Tachyon condensation and open string field theory

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  • 15 November 2001

    Physics Letters B 520 (2001) 385390www.elsevier.com/locate/npe

    Tachyon condensation and open string field theoryTaejin Lee

    Department of Physics, Kangwon National University, Chuncheon 200-701, South KoreaReceived 20 August 2001; accepted 9 September 2001

    Editor: T. Yanagida

    Abstract

    We perform canonical quantization of open string on a unstable D-brane in the background of tachyon condensation.Evaluating the Polyakov path-integral on a strip, we obtain a field theoretical propagator in the open string theory. Ascondensation occurs the string field theory is continuously deformed. At the infrared fixed point of condensation, the openstring field on the unstable D-brane transmutes to that on the lower-dimensional D-brane with the correct D-brane tension. 2001 Elsevier Science B.V. All rights reserved.

    PACS: 11.25.Sq; 11.25.-w; 04.60.Ds

    1. Introduction

    Tachyon condensation is a noble phenomenon instring theory, which determines the ultimate fates ofthe unstable D-branes and the DD-brane pairs. Theunstable systems in string theory are expected to re-duce to stable lower-dimensional D-brane systems ordisappear into vacuum, leaving only the closed stringspectrum behind. Since the celebrated Sens conjec-ture [1] on the tachyon condensation many importantaspects of this noble phenomena have been exploredby numerous authors. Since tachyon condensation isan off-shell phenomenon, the theoretical framework todeal with it should be the second quantized string the-ory. The main tools to discuss the tachyon condensa-tion are the open string field theory with the level trun-cation [2] and the boundary string field theory [3,4].The former one, which is based on the Wittens cubicopen string field theory [5], has been a useful practi-

    E-mail address: taejin@cc.kangwon.ac.kr (T. Lee).

    cal tool to describe the decay of the unstable D-branesto the bosonic string vacuum. The latter one, whichis based on the background independent string fieldtheory, has been useful to obtain the effective tachyonpotential. These two approaches are considered to becomplementary to each other.

    In a recent paper [6] we discuss the tachyon con-densation in a single D-brane, using the boundary stateformulation [7,8], which is closely related to the latterone. As we point out, the boundary state formulationcontains the boundary string field theory, since the nor-malization factor of the boundary state corresponds tothe disk partition function, which is the main ingredi-ent of the latter approach. Moreover, it provides an ex-plicit form of the quantum state of the system in termsof the closed string wavefunction. Thus, we may finda direct connection between the boundary state for-mulation and the former approach based on the stringfield theory if we appropriately utilize the openclosedstring duality. It suggests that the succinct boundarystate formulation of the tachyon condensation maybe transcribed into the open string field theory. The

    0370-2693/01/$ see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01) 01 13 2- 7

  • 386 T. Lee / Physics Letters B 520 (2001) 385390

    purpose of this Letter is to construct the open stringfield theory in the background of the tachyon conden-sation and to show that the descent relations amongthe D-branes is also well described in the frameworkof the open string field theory. To this end we per-form canonical quantization [9] of the open string ona unstable D-brane in the background of the tachyoncondensation. Then we evaluate the Polyakov stringpath-integral on a strip to obtain the field theoreticalpropagator of the open string in the background of thetachyon condensation. At the infrared fixed point, theopen string field on the unstable D-brane transmutesto that on the lower dimensional D-brane with the cor-rection D-brane tension.

    2. Canonical quantization

    It is well known that the field theoretical stringpropagator is obtained from the first quantized stringtheory, by evaluating the Polyakov path-integral overa strip, which is the world-sheet of the open string inthis case. Following the same steps, we will constructthe field theoretical open string propagator in thebackground of the tachyon condensation. To this endwe perform canonical quantization of the open stringattached on D-brane in the tachyon background. Thenintegration over the proper time yields the stringpropagator, therefore the kinetic part of the secondquantized string theory. As we vary the parameter ofthe tachyon profile, the field theoretical action for theopen string on a D-brane is continuously deformedand eventually reduced to that on a lower dimensionalD-brane. For the sake of simplicity we consider thebosonic string on a single D-brane. Extension to moregeneral cases is straightforward.

    The action for the open string in the background ofthe tachyon condensation is given as 1

    S = SM + ST= 1

    4

    M

    d dhhgXX

    1 Note that in the boundary state formulation we take ST =id T (X). The boundary state formulation is related to the

    canonical formulation of the open string theory under discussion bythe double Wick rotation.

    (1)+M

    d NT (X),

    where we consider a simple tachyon profile, T (X) =uijX

    iXj . Here N is an einbein on the world-line ofthe end points of the open string and its relation to theworld-sheet metric is given by

    (2)hh = 1

    N

    (1 00 N2

    ).

    That is, N is the lapse function of the world-sheetmetric. The string action is manifestly invariant underthe reparametrization. We may fix this reparametriza-tion invariance by choosing the proper-time gauge,dN/d = 0, equivalently N = T , constant. Hereafterwe confine our discussion to the proper-time gauge.The string propagator is defined as a Polyakov path-integral over a strip [10]

    (3)G[Xf ;Xi] =D[N]D[X] exp(iSM + iST ),

    where the path integral is subject to the boundarycondition X(f , )=Xf (), X(i, )=Xi ( ).

    In order to understand the structure of the openstring propagator on a D-brane, let us first consider aflat D-brane where ST = 0. Introducing the canonicalmomenta P, we find that the propagator is written as

    (4)

    G[Xf ;Xi] =

    0

    dT

    D[X,P ] ei

    T0(

    PXdH )d .

    If we expand the canonical variables in terms ofnormal modes

    X()=n

    Xn ein ,

    P()=n

    Pnein ,

    we find that the Hamiltonian is given as

    (5)

    H = 12n

    g

    ((2)Pn P n +

    n2

    (2)Xn X

    n).

    For the open string on a Dp-brane in d dimensions,we need to impose the Neumann boundary conditionfor Xi and the Dirichlet boundary condition for Xa ;X

    i |M = 0, Xa |M = 0, where i = 0,1, . . . , p,

  • T. Lee / Physics Letters B 520 (2001) 385390 387

    a = p + 1, . . . , d 1. These boundary conditionsresult in the following constraints:

    (6)Xin Xin = 0, P in P in = 0,(7)Xan +Xan = 0, P an + Pan = 0,

    xa = 0, pa = 0,for n = 1,2, . . . . Thus, the canonical variables arewritten as

    Xi = xi +2n=1

    Y in cosn,

    P i = pi +2n=1

    Kin cosn,

    Xa =2n=1

    Y an sinn, P a =

    2n=1

    Kan sinn,

    where (Yn,Kn) and (Yn, Kn) form canonical pairs,

    Yn =12(Xn +Xn

    ), Yn =

    i2(Xn Xn

    ),

    Kn =12(Pn + Pn

    ),

    Kn =i2(Pn Pn

    ).

    The procedure given above is equivalent to reducing afree closed string to an open string on the Dp-braneby imposing an orbifold condition:

    Xi()=Xi(), Xa( )=Xa(),P i( )= P i(), P a( )=Pa().If these constraints are imposed, the Hamiltonian isread as

    H = 12gij (2)pipj

    + 12n=1

    gij

    {(2)KinK

    jn + n

    2

    (2)Y inY

    jn

    }

    (8)

    + 12n=1

    gab

    {(2)Kan Kbn +

    n2

    (2)Y an Y bn

    }.

    Now the field theoretical propagator follows fromintegrating over the proper-time

    G[Xf ;Xi] =

    0

    dT Xf |eiT H |Xi

    = i

    D[][Xf ][Xi]

    (9) exp(iD[X][X]K[X]

    ),

    where [X] = [xi, Y i,Y a] and K = H . Hence, theHamiltonian in the first quantized theory correspondsto the kinetic operator for string field in the secondquantized theory.

    3. Background of tachyon condensation

    The background of the tachyon condensation altersthe boundary conditions for the open string on theD-brane. In order to have consistent equations ofmotion from the action Eq. (1) we need to impose thefollowing boundary conditions on M

    (10)( 1

    2gij X

    j + 2uijXj)

    == 0,

    (11)(

    12

    gij Xj + 2uijXj

    )=0

    = 0.

    If we rewrite these boundary conditions in terms ofnormal modes, we get

    (12)n

    nXin + i(2)2(g1u

    )ij

    n

    Xjn = 0,

    n

    nXin(1)n

    (13) i(2)2(g1u)i jn

    Xjn(1)n = 0.

    In the framework of the canonical quantization wetreat them as primary constraints. Let us denote thefirst constraint Eq. (12) as a primary constraint i0

    (14)i0 =n

    (nI + i(2)2g1u)i j Xjn = 0.

    Then the commutator of the primary constraint withthe Hamiltonian generates a secondary constraint i0,which is conjugate to the primary constraint i0

    (15)i0 =n

    (nI 2i(2)ug1)

    ijPjn = 0.

    The Dirac procedure requires further {H,i0} = 0and it generates yet another constraint i1. We maycontinue this procedure until it does not generates

  • 388 T. Lee / Physics Letters B 520 (2001) 385390

    additional new constraints. Repeating it we obtain acomplete set of constraints

    (16)im =

    n

    (n2m+1I + 2i(2)n2mg1u)i j Xjn = 0,

    (17)im =

    n

    (n2m+1I 2i(2)n2mug1)

    ij Pjn = 0,

    where m= 0,1,2, . . . . All these constraints belong tothe second class. We may apply the same procedure tothe primary constraint Eq. (13), but we only get a setof constraints equival