Open string field theory without open strings

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  • 12 July 2001

    Physics Letters B 512 (2001)

    Open string field theory without open stringsIan Ellwood, Washington Taylor 1

    Center for Theoretical Physics, MIT, Bldg. 6, Cambridge, MA 02139, USAReceived 16 May 2001; accepted 29 May 2001

    Editor: M. Cvetic


    Wittens 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 with M2 =1/. 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 Wittens cubic open string field

    E-mail addresses: (I. Ellwood),, (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 unstable

    vacuum and the perturbatively stable vacuumshould be E = VT25, where V is the volumeof spacetime and T25 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 [810]. 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) 00 67 3- 6

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    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 Wittens cubicformulation of open bosonic string field theory [5] (see[12,13] for reviews). The string field contains aninfinite family of spacetime 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) = (p)|0;p +A(p)1|0;p + ,where |0 is the ghost number one vacuum relatedto the SL(2,R)-invariant vacuum |0 through |0 =c1|0. Wittens cubic string field theory action is

    (2)S =12

    Q g



    where Q 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) =Q+ g( ),where is a ghost number zero string field.

    While there are an infinite number of componentfields in the string field , 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 BRSToperator Q and the star product [1416]. It has beenfound [1,17] that truncating the theory by includingonly fields up to a fixed level L 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 level L,the theory can be further simplified by only consid-ering interactions between fields whose levels total tosome number I < 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).

    Sens conjecture states that there is a Lorentzinvariant solution 0 of the full string field theoryequations of motion

    (4)Q0 =g0 0corresponding 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 122 + g3, where is a nu-merical constant. This potential gives a locally stablevacuum at 0 = 1/(3g). Evaluating the potential atthis point gives 68% of Sens conjectured value T25for 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 tachyon and take expectationvalues when 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 field0 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 = 0 satisfying Eq. (4) by shifting thestring field

    (5) =0 + .

  • I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181188 183

    In terms of the new field , the action becomes

    (6)S = S0 12

    Q g3


    where the new BRST operator Q acts on a string field of ghost number n through

    (7)Q =Q + g(0 (1)n 0).The identity

    (8)Q2 = 0for the new BRST operator follows from (4). TheBRST invariance of the level-truncated approximationto the vacuum 0 was studied in [18]

    We are interested in studying the physics of the newstring field theory defined through (6), (7). Accordingto Sen, the vacuum 0 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)i(0)j (p)k(p).In this Letter we restrict attention to scalar excitations,so we need to compute all terms of the form (9) wherei , j and k are scalar fields. Because j , k aremomentum-dependent, we must include among thesefields longitudinal polarizations of all higher-spintensor fields as well as the zero-momentum scalarsi(0) which can take nonzero vacuum expectationvalues in 0. 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 fieldsj (p) at even levels 6, including longitudinalpolarizations of tensor fields, and 31 momentum-independent scalar fields i(0), each of which takesa nonzero value in the vacuum 0. (Actually, thevacuum lies in a subspace H1 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 order g1 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,where Qn describes the action of the BRST operatorQ on a string field of ghost number n. From (8), itfollows that Q1Q0 = 0 in the full string field theory.The states associated with vanishing eigenvalues of thekinetic operator Q1 at a fixed value of p2 are the Q-closed states in the theory. Two Q-closed states arephysically equivalent if they differ by a Q-exact stateQ0 where is a string field of ghost number 0.

    Level truncation of the open string field theorybreaks the general gauge invariance (3) at order g2,although gauge invariance is preserved at order g0 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,where Q(L,I )n is the level (L, I) truncated approxima-tion to Qn. The inequality (11) means that Q-closedstates which are also Q-exact in the full string field

  • 184 I. Ellwood, W. Taylor / Physics Letters B 512 (2001) 181188

    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 computed Q-closed statesin the level-truncated theory by finding values ofM2 =p2 where

    (12)det Q(L,I )1 = 0and 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 field . The quadratic term for thetachyon field in the nontrivial vacuum is

    (13)(p)[p2 1

    2+ g


    )p2 3


    The determinant of Q(0,0)1 is simply the quantity insquare brackets. This quantity does not vanish for anyreal value of p2, so there are no Q-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, (1 1)|0;p,(0 2)|0;p, (0 1)2|0;p,b1c1|0;p, (0 1)b1c0|0;p,

    (14)b2c0|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 of p2 wheredet Q(2,6)1 = 0, associated with states havingM2 = 0.9067, 2.0032, 12.8566,

    (15)13.5478, 16.5998in units where M2 = 1 for the tachyon. 2 At level(4,12) we found 18 Q-closed states with M2 < 20, of

    2 Note: a similar calculation was done at level (2,6) inFeynmanSiegel 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 has M2 = 0.58817. At level (6,12)we found 33 Q-closed states with M2 < 20, of whichthe lightest has M2 = 0.85562. The complete set ofQ-closed states we found 3 is graphed in Fig. 1.

    To test the Q-exactness of a given Q-closed stateat level (L, I), we computed Q(L,I )0 i for each ghostnumber zero field i with level L. The span ofthe fields Q(L,I )0 i gives an approximation to thesubspace of Q-exact states at each level. Suppose that{ei} is an orthonormal basis for this subspace and sis one of the Q-closed states we found. We can thenmeasure the extent to which a state is Q-exact bythe norm squared of its projection onto the Q-exactsubspace. Explicitly,

    (16)fraction in exact subspace =

    i (s ei)2s s .

    There is no natural positive definite inner productdefined on the single string Hilbert space H, 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 the Q-closedstate at M2 = 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 with Q-closedstates proceeded by calculating the determinant of Q(L,I)1 at equallyspaced values of p (with p = 0.0001) and looking for changesof sign in the determinant. The spacing of our p values wassignificantly less than the smallest distance we observed betweenQ-closed states (0.0039), so we believe that we have found all t...