The phantom term in open string field theory

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  • JHEP06(2012)084

    Published for SISSA by Springer

    Received: February 14, 2012

    Accepted: May 18, 2012

    Published: June 14, 2012

    The phantom term in open string field theory

    Theodore Erlera and Carlo Maccaferria,b

    aInstitute of Physics of the ASCR, v.v.i.,

    Na Slovance 2, 182 21 Prague 8, Czech RepublicbINFN Sezione di Torino,

    Via Pietro Giuria 1, I-10125 Torino, Italy

    E-mail: tchovi@gmail.com, maccafer@gmail.com

    Abstract: We show that given any two classical solutions in open string field theory and

    a singular gauge transformation relating them, it is possible to write the second solution

    as a gauge transformation of the first plus a singular, projector-like state which describes

    the shift in the open string background between the two solutions. This is the phantom

    term. We give some applications in the computation of gauge invariant observables.

    Keywords: Tachyon Condensation, String Field Theory

    ArXiv ePrint: 1201.5122

    c SISSA 2012 doi:10.1007/JHEP06(2012)084

  • JHEP06(2012)084

    Contents

    1 Introduction 1

    2 The phantom term 2

    3 Relation to Schnabls phantom term 4

    4 Example 1: Identity-like marginals 5

    5 Example 2: Energy for Schnabls solution 7

    6 Example 3: Tadpole shift between two marginals 10

    1 Introduction

    Perhaps the most mysterious aspect of Schnabls analytic solution for tachyon conden-

    sation [1] is the so-called phantom term a singular and formally vanishing term in the

    solution which appears to be solely responsible for the disappearance of the D-brane. Much

    work has since shed light on the term, either specifically in the context of Schnabls solu-

    tion [13], or in some generalizations [49], but so far there has been limited understanding

    of why the phantom term should be present.

    In this paper we show that the phantom term is a consequence of a particular and

    generic property of string field theory solutions: Given any two solutions 1 and 2, it is

    always possible to find a ghost number zero string field U satisfying

    (Q+1)U = U2. (1.1)

    This is called a left gauge transformation from 1 to 2 [10]. Under some conditions,1

    the existence of U implies that 2 can be expressed as a gauge transformation of 1 plus

    a singular projector-like state which encapsulates the shift in the open string background

    between 1 and 2. This is the phantom term. The phantom term is proportional to a star

    algebra projector called the boundary condition changing projector, which is conjectured

    to describe a surface of stretched string connecting two BCFTs [10]. One consequence of

    this description is that phantom terms, in general, do not vanish in the Fock space, as is

    the case for Schnabls solution.

    This paper can be viewed as a companion to reference [10], to which we refer the

    reader for more detailed discussion of singular gauge transformations and boundary condi-

    tion changing projectors. Our main goal is to show how the phantom term can be used to

    1Specifically, U should have the property that it defines a boundary condition changing projector, which

    means that as a formal linear operator its image and kernel should be linearly independent and span the

    whole space [10].

    1

  • JHEP06(2012)084

    calculate physical observables, even for solutions where the existence of a phantom term

    was not previously suspected. We give three examples: The closed string tadpole ampli-

    tude [11] for identity-like marginal deformations [12]; the energy for Schnabls solution [1];

    and the shift in the closed string tadpole amplitude between two Schnabl-gauge marginal

    solutions [13, 14]. The last two computations reproduce results which have been obtained in

    other ways [1, 15], but our approach brings a different perspective and some simplifications.

    This description of the phantom term will be useful for the study of future solutions.

    2 The phantom term

    To start, lets review some concepts and terminology from [10]. Given a pair of classical

    solutions 1 and 2, a ghost number zero state U satisfying

    (Q+1)U = U2 (2.1)

    is called a left gauge transformation from 1 to 2. If U is invertible, then 1 and 2 are

    gauge equivalent solutions. However U does not need to be invertible. In this case, we say

    that the left gauge transformation is a singular gauge transformation.

    It is always possible to relate any pair of solutions by a left gauge transformation.

    Given a ghost number 1 field b, we can construct a left gauge transformation from 1 to2 explicitly with the formula:

    U = Qb+1b+ b2

    = Q12b. (2.2)

    Here, Q12 is the shifted kinetic operator for a stretched string between the solutions 1and 2. Equation (2.2) is not necessarily the most general left gauge transformation from

    1 to 2. This depends on whether Q12 has cohomology at ghost number zero [10].

    Even when U is not invertible, it often happens that U + is invertible for any suffi-

    ciently small > 0. However this raises a question: If U + is invertible, how is it possible

    that 1 and 2 are not automatically gauge equivalent in the 0 limit? The answer tothis question is contained in the following equation:

    2 =1

    + U

    [Q+1

    ](+ U)

    1()

    +

    + U(2 1)

    12()

    , (2.3)

    Computing the right hand side using (2.1), this equation is easily seen to be an identity.

    The first term 1() is a gauge transformation of 1, and the second term 12() is a

    remainder. Apparently, if 1 and 2 are not gauge equivalent, the remainder must be

    nontrivial in the 0 limit:

    lim0+

    12() = X(2 1). (2.4)

    2

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    This is the phantom term. Here we must assume that the 0 limit of +U in some senseconverges, defining the star algebra projector

    X = lim0+

    + U. (2.5)

    called the boundary condition changing projector [10]. The boundary condition changing

    projector is a subtle object, and is responsible for some of the mystery of the phantom

    term. Based on formal arguments and examples, it was argued in [10] that the boundary

    condition changing projector represents a surface of stretched string connecting the BCFTs

    of 2 and 1.

    The identity (2.3) offers a potentially useful method for computing gauge invariant

    quantities from classical solutions. The basic idea is that, with a judicious choice of U , the

    pure gauge term can be arranged to absorb all of the gauge-redundant artifacts of the

    solution, leaving the phantom term to to describe the shift in the open string background

    in a transparent manner. In later examples we will compute the on-shell action and the

    closed string tadpole [11]:

    S[] = 16Tr[Q

    ], A(V) = TrV []. (2.6)

    where is a tachyon vacuum solution and A(V) is the disk amplitude for a single on-shellclosed string V = ccVmatter to be emitted from the BCFT of .2 The change in the actionor tadpole between solutions 1 and 2 can be computed using the phantom term:

    S[2] S[1] = 16Tr[12()Q12()

    ] 1

    3Tr[12()Q2

    ](2.7)

    A2(V)A1(V) = TrV[12()

    ]. (2.8)

    These equations are exact for any , though they will be most useful in the 0 limit.It would be interesting to see whether the phantom term can also be used to compute the

    boundary state [17].

    The phantom term defined here is useful only if U+ is invertible3 and the 0 limitsensibly defines a boundary condition changing projector which simplifies the calculation

    of observables. The latter might not occur if the boundary condition changing projector

    does not exist (a possibility explained in [10]) or if it is degenerate. For example, when

    U = 0 the projector in the phantom term is the identity string field, and the resulting

    decomposition (2.3) into a pure gauge and phantom piece,

    2 = 11()

    + (2 1) 12()

    , (2.9)

    2We use Tr[] to denote the 1-string vertex (Witten integral) and TrV [] to denote the 1-string vertex

    with a midpoint insertion of V [11].3For example, U = Q(BK) = K2 is a singular gauge transformation from the perturbative vacuum to

    itself, but the inverse of K2 + would be defined as an integral over wedge states which at best converges

    in a distributional sense. In this case the identity (2.3) should be treated with care, or it may be more

    convenient to either make a different choice of U or a different type of splitting between pure gauge and

    phantom terms, analogous to, but different from, (2.3). An example of this is discussed in the next section.

    3

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    is too trivial to be useful. The examples we have analyzed concern a choice of U derived

    from b of the form

    b = Bf(K) (2.10)

    where f(K) is a state in the wedge algebra with f(K) > 0. (See [2, 18] and appendix A

    of [8] for definitions of K,B, c and the algebraic notation we use here.) In this case, we

    have always found that the phantom term is proportional to the sliver state, and the 0limit simplifies upon application of the L level expansion4 to fields inside the phantomterm while ignoring subleading levels which do not contribute. We will see how this works

    in later examples. For more exotic choices of U we do not know what form the boundary

    condition changing projector or the phantom term might take, or whether they will be

    useful for analytic calculations of observables.5

    Note that the phantom term is a property of a pair of solutions and a singular gauge

    transformation relating them. In this sense, a solution by itse