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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

JHEP06(2012)084

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

JHEP06(2012)084

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