Fundamental strings in open string theory at the tachyonic vacuum

Fundamental strings in open string theory at the tachyonic vacuumAshoke Sen Citation: Journal of Mathematical Physics 42, 2844 (2001); doi: 10.1063/1.1377037 View online: http://dx.doi.org/10.1063/1.1377037 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/42/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Neveu–Schwarz fivebrane and tachyon condensation J. Math. Phys. 46, 062301 (2005); 10.1063/1.1922069 Magnetic backgrounds and tachyonic instabilities in closed string theory AIP Conf. Proc. 607, 269 (2002); 10.1063/1.1454381 Noncommutative tachyons and K-theory J. Math. Phys. 42, 2765 (2001); 10.1063/1.1377270 D=4 chiral string compactifications from intersecting branes J. Math. Phys. 42, 3103 (2001); 10.1063/1.1376157 The Hagedorn transition in noncommutative open string theory J. Math. Phys. 42, 2749 (2001); 10.1063/1.1372176

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Fundamental strings in open string theoryat the tachyonic vacuum

Ashoke Sena)

Harish-Chandra Research Institute,b) Chhatnag Road, Jhusi, Allahabad 211019, India

~Received 2 January 2001; accepted for publication 13 February 2001!

We show that the world-volume theory on a D-p-brane at the tachyonic vacuum hassolitonic string solutions whose dynamics is governed by the Nambu–Goto actionof a string moving in (2511) dimensional space–time. This provides strong evi-dence for the conjecture that at this vacuum the full (2511) dimensional Poincare´invariance is restored. We also use this result to argue that the open string fieldtheory at the tachyonic vacuum must contain closed string excitations. ©2001American Institute of Physics.@DOI: 10.1063/1.1377037#

I. INTRODUCTION AND SUMMARY

It has been conjectured that the tachyonic vacuum in open bosonic string theory on a D-branedescribes the closed string vacuum without D-branes, and that various soliton solutions in thistheory describe D-branes of lower dimension.1 Similar conjectures have also been put forward forsuperstring theories.2–4 Evidence for these conjectures come from both first5,6 and second7–10

quantized string theories.Given that all D-branes can be regarded as solitons in the open string field theory,11 one might

wonder if the open string field theory could be used for a nonperturbative formulation of stringtheory.12 For this one needs to show that not only the D-branes, but other known objects in stringtheory, namely the fundamental closed strings and the NS five-branes are also present in this openstring field theory. Progress in identifying the fundamental string has been made in Refs. 13, 14,8, 15, and 16. In particular, in Refs. 14, 8, 15, and 16 it was shown that the effective action17

describing the dynamics of the D-brane around the tachyonic vacuum admits stringlike classicalsolution whose tension matches that of a fundamental string. It was also established that on aD-25-brane world-volume, the dynamics of these strings is described by that of a Nambu–Gotostring moving in (2511) dimensions.

On the world-volume of a D-p-brane embedded in the (2511) dimensional space–time, thefull (2511) dimensional Poincare´ invariance is spontaneously broken to the product of (p11)dimensional Poincare´ group, and the (252p) dimensional rotation group. However, if the tachy-onic ground state really represents the vacuum without a D-brane, then we expect that in thisvacuum the full (2511) dimensional Poincare´ invariance should be restored. Thus the dynamicsof the stringlike solutions should be described by a Nambu–Goto action with (2511) dimensionaltarget space rather than a (p11) dimensional target space. This is what we shall demonstrate inthis paper.~This question was partially addressed in Ref. 15 where it was shown that in theapproximation where the contribution to the Hamiltonian is dominated by the electric flux on theD-brane world-volume, there is a symmetry that exchanges the velocity tangential to the D-branewith the velocity transverse to the D-brane.! Since the Nambu–Goto action in (2511) dimen-sional target space has full (2511) dimensional Poincare´ invariance, this result provides strongsupport to the conjecture that at the tachyonic vacuum of the D-p brane the full (2511) dimen-sional Poincare´ invariance is restored. Earlier string field theory analysis has provided evidence for

a!Electronic mail: [email protected], [email protected]!Formerly Mehta Research Institute of Mathematics & Mathematical Physics.

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 42, NUMBER 7 JULY 2001

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the restoration of the translational invariance along directions transverse to the D-braneworld-volume.18,19

Since the D-p-brane world-volume is (p11) dimensional, and the string solution lives on theD-p-brane, it may sound strange at first that this string actually moves in (2511) dimensions. Thereason it can happen is that at the tachyonic vacuum the D-brane has vanishing tension, and henceit does not cost the D-brane any additional energy to adjust its world-volume to contain any givenfundamental string world-sheet embedded in (2511) dimensional space–time. Thus the stringworld-sheet always lies inside the D-p-brane world-volume, as should be the case. The nontrivialfact here is that the dynamics of the string tangential to the D-brane world-volume, which isdescribed by the gauge fields, and the dynamics transverse to the D-brane world-volume, which isdescribed by the massless scalar fields associated with the transverse motion of the D-brane, aretogether described by the Nambu–Goto action in the full (2511) dimensional target space–time.

Although the dynamics of the string soliton constructed this way agrees with that of thefundamental string, there are some caveats. First of all the tension of the string is governed by thetotal amount of electric flux it carries, and only after properly taking into account the quantizationrule for the electric flux one can show that the tension matches that of the fundamental string.Within the classical field theory which we shall be studying, there is no rationale for this quanti-zation law. A related problem is as follows. Although the string solution constructed here has thecorrect degrees of freedom describing the dynamics of a fundamental string, it also has additionaldegrees of freedom corresponding to the energy density spreading out in the direction transverseto the original string solution instead of being confined in a thin tube along the string. We showthat these problems can be avoided by making the solitonic string driven by an external openstring. For this we consider the case where one of the directions transverse to the D-p-brane iscompact, and we begin with a configuration of open strings starting on the D-brane, and ending onits image under translation along the compact direction. We then ask what happens when thetachyon on the D-brane rolls down to its ground state. We argue that at the tachyonic vacuum, thetwo ends of the original open string are connected by a flux line on the D-brane, with the totalamount of flux fixed by the source~and the sink! of flux, namely the end points of the originalopen string on the brane. Furthermore, the condition for minimum energy prevents the flux fromspreading, since the source and the sink of flux are pointlike objects on the D-p-brane world-volume. The net result is a single fundamental string winding along the compact direction. Usinga T-duality transformation along the compact direction we can then argue that theT-dual D-~p11)-brane at the tachyonic vacuum must contain closed string excitations carrying momentumalong the compact direction.

Related earlier work in Ref. 20 analyzed the dynamics of tensionless D-branes in a differentformalism and found that the D-brane world-volume is foliated by string world-sheet. It will beinteresting to explore the precise relation between these results and the static gauge results of Refs.14, 8, 15, 16 and the present paper.

The paper is organized as follows. In Sec. II we review the result for the effective action onthe D-brane world-volume at the tachyonic vacuum17 and its Hamiltonian formulation.15 In Sec.III we show that given any solution of the equations of motion of a Nambu–Goto string movingin (2511) dimensional space–time, we can construct a solution of the equations of motion of theD-p-brane world-volume theory with energy density localized along the world-sheet of the corre-sponding Nambu–Goto string solution. This establishes that the D-p-brane world-volume theoryadmits stringlike soliton solutions whose dynamics is governed by the Nambu–Goto action in(2511) dimensions. In Sec. IV we use this result to argue that the open string field theory,describing the D-brane world-volume theory at the tachyonic vacuum, must contain closed stringexcitations.

II. LOW ENERGY EFFECTIVE FIELD THEORY ON THE D-BRANE AT THE TACHYONICVACUUM

We shall analyze the dynamics of massless fields living on a D-p brane at the tachyonicvacuum in the static gauge. Let us denote byxm(0<m<p) the world-volume coordinates on the

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D-brane, byAm the U~1! gauge field living on the D-brane, and byYI (p11<I<25) the masslessscalars representing the transverse coordinates of the brane. The action is given by17

S52V~T!E dp11xA2det~hmn1Fmn1]mYI]nYI !, ~2.1!

whereV(T) is the tachyon potential which vanishes at the tachyonic vacuumT5T0 . We shallwork in the gaugeA050, and denote byp i(x) and pI(x) the momenta conjugate toAi andYI ,respectively, for 1< i<p. As was shown in Ref. 15, the dynamics of the brane at the tachyonicvacuum is best described in the Hamiltonian formalism, with the Hamiltonian

H5E dpxH, ~2.2!

with

H5Ap ip i1pIpI1~p i] iYI !21bibi , ~2.3!

where

bi[Fi j pj1] iY

IpI . ~2.4!

Fi j 5] iAj2] jAi is the magnetic field strength. Thep i ’s satisfy a constraint:

] ipi50. ~2.5!

In writing down the Hamiltonian~2.2!, ~2.3! we have taken the tachyon fieldT to be frozen at itsminimumT5T0 . Proposals for the effective action including tachyon kinetic term have been putforward in Ref. 21.

Let us denote byEi5]0Ai the electric field strength. Then the Bianchi identities and theequations of motion derived from the Hamiltonian given in~2.2!, ~2.3! are given by

] [ iF jk]50, ]0Fi j 5] iEj2] jEi , ~2.6!

Ei51

H ~p i1] iYIp j] jY

I2Fi j bj !, ~2.7!

]0p i1] j S 1

H ~p jbi2p ibj ! D50, ~2.8!

]0YI51

H ~pI1] iYIbi !, ~2.9!

]0pI5] i S 1

H ~p ip j] jYI1bip

I ! D . ~2.10!

For this system, there are conserved Noether currentsTmn andTmI (0<m,n<p, (p11)<I<25) associated with the translation along the spatial coordinatesxm labeling the D-p-braneworld-volume, as well as translation along the coordinatesYI transverse to the world-volume.These are given by

T005H, Tk052bk , T0i52bi , Tki51

H ~pkp i2bkbi !,

~2.11!

2846 J. Math. Phys., Vol. 42, No. 7, July 2001 Ashoke Sen


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T0I5pI , TkI51

H ~pkp j] jYI1bkp

I !,

and satisfy

hmn]mTnr50, hmn]mTnI50. ~2.12!

III. FUNDAMENTAL STRING SOLUTION

In this section we shall demonstrate that the equations of motion discussed in Sec. II admitfundamental string solutions whose dynamics is identical to that of a Nambu–Goto string movingin (2511) dimensional space–time. Using the results of Ref. 22, Ref. 15 showed that if we set theYI ’s to 0, then the dynamics of the solitonic string is described by the Nambu–Goto action in(p11) dimensional space–time. The new result is the incorporation of theYI ’s. Since the dy-namics of a Nambu–Goto string in (2511) dimensional space–time has full (2511) dimensionalPoincare´ invariance, our result gives strong support to the conjecture that the tachyonic vacuum ofthe D-p-brane represents a configuration where the full (2511) dimensional Poincare´ invarianceis restored.

Our strategy will be as follows. We shall show that for every configuration of a Nambu–Gotostring satisfying the string equations of motion we can construct a solution of the equations ofmotion ~2.5!–~2.10!, with energy density localized along the string. For this we start by writingdown the action of the Nambu–Goto string in (2511) dimensional space–time:

SNG52E dt dsA2det~hMN]aZM]bZN!, ~3.1!

whereja for a50,1 denote the world-volume coordinates of the string: (j0,j1)[(t,s), ZM (0<M<25) denote the space–time coordinates of the string, andhMN is the Minkowski metricdiag(21,1,1,...,1). We shall choose the static gauge: (Z05t,Z15s) and go to the Hamiltonianformalism. If we denote byPs the momenta conjugate toZs for 2<s<25, the Hamiltonian isgiven by

HNG[E ds HNG5E ds A11PsPs1]sZs]sZs1~Ps]sZs!2. ~3.2!

The equations of motion following from this Hamiltonian are given by

]tZs5

1

HNG~Ps1]sZsPt]sZt!, ~3.3!

]tPs5]sS 1

HNG~]sZs1PsPt]sZt! D . ~3.4!

In these equationss and t indices take values 2,3,...,25. Forfuture use, we shall define

P152(s52

25

Ps]sZs, Z1~t,s!5s. ~3.5!

With these definitions, it is straightforward to verify that Eqs.~3.3! and~3.4! are satisfied also fors51. ~The sum overt in these equations still runs from 2 to 25!.

Let (Zs(t,s),Ps(t,s)) for 2<s<25 be a solution of Eqs.~3.3! and~3.4!. Now consider thefollowing field configuration on the D-p-brane:

p i~x0,...,xp!5]sZi~t,s! f ~x0,...,xp!u(t,s)5(x0,x1) ,~3.6!



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pI~x0,...,xp!5PI~t,s! f ~x0,...,xp!u(t,s)5(x0,x1),

where we have used the convention that the indicesi , j ,k run from 1 top, the indicesI ,J,K runfrom (p11) to 25, and the indicess,t run from 2 to 25.f (x0,...,xp) is an arbitrary function of thevariables (xm2Zm(x0,x1)) for 2<m<p, and hence satisfies

]sZi] i f u(t,s)5(x0,x1)50, ~]tZi] i f 1]0f !u(t,s)5(x0,x1)50. ~3.7!

The fieldsYI(x0,...,xp) andFi j (x0,...,xp) are subject to the following set of conditions:

~]sZj] jYI2]sZI !u(t,s)5(x0,x1)50, ~3.8!

and

~Fi j ]sZj1] iYI PI1Pi !u(t,s)5(x0,x1)50, ~3.9!

but are otherwise unspecified. Using Eqs.~3.6!, ~3.8!, and~3.9! we can easily verify that for thisbackground,

H~x0,...,xp!5HNG~t5x0,s5x1! f ~x0,...,xp!,

bi~x0,...,xp!52Pi~t5x0,s5x1! f ~x0,...,xp!, ~3.10!

p j] jYI~x0,...,xp!5]sZI~t5x0,s5x1! f ~x0,...,xp!.

Using Eqs.~3.3!–~3.7! and~3.10! we can now verify that Eqs.~2.5!, ~2.8!, and~2.10! are satisfiedby this background. Thus in order to construct a solution of the full set of equations of motion~2.5!–~2.10! we need to show that it is possible to findFmn andYI satisfying the constraints~2.6!,~2.7!, ~2.9!, ~3.8!, and~3.9!.

First we shall establish the existence ofYI ’s satisfying Eqs.~2.9! and~3.8!. @Note that the Eq.~3.9! imposes a constraint onYI of the form ]sZi(] iY

I PI1Pi)u(t,s)5(x0,x1)50, but due to Eq.~3.5!, this is automatically satisfied once Eq.~3.8! is satisfied.# Using Eqs.~3.6! and ~3.10!, weshall now write Eqs.~2.9! and ~3.8! as follows:

]0YI51

HNG~PI2] iY

I Pi !,

~3.11!]1YI5~2]sZm]mYI1]sZI !,

where the indicesm,n,q run from 2 top, and it will be understood from now on thatt ands areto be identified withx0 and x1, respectively. We can now treat Eqs.~3.11! as the equationsdetermining thex0 andx1 evolution of the functionsYI . ~We replace the]1YI appearing on theright-hand side of the first equation by the right-hand side of the second equation.! Existence of asolution to these equations requires the integrability condition:

]1S 1

HNG~PI2]mYI Pm1P1~]sZm]mYI2]sZI !! D2]0~~2]sZm]mYI1]sZI !!50. ~3.12!

It is a straightforward although tedious exercise to show that once Eqs.~3.3! and~3.4! are satisfied,Eq. ~3.12! is satisfied.

Thus it remains to show the existence of a set ofFmn satisfying Eqs.~2.6!, ~2.7!, and~3.9!. Webegin with theFmn’s (2<m,n,q<p). We take them to satisfy the following identities:

] [mFnq]50, ~3.13!

and



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]1Fmn1]1Zq@x0,x1#]qFmn50,~3.14!

]0Fmn1]0Zq@x0,x1#]qFmn50.

To see that it is possible to chooseFmn’s satisfying these conditions, we regard Eq.~3.14! as theevolution equation forFmn in x0 and x1 from an initial configuration satisfying the Bianchiidentities ~3.13!. It is easy to verify that the evolution equations~3.14! preserve the Bianchiidentities at all values ofx0 andx1. It is also easy to verify the integrability of Eq.~3.14!:

]0~]1Zq@x0,x1#]qFmn!2]1~]0Zq@x0,x1#]qFmn!50. ~3.15!Given Fmn satisfying ~3.13! and ~3.14!, we can use~3.10! to write Eqs.~2.7! and ~3.9! as

follows:

F0i51

HNG~]sZi1] iY

I]sZI1Fi j Pj !, ~3.16!

and

Fi152Fin]sZn2] iYI PI2Pi . ~3.17!

If Eq. ~3.17! is satisfied fori 5m, then it is also automatically satisfied fori 51 with the help ofEqs.~3.5! and ~3.11!. Thus the independent equations in~3.17! are

Fm152Fmn]sZn2]mYI PI2Pm . ~3.18!

This givesFm1 in terms ofFmn and other known quantities. Replacing theFm1’s appearing on theright-hand side of Eq.~3.16! by the right-hand side of Eq.~3.18!, we can now regard Eq.~3.16! asexpressions forF01 andF0m in terms ofFmn and other known quantities.

We now need to check thatF0i andFm1 defined through Eqs.~3.16! and ~3.18!, satisfy theremaining Bianchi identities:

]0Fmn1]mFn01]nF0m50,

]1Fmn1]mFn11]nF1m50, ~3.19!

]0Fm11]mF101]1F0m50.

It is straightforward to verify that all of these identities are consequences of Eqs.~3.3!, ~3.4!,~3.11!, ~3.13!, and ~3.14!. This completes the construction of a solution of the complete set ofequations of motion~2.5!–~2.10! of the D-p-brane world-volume field theory.

We shall now make a special choice of the functionf :

f ~x0,...,xp!5 )m52

p

d~xm2Zm~x0,x1!!, ~3.20!

which satisfies Eq.~3.7!. Furthermore we take

YI~x0,x1,xm5Zm~x0,x1!!5ZI~x0,x1!, ~3.21!

which can be seen to be compatible with Eq.~3.11!. Indeed, Eqs.~3.11! and ~3.14! can beinterpreted as the requirement of vanishing of the derivatives of (YI2ZI) and Fmn along direc-tions tangential to the string world-sheet. Thus we can solve these equations by takingYI2ZI andFmn to be arbitrary functionsgI andgmn , respectively, ofx22Z2(x0,x1),...xp2Zp(x0,x1). Equa-tion ~3.21! can then be satisfied by requiring thatgI(0,...,0)50. The Bianchi identities~3.13! aresatisfied by requiring that the functionsgmn satisfy] [qgmn]50.



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As can be seen from Eq.~3.10!, for the choice off given in Eq.~3.20!, the energy density islocalized along the surfacexm5Zm(x0,x1) for 2<m<p. Using Eq.~3.21! we see that in the full(2511) dimensional space–time, this describes the surfacexs5Zs(x0,x1) for 2<s<25. This isprecisely the world-sheet of the string described by the Nambu–Goto action~3.1!. Thus ouranalysis shows that whenever the Nambu–Goto equation has a solution described byZs(s,t),there is a corresponding solution in the D-p-brane world-volume field theory with energy densitylocalized along the world-sheet of the string. In other words, the D-p-brane world-volume theorycontains a solution whose dynamics is exactly that of a Nambu–Goto string in(2511)-dimensions.

Note, however, that the freedom of replacing thed function by an arbitrary function ofxm

2Zm(x0,x1) means that besides the usual degrees of freedom of the fundamental string, oursolution has additional degrees of freedom which corresponds to the freedom of spreading out theelectric flux in directions transverse to the string. Also the overall normalization on the right-handside of Eq.~3.20!, which fixes the tension/charge of the string, is arbitrary. We shall return to thesequestions in Sec. IV. There are also additional degrees of freedom stemming from the fact thatEqs.~3.11!, ~3.13!, and~3.14! do not determineYI andFmn completely for a given configurationof the Nambu–Goto string. This is analogous to the spurious degeneracy found in Ref. 23. It hasbeen argued in Ref. 24 that this apparent degeneracy is due to the wrong choice of variables indescribing the theory, and will disappear once we use the correct set of variables.

We shall end this section by writing down the expressions for the conserved Noether currentsfor the background described previously. This is a straightforward exercise, and the results are asfollows:

T00~x0,...,xp!5HNG~t,s! )m52

p

d~xm2Zm~t,s!!u(t,s)5(x0,x1) ,

T0k~x0,...,xp!5Tk0~x0,...,xp!5Pk~t,s! )m52

p


Tki~x0,...,xp!51

HNG~]sZk]sZi2PkPi ! )

m52

p

d~xm2Zm~t,s!!u(t,s)5(x0,x1) , ~3.22!

T0I~x0,...,xp!5PI~t,s! )m52

p


TkI~x0,...,xp!51

HNG~]sZk]sZI2PkPI ! )

m52

p

d~xm2Zm~t,s!!u(t,s)5(x0,x1) .

Verification of the conservation laws~2.12! for Tmn and TmI is a straightforward application ofEqs.~3.3! and~3.4!. It is also a simple exercise to verify that the corresponding conserved charges*dpx T00, *dpxT0i and *dpx T0I agree with the Noether charges of the Nambu–Goto stringassociated with translation invariance alongx0, xi , andxI directions, respectively.

IV. CLOSED STRINGS IN THE D-BRANE WORLD-VOLUME THEORY

In this section we shall use the results of Sec. III to argue that at the tachyonic vacuum theD-brane world-volume theory must contain closed string excitations. Identification of closedstrings as closed flux lines has been discussed earlier in Refs. 13, 14, 8, 15, and 16. The presentconstruction is closely related, but differs in that here part of the closed string is formed by anexternal open string.

We begin with a thought experiment. Consider three well-separated D-branesA, B, andC,and a state on the world volume of this system consisting of a fundamental string stretched from



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A to B, and another fundamental string stretched fromB to C. Let us now ask: What happens tothis state when the tachyon field on the braneB rolls down to its~local! minimum, but the branesA andC remain unchanged. The D-brane world-volume field theory analysis tells us that the endsof the AB and BC strings are source and sink of one unit of electric flux~measured in naturalunits! on the world-volume of the braneB. Thus the fate of the system is clear: The finalconfiguration will consist of theAB string, theBC string, and an electric flux line~described bythe solution given in Sec. III! on the B-brane world-volume connecting the end point of theABstring to the starting point of theBC string. Note that the condition for minimum energy willprevent the flux from spreading out as its source and sink are point objects.~Of course, one wouldstill need to understand why local fluctuations on the string involving the spreading of the flux isabsent. Some discussion on this can be found in Refs. 14 and 8.! Furthermore there is preciselyone unit of electric flux and hence its tension is equal to that of a fundamental string.14 Thus it isnatural to interpret the flux line as a fundamental string stretched from the end point of theABstring to the starting point of theBC string. ~This string, as well as the externalAB and BCstrings, can adjust their positions and orientations so as to minimize the energy of the configura-tion.! Thus the net result of this process is a single open string stretched betweenA andC. It is asif the tachyon condensation on the world-volume of the braneB joins the ends of theAB andBCstrings by a fundamental string. Even before tachyon condensation on the braneB, the ends ofABand BC strings could join to produce anAC string. But there it was a perturbative quantumprocess, whereas the process described here is a nonperturbative classical process.

Now we consider a different system: Take a single D-brane and an open string with both endson this D-brane. One can use the same argument to conclude that when the tachyon condenses toits ground state, the two ends of the open string will be connected by a flux line, and once weidentify the flux line as the fundamental string, we get a closed string state! This argument can bemade more concrete as follows. Take a D-brane with one of its transverse directions compact, andconsider an open string stretched from the D-brane to its image under translation along thecompact direction. Now let us ask what happens to this open string state when the tachyon on theD-brane condenses to its ground state. Since the original state carries fundamental string windingcharge this state cannot disappear. To see what happens it is simplest to go to the infinite cover;in this case we have initially an infinite number of parallel D-branes at regular spacing, andbetween any two neighboring D-branes we have an open string suspended between the two. Thuson any of the D-branes we have an open string ending and another open string starting giving asource and a sink of electric flux. When the tachyon condenses to its ground state, each D-branedevelops a flux line joining the source to the sink. If we identify this flux line as a fundamentalstring as before, the result is a single infinitely long string. After modding out by the discretetranslation symmetry to compactify the direction, this is nothing but a closed string wrappedaround the compact direction!

Thus we conclude that if we start with a D-brane with a transverse direction compact, thenafter tachyon condensation the open string field theory on the D-brane world-volume must containexcitations corresponding to closed string winding states along the compact direction. Let us nowmake aT-duality transformation along the compact direction. This converts the D-p brane to aD-~p11) brane, but is otherwise a symmetry of the open string field theory order by order in openstring perturbation theory. On the other hand this transforms the closed string winding modes toclosed string momentum modes. Thus if we start with a D-brane with one of its tangentialdirections compact, then after tachyon condensation the corresponding open string field theorywill contain excitations corresponding to closed string states carrying momentum along the com-pact direction.

It will be interesting to see if we can study this phenomenon directly in open string fieldtheory.

ACKNOWLEDGMENTS

I would like to thank S. Das, D. Ghoshal, D. Jatkar, J. Majumder, and P. Mukhopadhyay foruseful discussions.



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Fundamental strings in open string theory at the tachyonic vacuum