12 July 2001

Physics Letters B 512 (2001) 181–188www.elsevier.com/locate/npe

Open string field theory without open strings

Ian Ellwood, Washington Taylor1

Center for Theoretical Physics, MIT, Bldg. 6, Cambridge, MA 02139, USA

Received 16 May 2001; accepted 29 May 2001Editor: M. Cvetic

Abstract

Witten’s cubic open string field theory is expanded around the perturbatively stable vacuum, including all scalar fields atlevels 0, 2, 4 and 6. The (approximate) BRST cohomology of the theory is computed, giving strong evidence for the absence ofphysical open string states in this vacuum. 2001 Published by Elsevier Science B.V.

1. Introduction

The 26-dimensional open bosonic string has atachyon in its spectrum withM2 = −1/α′. The pres-ence of this tachyon indicates that the perturbativevacuum of the theory is unstable. While some earlywork [1] indicated the possible existence of a more sta-ble vacuum at lower energy (see also [2,3]), until fairlyrecently the significance of this other vacuum was notunderstood, and the tachyon was taken to be an indi-cation of fundamental problems with the open bosonicstring.

In 1999, Sen suggested that the open bosonic stringshould be interpreted as ending on an unstable space-filling D25-brane [4]. Sen argued that the condensa-tion of the tachyon should correspond to the decay ofthe D25-brane, and that it should be possible to give ananalytic description of this condensation process us-ing the language of Witten’s cubic open string field

E-mail addresses: iellwood@mit.edu (I. Ellwood),wati@mit.edu, wati@itp.ucsb.edu (W. Taylor).

1 Current address: Institute for Theoretical Physics, Universityof California, Santa Barbara, CA 93106-4030, USA.

theory [5]. In particular, Sen made three concrete con-jectures:

(a) The difference in the action between the unstablevacuum and the perturbatively stable vacuumshould be�E = V T25, whereV is the volumeof space–time andT25 is the tension of the D25-brane.

(b) Lower-dimensional Dp-branes should be realizedas soliton configurations of the tachyon and otherstring fields.

(c) The perturbatively stable vacuum should corre-spond to the closed string vacuum. In particular,there should be no physical open string excitationsaround this vacuum.

Conjecture (a) has been verified to a high degreeof precision in level-truncated cubic open string fieldtheory [6,7] and has been shown exactly using back-ground independent string field theory [8–10]. Con-jecture (b) has been verified for a wide range of sin-gle and multiple Dp-brane configurations, using boththe cubic and background independent formulationsof SFT (see [11] for a review and further references).To date, however, little concrete evidence has beenput forth either for the decoupling of open stringsin the perturbatively stable vacuum or for the inter-

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00673-6

182 I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181–188

pretation of this state as the closed string vacuum.In this Letter we explicitly compute the scalar openstring spectrum in the cubic open string field theoryexpanded around the perturbatively stable vacuum, us-ing the level-truncation approximation. The results ofthis computation give strong evidence that there are nophysical open string states in this vacuum, and that theopen strings are removed from the spectrum by purelyclassical effects in the string field theory.

2. String field theory in the stable vacuum

We begin with a brief summary of Witten’s cubicformulation of open bosonic string field theory [5] (see[12,13] for reviews). The string fieldΦ contains aninfinite family of space–time fields, one field beingassociated with each state in the open string Fockspace. Physical fields are associated with states in theHilbert space of ghost number one. The string fieldmay be formally written as

(1)Φ = φ(p)|0;p〉 + Aµ(p)αµ−1|0;p〉 + · · · ,

where |0〉 is the ghost number one vacuum relatedto the SL(2,R)-invariant vacuum|0〉 through |0〉 =c1|0〉. Witten’s cubic string field theory action is

(2)S = −1

2

∫Φ � QΦ − g

3

∫Φ � Φ � Φ,

whereQ is the BRST operator of the string theory andthe “star product”� is defined by dividing each stringevenly into two halves and “gluing” the right side ofone string to the left side of the other through a deltafunction interaction. The action (2) is invariant underthe stringy gauge transformations

(3)δΦ = QΛ + g(Φ � Λ − Λ � Φ),

whereΛ is a ghost number zero string field.While there are an infinite number of component

fields in the string fieldΦ, for any particular com-ponent fields the quadratic and cubic interactions in(2) and the related terms in the gauge transforma-tions (3) can be computed in a straightforward fash-ion using a Fock space representation of the BRSToperatorQ and the star product [14–16]. It has beenfound [1,17] that truncating the theory by includingonly fields up to a fixed levelL is an effective ap-proximation technique for many questions relevant to

the tachyon condensation problem. (By convention thetachyon is taken to have level zero.) At fixed levelL,the theory can be further simplified by only consid-ering interactions between fields whose levels total tosome numberI < 3L. Empirical evidence [1,7] indi-cates that truncating at level(L, I) = (L,2L) is themost effective cutoff to maximize accuracy for a fixednumber of computations and that calculations at trun-cation level(L,2L) give similar results to calculationsat truncation level(L,3L).

Sen’s conjecture states that there is a Lorentzinvariant solutionΦ0 of the full string field theoryequations of motion

(4)QΦ0 = −gΦ0 � Φ0

corresponding to the closed string vacuum withouta D25-brane. The existence of a nontrivial solutionto (4) has been analyzed in the level-truncated the-ory [1,6,7]. In the level(0,0) truncation, the tachyonpotential is simply−1

2φ2 + gκφ3, whereκ is a nu-merical constant. This potential gives a locally stablevacuum atφ0 = 1/(3gκ). Evaluating the potential atthis point gives 68% of Sen’s conjectured valueT25for the energy gap between the unstable vacuum andthe perturbatively stable vacuum. When other scalarfields are included in the string field by raising thelevel at which the theory is truncated, many of thesefields couple to the tachyonφ and take expectationvalues whenφ becomes nonzero, but similar solutionsto the level-truncated string field theory equations ofmotion continue to exist. In the level(4,8) truncation,the energy gap between the two vacua becomes 98.6%of the predicted value [6], and in the level(10,20)truncation the energy gap becomes 99.91% of the pre-dicted value [7]. As the level of truncation is increased,the vacuum expectation values of the scalar fields con-verge rapidly, so that the level(10,20) values for thefield values in the vacuum appear to be within less than1% of their exact values for low-level fields. These re-sults provide us with a close approximation to a fieldΦ0 satisfying (4), which we can use to study the per-turbatively stable vacuum in the level truncated theory.

We can describe the physics around a nontrivialvacuum〈Φ〉 = Φ0 satisfying Eq. (4) by shifting thestring field

(5)Φ = Φ0 + Φ.

I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181–188 183

In terms of the new fieldΦ , the action becomes

(6)S = S0 − 1

2

∫Φ � QΦ − g

3

∫Φ � Φ � Φ,

where the new BRST operatorQ acts on a string fieldΨ of ghost numbern through

(7)QΨ = QΨ + g(Φ0 � Ψ − (−1)nΨ � Φ0

).

The identity

(8)Q2 = 0

for the new BRST operator follows from (4). TheBRST invariance of the level-truncated approximationto the vacuumΦ0 was studied in [18]

We are interested in studying the physics of the newstring field theory defined through (6), (7). Accordingto Sen, the vacuumΦ0 should be the closed stringvacuum, and should not admit open string excitations.To study the spectrum of excitations of the theory,we need to explicitly calculate the quadratic terms inthe action, or equivalently to compute the action ofthe new BRST operator (7) on a general string field.This requires us to compute all cubic couplings in theoriginal string field theory of the form

(9)tijk(p)φi(0)φj (−p)φk(p).

In this Letter we restrict attention to scalar excitations,so we need to compute all terms of the form (9) whereφi , φj and φk are scalar fields. Becauseφj , φk aremomentum-dependent, we must include among thesefields longitudinal polarizations of all higher-spintensor fields as well as the zero-momentum scalarsφi(0) which can take nonzero vacuum expectationvalues inΦ0. We restrict attention in this Letter toscalars at even levels, which decouple from odd-levelscalars at quadratic order due to the twist symmetry ofthe theory [1,6].

With the assistance of the symbolic manipulationprogram MATHEMATICA we have computed all 58481scalar interactions of the form (9) in the level(6,12)truncation of the theory. There are 160 scalar fieldsφj (p) at even levels� 6, including longitudinalpolarizations of tensor fields, and 31 momentum-independent scalar fieldsφi(0), each of which takesa nonzero value in the vacuumΦ0. (Actually, thevacuum lies in a subspaceH1 of the full scalar fieldspace [4], but we do not use this decomposition in

our analysis.) As a check on our calculations wehave also computed all the coefficients associatedwith gauge transformations (3) where one of the threefields involved has vanishing momentum. We haveverified that our terms of the form (9) give rise to anaction (in the perturbative vacuum) which is invariantat orderg1 under arbitrary momentum-independentgauge transformations and a random sampling ofmomentum-dependent gauge transformations.

Using the complete set of terms of the form (9) inthe level(6,12) truncation, we have calculated all thequadratic terms for even-level scalars in the action (6)around the perturbatively stable vacuum. The detailsof this action are far too lengthy to appear in print,but will be made available in the future in electronicform. In the remainder of this Letter we summarizethe results of using this quadratic action to study thespectrum of open string states in the theory (6). Earlierattempts to study the spectrum of physical states in asubset of the level(2,6) truncation appeared in [1,19].

3. BRST cohomology

The spectrum of physical states in the theory (6) isgiven by the BRST cohomology

(10)KerQ1/ ImQ0,

whereQn describes the action of the BRST operatorQ on a string field of ghost numbern. From (8), itfollows thatQ1Q0 = 0 in the full string field theory.The states associated with vanishing eigenvalues of thekinetic operatorQ1 at a fixed value ofp2 are theQ-closed states in the theory. TwoQ-closed states arephysically equivalent if they differ by aQ-exact stateQ0Λ whereΛ is a string field of ghost number 0.

Level truncation of the open string field theorybreaks the general gauge invariance (3) at orderg2,although gauge invariance is preserved at orderg0 andg1. This breaking of gauge invariance means that thelevel-truncated BRST operator no longer squares tozero. In other words,

(11)Q(L,I )1 Q

(L,I )0 �= 0,

whereQ(L,I )n is the level(L, I) truncated approxima-

tion to Qn. The inequality (11) means thatQ-closedstates which are alsoQ-exact in the full string field

184 I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181–188

theory (6) will be approximated in the level truncatedtheory by Q-closed states which are not preciselyQ-exact. This fact makes the identification of physi-cal states in the theory only possible in an approximatesense.

We have systematically computedQ-closed statesin the level-truncated theory by finding values ofM2 = −p2 where

(12)detQ(L,I )1 = 0

and then computing the eigenvectors associated withthe vanishing eigenvalues.

As an example of this computation, consider thelevel (0,0) truncation of the theory, which includesonly the tachyon fieldφ. The quadratic term for thetachyon field in the nontrivial vacuum is

(13)φ(−p)

[p2 − 1

2+ gκ

(16

27

)p2

· 3〈φ〉]φ(p).

The determinant ofQ(0,0)1 is simply the quantity in

square brackets. This quantity does not vanish for anyreal value ofp2, so there are noQ-closed states in thespectrum at this level [1].

In the level(2,6) truncation there are seven scalarfields to be considered, associated with the Fock spacestates

|0;p〉, (α−1 · α−1)|0;p〉,(α0 · α−2)|0;p〉, (α0 · α−1)2|0;p〉,b−1c−1|0;p〉, (α0 · α−1)b−1c0|0;p〉,

(14)b−2c−0|0;p〉.At this level of truncation, using the vacuum ex-pectation values determined in [7] with the level(10,20) truncation, we found five values ofp2 wheredetQ(2,6)

1 = 0, associated with states having

M2 = 0.9067, 2.0032, 12.8566,

(15)13.5478, 16.5998

in units whereM2 = −1 for the tachyon.2 At level(4,12) we found 18Q-closed states withM2 < 20, of

2 Note: a similar calculation was done at level(2,6) inFeynman–Siegel gauge by Kostelecky and Samuel [1]. They did notcheck, however, whether their spectrum was associated with phys-ical or exact states. Our calculation can be restricted to this gauge,where it agrees for the most part (but not exactly) with their calcu-lation.

which the lightest hasM2 = 0.58817. At level(6,12)we found 33Q-closed states withM2 < 20, of whichthe lightest hasM2 = 0.85562. The complete set ofQ-closed states we found3 is graphed in Fig. 1.

To test theQ-exactness of a givenQ-closed stateat level(L, I), we computedQ(L,I)

0 Λi for each ghostnumber zero fieldΛi with level � L. The span ofthe fields Q

(L,I )0 Λi gives an approximation to the

subspace ofQ-exact states at each level. Suppose that{ei} is an orthonormal basis for this subspace ands

is one of theQ-closed states we found. We can thenmeasure the extent to which a state isQ-exact bythe norm squared of its projection onto theQ-exactsubspace. Explicitly,

(16)fraction in exact subspace=∑

i (s · ei)2

s · s.

There is no natural positive definite inner productdefined on the single string Hilbert spaceH, so tocompute (16) we had to make an ad hoc choice ofsuch an inner product. We did the calculation usingtwo choices for this inner product, and found similarresults in both cases. The first choice, which seemsmost natural, is to take the inner product〈s|s〉 withp → |p| in the matter sector and a Kronecker deltafunction in the ghost sector. The second inner productwe tried was simply defined by a Kronecker deltafunction on a basis of states spanned by all possiblescalar products of matter and ghost operators (suchas (14) at level (2, 6), giving a unit normalization toeach of these states). Using these two definitions of theinner product, we find for example that theQ-closedstate atM2 = 0.9067 found in the (2, 6) truncationlies 97.90% in the exact subspace using the first innerproduct, and 95.24% in the exact subspace using thesecond inner product. In the remainder of this Letter

3 Our algorithm for locating momenta associated withQ-closed

states proceeded by calculating the determinant ofQ(L,I)1 at equally

spaced values ofp (with �p = 0.0001) and looking for changesof sign in the determinant. The spacing of ourp values wassignificantly less than the smallest distance we observed betweenQ-closed states (0.0039), so we believe that we have found all theQ-closed states atM2 < 20. Some possibility remains that we havemissed pairs ofQ-closed states which are very close in momentum.It is remotely possible that physical states are hiding in such closelyspaced pairs ofQ-closed states.

I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181–188 185

Fig. 1. Spectrum ofQ-closed states in level truncations(0,0), (2,6), (4,12) and(6,12). States below the cutoffM2 = L − 1 lie mostly in theexact subspace, confirming Sen’s conjecture.

all calculations use the first definition of the innerproduct.

In the full string field theory, there are continuousfamilies of Q-closed states which are alsoQ-exactat all p, given by states of the formQ0|s;p〉. In thelevel truncation approximation we expect these contin-uous families to be replaced by a discrete spectrum ofalmost-exact states, approaching a continuous distrib-ution as the level of truncation is increased. The extentto which we see a continuous distribution ofQ-exactstates arising in the level-truncation approximation tothe theory around the vacuumΦ0 is a measure of howwell level truncation works in the new vacuum, andhow close the level-truncation approximation comes togiving a BRST operator satisfyingQ1Q0 = 0. A com-

plete list of Q-closed states atM2 < 20 and the ex-actness of these states is given in Table 1; qualitativeresults for the exactness of allQ-closed states are de-picted in Fig. 1. As we would hope, as the level oftruncation is increased we see a discrete distribution ofalmost-exact states which become both more exact andmore closely spaced as the level of truncation is lifted.We interpret these almost-exact states as the remnantin the level-truncated theory of the continuous familiesof Q-exact states in the full theory.

Physical states in the theory correspond toQ-closed states satisfyingQ1|s;p〉 = 0 which are notQ-exact. Because states in the cohomology ofQ willnot be removed by a generic small perturbation, allphysical states in the theory should appear in the level

186 I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181–188

Table 1Masses and exactness of allQ-closed states found in level truncations(2,6), (4,12), and(6,12) with M2 < 20

(L, I ) M2 % exact M2 % exact M2 % exact

(2,6) 0.9067 97.90% 2.0032 93.79% 12.8566 64.17%

13.5478 5.50% 16.5998 2.56%

(4,12) 0.5882 99.99% 2.9412 99.94% 3.1163 99.97%

3.9757 98.51% 4.3462 98.92% 4.5429 99.07%

5.7318 98.28% 10.1466 80.96% 10.6907 98.06%

12.7265 73.42% 13.0284 52.71% 13.4834 37.09%

13.9911 12.86% 15.2853 58.25% 16.2490 66.20%

17.0407 13.88% 17.7912 14.72% 19.2337 35.80%

(6,12) 0.8632 99.997% 2.0525 99.982% 2.3355 99.976%

2.9664 99.997% 3.1800 99.998% 4.0961 99.999%

4.2023 99.999% 4.5645 99.999% 4.5869 99.999%

4.7265 99.999% 4.7841 99.993% 4.9703 99.994%

5.4552 99.984% 5.5382 99.976% 5.6285 99.992%

5.8999 99.988% 6.3008 99.986% 6.3204 99.265%

6.5285 99.986% 6.7381 98.328% 7.6480 97.672%

8.2205 99.936% 8.2441 98.748% 8.6604 96.683%

11.5289 98.958% 11.7778 99.652% 12.1027 99.529%

13.3346 88.497% 14.5313 94.919% 16.3295 52.298%

16.9177 90.786% 16.9991 79.305% 18.2649 86.453%

truncation asQ-closed states with−p2 approachingsome fixed valueM2 as the level of truncation is in-creased. To verify Sen’s conjecture, we would hope tofind that the states lying below the cutoffM2 = L − 1are all approximatelyQ-exact, to the precision al-lowed by the level-truncation approximation, so thatno physical states appear in the limiting theory. In-deed, we find that beyond the level(2,6) truncationall states below the cutoff lie more than 99% in theexact subspace, using either choice of inner productdescribed above. For example, the lowest lying statesmentioned above in the level truncations(2,6), (4,12)and(6,12) lie 97.90%, 99.990% and 99.997% in theexact subspace. We would expect physical states in thetheory to appear consistently in each level truncation

as states with significant components outside the exactsubspace, since the average state in the level-truncatedspace lies less than 35% in the exact space. We seeno sign in our data of such physical open string statesat low levels, even up to values ofM2 several timeslarger than the cutoff. We interpret this result as strongevidence for Sen’s conjecture that there are no physi-cal open string excitations around the vacuumΦ0, andthat this string field configuration should be identifiedwith the closed string vacuum.

It may seem surprising that we expect to see thephysical states in the cohomology ofQ, which forma set of measure zero in the full space ofQ-closedstates, through this approach. The difference betweenthe behavior of physical and exact states under level

I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181–188 187

truncation ofQ can be understood by considering thebehavior of the zeros of the functionsf (x) = x − 1and g(x) = 0 under a small perturbation by a noisefunction η(x). In the first case, genericallyf (x) +η(x) will have a single zero nearx = 1. In the secondcase,g(x) + η(x) will develop a discrete spectrumof randomly spaced zeros. The physical states inthe cohomology ofQ are controlled by functionslike f (x), while the continuous families of exactstates are controlled by functions likeg(x). Whilethis argument suggests that physical states shouldindeed continue to be present in the level-truncationapproximation, as a check on our methodology wehave used the same method we used to computethe approximate cohomology ofQ to compute theapproximate cohomology of the BRST operatorQ inthe perturbative vacuum, after adding a small randomperturbationQ = Q + η. We find that unless theperturbationη is large enough to dominate the system(e.g., by pushing the exactness of a genericQ-closedstate below 90%), the physical states atM2 = −1 andM2 = 3 are easily distinguishable in a level(4,12)truncation of the theory.

4. Discussion

We have explicitly calculated the quadratic termsin the open string field theory action around the non-perturbative vacuumΦ0 in the (6,12) level trunca-tion. We computed the BRST cohomology by comput-ing all closed states under the truncated BRST opera-tor Q

(L,I )1 , and comparing with the subspace of exact

states formed by the operatorQ(L,I)0 . We found ev-

idence that allQ-closed states in the theory becomeQ-exact in the limit when fields of all levels are in-cluded.

There are several directions in which it would beinteresting to proceed, given the results in this Letter.For one thing, it would be very nice to have a betterconceptual understanding of the decoupling of openstring states in the perturbatively stable vacuum. Whilesome interesting perspectives on this phenomenonhave been given [8,20–26], a convincing picture whichexplains the classical decoupling of open strings in thecubic string field theory picture has yet to be given.An intriguing suggestion for the form of the string

field theory in the nontrivial vacuum was made in[27,28], where it was suggested that the new BRSToperatorQ can be related through a field definitionto a pure ghost operator such asc0 or more generally∑

an(cn + (−1)nc−n). It would be very interesting touse the explicit form ofQ which we have computed inlevel truncation to prove or disprove this conjecture.Finally, a question of fundamental importance is towhat extent the cubic open string field theory in theperturbatively stable vacuum contains closed stringexcitations. Sen’s conjectures suggest that it mightbe possible to give a direct description of asymptoticclosed string states in terms of the open string fieldtheory degrees of freedom in the vacuumΦ0. If sucha description could be made explicit, it would lead tonew insight into the nature of closed string field theoryand the structure of D-branes as closed string solitons.

Acknowledgements

We would like to thank D. Gross, N. Moeller,J. Polchinski, A. Sen and B. Zwiebach for helpful dis-cussions in the course of this work. Particular thanksto A. Sen and B. Zwiebach for constructive commentson an early version of this manuscript. WT wouldlike to thank the Institute for Theoretical Physics inSanta Barbara for hospitality during the latter part ofthis work. The work of IE was supported in part bya National Science Foundation Graduate Fellowshipand in part by the DOE through contract #DE-FC02-94ER40818. The work of WT was supported in partby the A.P. Sloan Foundation and in part by the DOEthrough contract #DE-FC02-94ER40818.

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