunder which,V → V h(x), h(x) ∈ O(2)×O(2). Let


us choose the 4× 4 matrixh to be space–time inde-pendent and of the form

(34)h=(γ 1 δε

δε γ 1

).

It is easy to check that,hhT = hT h = 1 provided,γ 2 + δ2 = 1. TheO(2) × O(2) nature of the matrixbecomes more transparent, if we go to the diagonalbasis for the metric ofO(2,2), where we see thath → LT hL = diag(γ 1 + δε, γ 1 − δε) and the off-diagonal elements are zero. AfterV gets transformedunder bothΩ andh, we finally get

V →ΩT Vh

(35)

=(aγE−1 + cδET ε aδE−1ε + cγ εET

−bγ εE−1 + dδET ε −bδεE−1ε + dγET).

If this transformedV is to be of triangular form, thenits “12” component is to be set to zero,aδE−1ε +cγ εET = 0 which determines,γ andδ to be

γ = a√a2 + c2E2

1E22

,

(36)δ = − cE1E2√a2 + c2E2

1E22

,

and they automatically satisfy the requirementγ 2+δ2 = 1. The new backgrounds, denoted by primes(not to be confused with derivatives) are

(37)G′ = (a2G−1 − c2εGε

)−1,

(38)B ′ = −ab+ cdE21E

22

a2 + c2E21E

22

ε,

and we note that the new moduliB ′ij are space–time-

dependent. Furthermore,G′ij depends onG andG−1

and is diagonal. For this choice ofΩ andh we havegenerated nontrivial backgrounds.

Let us now discuss some of the essential proper-ties of the solutions in region IV, since this is the re-gion where collision occurs. When we implement anO(d,d) transformation, the shifted dilatonφ remainsinvariant. The solutions to the equation of motion, fordiagonal metric andB = 0, has been discussed in[1,2], where they appear in form of integrals. Theseresults are generalization of the case of pure gravityconsidered by Szekeres [17] and subsequently by Yurt-sever [18]. The asymptotic behavior of the solutions

are of interest, since one of the aims is to examinewhether a singularity appears in the backgrounds. Letus express [1] the diagonal elementsGij , i, j = 2,3,corresponding to directionsx2 andx3, in the follow-ing formG22 = exp(λ + ψ) andG33 = exp(λ − ψ).Thusλ= 1

2 logdetG. The dilaton, in region IV, can bewritten as

(39)e−φ(u,v) = e−φII (u) + e−φIII (v) − 1.

That it is a sum of a function ofu and ofv in region IV,follows from the equation of motion forφ as wellas the fact that the solutions should match smoothlyacross the boundaries of the different regions. It isconvenient to introduce a new set of variables

r = 2 exp(−φIII (v)

) − 1,

(40)s = 2 exp(−φII (u)

) − 1.

Thusr and s are functions of the variablesv andu,respectively. Let us define a set of coordinates

ξ = (r + s)2

= exp(−φ(u, v)),

(41)z= (r − s)2

= e−φII (u) − e−φIII (v).

Obviously,φ(u, v) is defined only in region IV. Fromnow on, we are going to concentrate on the back-grounds only in region IV and therefore we suppressthe subscript IV from all of them. Under the change ofvariables, the conformal conformal factor appearing inthe space–time metric gets affected. The line elementin region IV takes the form (subscript not displayed)

(42)

ds2 = −ef dξ2 + ef dz2 +G22(dx2)2 +G33

(dx3)2

,

wheref = f (r, s) is the conformal factor and

(43)φ = −1

2log(r + s)= − logξ.

The asymptotic behavior of the functionsλ,ψ , appear-ing in Gij , the conformal factor,f and dilatonφ aregiven by

(44)λ∼ κ(z) logξ, ψ ∼A(z) logξ,

(45)f ∼ B(z) logξ, φ ∼ κ(z) logξ − 1,


where

κ(z)= 1

π√

1+ z

×1∫z

ds[(1+ s)1/2λ(1, s)],s ( s + 1

s − z)1/2

+ 1

π√

1− z

(46)×1∫

−zdr

[(1+ r)1/2λ(r,1)],r ( r + 1

r + z)1/2

,

A(z)= 1

π√

1+ z

×1∫z

ds[(1+ s)1/2ψ(1, s)],s ( s + 1

s − z)1/2

+ 1

π√

1− z

(47)×1∫

−zdr

[(1+ r)1/2ψ(r,1)],r ( r + 1

r + z)1/2

,

(48)B(z)= 1

2

(A(z)2 + κ(z)2) − 1.

We can now read off the line element in the asymptoticlimit of the metric

ds2 = −ξB(z) dξ2 + ξB(z) dz2(49)+ ξκ(z)+A(z)(dx2)2 + ξκ(z)−A(z)(dx3)2

.

If we define the cosmic timet = ξ(B(z)+2)/2, then theKasner exponents can be read off directly to be [1,2]

p1 = B

B + 2, p2 = κ +A

B + 2,

(50)p3 = κ −AB + 2

.

The exponents satisfy the constraints,

p21 + p2

2 + p23 = 1,

(51)φ = (p1 + p2 + p3 − 1) logξ.

There is a region in the parameter space such that allthe exponents are allowed to take negative values and

at the same time they can satisfy the constraints as hasbeen discussed for the four-dimensional case [1]. Forsuch a case, there is a curvature singularity, and thissingularity has the characteristics of a cosmologicalsingularity, which is very interesting in the PBBscenario.

Now we would like to examine, the singularitystructure for the new backgrounds generated throughspecificO(d,d) transformations introduced earlier.Note that the shifted dilaton,φ remains invariantunderO(d,d) transformation. The two elements ofthe vielbein,E are given by

E1 = exp

[1

2(λ+ψ)

]= ξ 1

2 (κ+A),

(52)E2 = exp

[1

2(λ−ψ)

]= ξ 1

2 (κ−A).

Now the transformed metric becomes

G′22 = E1

(a2 + c2E1E2),

(53)G′33 = E2

(a2 + c2E1E2).

The explicit forms of the elements,E1 andE2 aregiven above. Thus, near the singularity, whenE1,E2go to large values, the elements of the transformedmetric remain finite. Similarly, whenE1,E2 tend tozero, the structure ofGij tells us that its matrixelements also tend to zero asE1 andE2, respectively.Of course, the diagonal structure is maintained. Nowturning to the dilaton in the transformed theory,

(54)φ′ = φ − 1

2log

(detG

detG′

)= φ − log

(a2 + c2eλ

).

Note, however, that in the two extreme limits ofE1,E2either tending to zero or to infinity,Bij tends to aconstant as is evident from (38).

Let us proceed to derive the monodromy matrixfor the aboveD = 4 case which in turn will provideconnections with integrable system. The relation (24)suggests thatV is a 4× 4 matrix for our case.Moreover, the requirement of factorization of themonodromy matrix, alluded to earlier, implies thatV (t) will have a simple pole structure in the spectralparametert . In turn, it ensures thatM(w) also hasonly simple poles. Now one can suitably modify thesteps indicated in [11] to construct theV matrixand eventually the monodromy matrix, which has


the form

(55)M(w)=

w1−ww1+w 0 0 0

0 w3−ww3+w 0 0

0 0 w1+ww1−w 0

0 0 0 w3+ww3−w

.This form of the monodromy matrix satisfies all theexpected properties. Note from (39) that the shifteddilaton, in region IV, can be written as sum of twofunctions: one depends onu and other onv, whichis similar to the decomposition employed in [11].As a consistency check, we have considered thecase whenψ = 0, that appears in metricGij andone could recover the known results as discussedin [11]. Note that our discussion can be generalizedto the case whenG is a d × d matrix as wasconsidered in [2]; however, the number of polesfor the (2d × 2d)-dimensional V matrix will bemore.

To summarize, in this note, we have presented acoset space reformulation of the string effective ac-tion to study the collision of plane-fronted wavescorresponding to massless states of closed string.In this process, we have expressed the action in aform which makes it suitable to study the integra-bility properties of the theory. We have shown thatthe techniques employed in the study of integrablesystems in the presence of gravity can be general-ized to the case of plane wave scatterings in stringtheory. The monodromy matrix is derived explicitlyfor the D = 4 case. We have presented an illus-trative example, in the case of four space–time di-mensions, of how one can generate new backgroundconfigurations from known solutions in the collisionprocess. We have exploited the properties of the cosetspace reformulation to generate a space–time depen-dent B-field keeping the transformed metric diago-nal. We have only used a special type ofO(2,2)and O(2) × O(2) transformation to generate newbackgrounds. One can employ more general form,such as space–time dependentO(2) × O(2) func-tion h(x), for generating a wide variety of backgroundconfigurations. The approach presented here can beapplied to more general situations while consider-ing plane wave scatterings in the context of stringtheory.

Acknowledgements

One of us (J.M.) would like to thank G. Venezianofor discussions on their work [2]. We would liketo thank H. Nicolai for discussions and suggestions.One of us (J.M.) acknowledges gracious hospitality ofH. Nicolai and the Albert Einstein Institute. This workis supported in part by US DOE Grant No. DE-FG 02-91ER40685.

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[5] A. Buonano, T. Damour, G. Veneziano, Nucl. Phys. 543 (1999)275, hep-th/9806230.

[6] D. Clancy, J.E. Lidsey, R. Tavakol, Phys. Rev. D 58 (1998)044017;D. Clancy, J.E. Lidsey, R. Tavakol, Phys. Rev. D 59 (1999)063511;D. Clancy, A. Feinstein, J.E. Lidsey, R. Tavakol, Phys.Rev. D 60 (1999) 043593;D. Clancy, A. Feinstein, J.E. Lidsey, R. Tavakol, Phys.Lett. B 451 (1999) 303;K.E. Kunze, Class. Quantum Grav. 16 (1999) 3795.

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[9] For discussion on colliding waves see: J.B. Griffiths, CollidingPlane Waves in General Relativity, Oxford Univ. Press, Ox-ford, UK, 1991.

[10] A. Das, Integrable Models, World Scientific, Singapore, 1989.[11] There is a vast literature on the subject; we found following

two articles very useful which also contain extensive refer-ences.H. Nicolai, Recent aspects of quantum fields, in: H. Mitter,H. Gausterer (Eds.), Schladming Lectures, Springer Verlag,Berlin, 1991;H. Nicolai, D. Korotkin, H. Samtleben, Integrable classicaland quantum gravity, NATO Advanced Study Institute onQuantum Fields and Quantum Spacetime, Cargese, 1996, hep-th/9612065.

[12] J. Maharana, Phys. Rev. Lett. 75 (1995) 205, hep-th/9502001;


J. Maharana, Mod. Phys. Lett. A 11 (1996), hep-th/9502002.[13] J.H. Schwarz, Nucl. Phys. B 447 (1995) 137, hep-th/9503078;

J.H. Schwarz, Classical duality symmetries in two dimensions,STRINGS 95, hep-th/9505170;J.H. Schwarz, Nucl. Phys. B 454 (1995) 427.

[14] A. Sen, Nucl. Phys. B 447 (1995) 62.[15] I. Bakas, Nucl. Phys. 428 (1994) 374;

S. Mizoguchi, Nucl. Phys. B 461 (1996) 155;E. Abdalla, M.C.B. Abdalla, Phys. Lett. B 365 (1996) 41;

A.K. Biswas, A. Kumar, K. Ray, Nucl. Phys. B 453 (1995)181;A.A. Kehagias, Phys. Lett. B 360 (1995) 19.

[16] J. Maharana, J.H. Schwarz, Nucl. Phys. B 390 (1993) 3, hep-th/9207016.

[17] P. Szekeres, J. Math. Phys. 13 (1972) 286.[18] U. Yurtsever, Phys. Rev. D 37 (1988) 2803.

colliding string waves and duality

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