Prog. Theor. Phys. 2012 Okawa 1001 60

INVITED PAPERS 1001

Progress of Theoretical Physics, Vol. 128, No. 6, December 2012

Analytic Methods in Open String Field Theory

Yuji Okawa∗)

Institute of Physics, The University of Tokyo, Tokyo 153-8902, Japan

(Received October 29, 2012)

We review the basics of recent developments of analytic methods in open string fieldtheory. We in particular explain Schnabl’s analytic solution for tachyon condensation indetail, assuming only the basic knowledge on conformal field theory.

Subject Index: 128

Contents

1. Introduction 10012. The basics of open string field theory 1003

2.1. Construction of the free theory . . . . . . . . . . . . . . . . . . . . . 10032.2. The interacting theory . . . . . . . . . . . . . . . . . . . . . . . . . . 10062.3. CFT description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10092.4. BPZ inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10142.5. Star product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019

3. Tachyon condensation 10213.1. Approximate solutions in level truncation . . . . . . . . . . . . . . . 10213.2. Sliver frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10243.3. Wedge state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10263.4. Derivatives of the wedge state . . . . . . . . . . . . . . . . . . . . . 10273.5. Eigenstates of the operator L . . . . . . . . . . . . . . . . . . . . . . 10303.6. Schnabl gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10343.7. Schnabl’s analytic solution . . . . . . . . . . . . . . . . . . . . . . . 10383.8. KBc subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10433.9. Phantomless solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

4. Analytic solutions for marginal deformations 10514.1. Marginal deformations with regular operator products . . . . . . . . 10514.2. Solutions for general marginal deformations . . . . . . . . . . . . . . 1055

5. Future directions 1057

§1. Introduction

One of the most important problems in theoretical physics today is to construct aconsistent quantum theory including gravity. String theory is expected to provide animportant clue to this problem, but its current formulation is not yet complete. First,it is defined only perturbatively with respect to a coupling constant, and we do not

∗) E-mail: [email protected]

at Akadem

ia Rolnicza on July 3, 2014

http://ptp.oxfordjournals.org/D

ownloaded from

http://ptp.oxfordjournals.org/

1002 Y. Okawa

expect that the perturbative expansion has a finite radius of convergence. Second,only on-shell scattering amplitudes can be calculated and off-shell quantities such ascorrelation functions are not well defined in string theory. Third, since string theoryis independently defined for each consistent background, it is not clear whether thereis a universal set of degrees of freedom describing different backgrounds. We couldsay that we have not yet identified fundamental degrees of freedom of string theory.

One possible approach to these problems is to construct a field theory of strings,which is called string field theory. While it is a natural approach in view of thesuccess of quantum field theory, it is possible, and perhaps likely, that string theorywill turn out to be formulated nonperturbatively in a completely different way. Forexample, there has been a long history of developments in formulating string theoryusing matrix models. We also learned from the AdS/CFT correspondence that stringtheory on a class of backgrounds can be defined in terms of ordinary field theorywithout gravity. The quantum formulation of string field theory by path integraldoes not seem to be the best candidate for the final formulation of string theory.

On the other hand, string field theory can play a role complementary to otherapproaches in our quest for a consistent formulation of string theory. While stringtheory as a perturbation theory can only explore an infinitesimal neighborhood ofeach consistent background, we can in principle discuss the relationship of variousbackgrounds using string field theory. String field theory can also be thought of as auniversal, effective theory when elementary excitations are string-like. To eliminateghosts from such string-like excitations in unphysical directions, any covariant de-scription in terms of a spacetime field theory would require gauge invariance. Thenthe gauge invariance seems to determine the interacting theory uniquely. This hasbeen a guiding principle in constructing covariant string field theory. Since consis-tent backgrounds of string theory are described by conformal field theories in twodimensions, the classical equation of motion of string field theory determined by thespacetime gauge invariance should therefore reproduce this requirement of confor-mal invariance in the world-sheet perspective. We hope that deeper understanding ofthe relation between the world-sheet conformal invariance and the spacetime gaugeinvariance will help us reveal aspects of the non-perturbative theory behind the per-turbative string theory.

Recently there have been impressive developments in the research of open stringfield theory since the construction of an analytic solution by Schnabl.1) This review isintended to provide the gateway to this exciting area of research without assumingmuch background.∗) In fact, it is not even necessary to have detailed knowledgeon string theory to appreciate the recent analytic development in open string fieldtheory. On the other hand, the basic knowledge of conformal field theory (CFT) isrequired, but it is sufficient to be familiar with the contents explained in Chapter 2 ofthe textbook by Polchinski.3) This review will be understood without much difficultyafter learning the chapter, and we mostly follow the conventions of the textbook.

∗) For an exhaustive list of references, see the review by Fuchs and Kroyter.2)

at Akadem



ownloaded from


Analytic Methods in Open String Field Theory 1003

§2. The basics of open string field theory

2.1. Construction of the free theory

Quantum field theory is defined by an action such as

S =∫d4x

[−1

4FμνF

μν + iψD/ψ + . . .]. (2.1)

If the coupling is weak, we can use perturbation theory. Perturbation theory foron-shell scattering amplitudes takes a form of an expansion in terms of Feynmandiagrams. String theory is perturbation theory for on-shell scattering amplitudeswith particles in Feynman diagrams replaced by strings. It provides a consistentperturbation theory including gravity. However, we generally do not know wherethis perturbation theory comes from. As we mentioned in the introduction, stringfield theory is one natural approach, and it is formulated such that the spectrum ofstring theory is reproduced from an action with gauge invariance.

Let us consider open bosonic string theory on a flat spacetime in 26 dimensionswith Neumann boundary conditions. When the open string is in the ground stateof the internal oscillators, it behaves like a tachyonic scalar particle of mass m givenby

m2 = − 1α′ , (2.2)

where the constant α′ is related to the string tension T by

T =1

2πα′ . (2.3)

The open string in the ground state is thus described by a tachyonic scalar field T (k),where we write it in momentum space. When the open string is in the first-excitedstates, it behaves like a massless particle described by a massless vector field Aμ(k).When the open string is in a higher-excited state, it behaves like a massive particlewith mass m given by

m2 =1α′ ,

2α′ ,

3α′ , . . . . (2.4)

Therefore, one way to think about the degrees of freedom of string field theory is toconsider a set of an infinite number of spacetime fields {T (k) , Aμ(k) , . . . }, and theaction will be a functional of these fields: S[T (k) , Aμ(k) , . . . ].

Let us try to construct such an action for the free theory. First construct anaction for T (k) and Aμ(k). It should be

S = −∫

d26k

(2π)26

[12T (−k)

(k2 − 1

α′)T (k) +

12Aμ(−k) ( k2ημν − kμkν )Aν(k)

].

(2.5)Or this action can also be written as

S = −∫

d26k

(2π)26

[12T (−k)

(k2 − 1

α′)T (k) +

12Aμ(−k) k2Aμ(k)

+ iB(−k) kμAμ(k) +12B(−k)B(k)

](2.6)

at Akadem



ownloaded from


1004 Y. Okawa

by introducing an auxiliary field B(k). The equations of motion derived from thesecond form of the action are(

k2 − 1α′

)T (k) = 0 , k2Aμ(k)− ikμB(k) = 0 , B(k) + ikμAμ(k) = 0 . (2.7)

If we eliminate B(k) using the last equation, we recover the first form of the action.The two actions are therefore equivalent. The second form of the action is invariantunder the gauge transformations given by

δΛAμ(k) = ikμΛ(k) , δΛB(k) = k2Λ(k) . (2.8)

We could in principle proceed to the massive fields and construct the free actionof those fields, but it would be formidably complicated. Actually, the action in thesecond form (2.6), the equations of motion (2.7), and the gauge transformations (2.8)can be written compactly by introducing an object called string field. Just as theSU(2) gauge fields Aa

μ(x) with a = 1, 2, 3 can be incorporated into a single 2 × 2matrix field Aμ(x) using the Pauli matrices σa as

Aμ(x) =12

3∑a=1

Aaμ(x)σa , (2.9)

the component fields {T (k) , Aμ(k) , . . . } can be incorporated into a string field Ψ ,which is a state in a boundary CFT in two dimensions. For the open string on aflat spacetime in 26 dimensions with Neumann boundary conditions, the Fock spaceof the boundary CFT is constructed by the bosonic operators αμ

n and the fermionicoperators bn and cn satisfying

[αμn, α

νm ] = nημνδn+m,0 , { bn, cm } = δn+m,0 (2.10)

with the state | 0; k 〉 defined by

αμn| 0; k 〉 = 0 for n > 0 ,bn| 0; k 〉 = 0 for n > −2 ,cn| 0; k 〉 = 0 for n > 1 ,pμ| 0; k 〉 = kμ| 0; k 〉 . (2.11)

The component fields T (k), Aμ(k), and B(k) are incorporated into Ψ as follows:

Ψ =∫

d26k

(2π)26

[1√α′ T (k) c1| 0; k 〉+ 1√

α′ Aμ(k)αμ−1c1| 0; k 〉+ i√

2B(k) c0| 0; k 〉

].

(2.12)The equations of motion can be written compactly as

QB Ψ = 0 , (2.13)

where the BRST operator QB is given by

QB =∞∑

n=−∞cnL

(m)−n +

12

∞∑n=−∞

∞∑m=−∞

(m− n) ◦◦ cmcnb−m−n

◦◦−c0 . (2.14)

at Akadem



ownloaded from



We denote the creation-annihilation normal ordering of O by ◦◦O ◦◦. The creation-annihilation normal ordering for the bc oscillators is defined by

◦◦ cnb−n

◦◦ =

{cnb−n for n ≤ 0 ,− b−ncn for n > 0 , (2.15)

and the Virasoro operators in the matter sector L(m)n are given by

L(m)n =

12

∞∑m=−∞

◦◦ α

μmα

νn−m

◦◦ ημν with αμ

0 =√

2α′ pμ . (2.16)

The gauge transformations can also be written compactly as

δΛΨ = QBΛ (2.17)

with

Λ =i√2α′

∫d26k

(2π)26Λ(k) | 0; k 〉 . (2.18)

The action can be written as

S = −12〈Ψ,QBΨ 〉 (2.19)

using the BPZ inner product. We will give a general definition of the BPZ innerproduct based on the CFT description later, but for the current particular casethe BPZ inner product 〈A,B 〉 can be defined as follows. First, we associate a bra〈A | with each state |A 〉. Its rule is defined recursively. For the state | 0; k 〉, thecorresponding bra is 〈 0; k |:

| 0; k 〉 → 〈 0; k | . (2.20)

Note that this is different from the usual Hermitian conjugation where 〈 0;−k | isassociated with | 0; k 〉. Then for the state O |A 〉, the corresponding bra is associatedas follows:

O |A 〉 →{ 〈A | O� when O is Grassmann even ,

(−1)A 〈A | O� when O is Grassmann odd ,(2.21)

where the BPZ conjugation O� of the operator O is defined by

(αμn)� = (−1)n+1αμ

−n , (bn)� = (−1)nb−n , (cn)� = (−1)n+1c−n . (2.22)

Here and in what follows a string field in the exponent of −1 denotes its Grassmannproperty: it is 0 mod 2 for a Grassmann-even string field and it is 1 mod 2 for aGrassmann-odd string field. It follows from this definition that

(O1O2)� ={ −O�

2 O�1 when both O1 and O2 are Grassmann odd ,

O�2 O�

1 when at least one of O1 and O2 is Grassmann even .(2.23)

at Akadem



ownloaded from


1006 Y. Okawa

The bra 〈A | for an arbitrary state |A 〉 has now been defined. Then the contraction〈A |B 〉 can be reduced to 〈 0; k |c−1c0c1| 0; k′ 〉 using the commutation and anticom-mutation relations in (2.10) as well as the definition (2.11) of | 0; k 〉. We define

〈 0; k |c−1c0c1| 0; k′ 〉 = (2π)26δ(26)(k + k′) . (2.24)

Now that the contraction 〈A |B 〉 for an arbitrary pair A and B is determined, theBPZ inner product 〈A,B 〉 is defined by

〈A,B 〉 = 〈A |B 〉 . (2.25)

It follows from the definition that

〈OA,B 〉 ={ 〈A,O�B 〉 when O is Grassmann even ,

(−1)A〈A,O�B 〉 when O is Grassmann odd . (2.26)

Important properties of the BRST operator and the BPZ inner product are

〈A,B 〉 = (−1)AB〈B,A 〉 , 〈QBA,B 〉 = −(−1)A 〈A,QBB 〉 , Q2B = 0 . (2.27)

The first property is not manifest in the definition we just presented, but we willlater give a more general definition of the BPZ inner product and prove this property.The second property can also be stated as Q�

B = −QB . Namely, the BRST operatoris BPZ odd. The nilpotency of the BRST operator, Q2

B = 0, can be confirmedusing the commutation and anticommutation relations in (2.10), but we will givea definition of the BRST operator for a general boundary CFT and the propertiesQ�

B = −QB and Q2B = 0 hold for any boundary CFT with central charge c = 26.

In this description using the string field, it is straightforward to generalize theconstruction of the free action to the massive fields. We generalize the string field Ψin (2.12) by incorporating all the states of ghost number 1, where the ghost numberis defined by the number of c oscillators minus the number of b oscillators. We thenintroduce spacetime fields as coefficients in front of these states. The action (2.19)constructed from the resulting string field is guaranteed to be gauge invariant becauseof the properties (2.27).

At the end of this subsection, let us briefly discuss gauge fixing. One possiblegauge-fixing condition of the action (2.6) is to set B(k) = 0. In terms of the stringfield Ψ in (2.12), this condition can be stated as b0 Ψ = 0. In fact, this conditionworks as a gauge-fixing condition for the whole string field and is called the Siegelgauge condition.

2.2. The interacting theory

We have completed the construction of the free action for string field theory. Letus next consider the interacting theory. An important point of string perturbationtheory is that there are no Lorentz-invariant interaction points. This means thatthe form of the interactions is uniquely determined for a given free theory. This iscrucially different from ordinary field theories where many interacting theories arepossible for a given free theory. In the context of string field theory, we look foran action for the interacting theory which is invariant under nonlinearly extended

at Akadem



ownloaded from



Fig. 1. Witten’s star product.

gauge transformations. Just as the gauge transformation of the free U(N) Yang-Mills theory δAμ = ∂μΛ is nonlinearly extended in the interacting theory as δAμ =∂μΛ + i(ΛAμ − AμΛ) and the Yang-Mills action is invariant under this extendedtransformation, we look for a nonlinear extension of the gauge transformation δΛΨ =QBΛ of the free theory and an action which is invariant under the nonlinear gaugetransformation. Such an action was constructed by Witten.4) Witten introduced aproduct A ∗B of string fields A and B called Witten’s star product and constructedthe following action:

S = − 1α′ 3g2

T

[12〈Ψ,QBΨ 〉+ 1

3〈Ψ, Ψ ∗ Ψ 〉

], (2.28)

where gT is the open string coupling constant.∗) The string field is a state in aboundary CFT and is defined on a segment. This is not manifest in the representationusing the oscillators, but we will later make it manifest in a different representation.The star product A ∗ B is then defined by taking the BPZ inner product for theright half of the string field A and the left half of the string field B. The resultingstate defined on the combined segment for the left half of A and the right half of Bis A ∗ B. See Fig. 1. Again from our definition of the BPZ inner product using theoscillators it is not clear how to take the BPZ inner product only for a part of thestates, but we will explain it later using a different representation. For the moment,it is sufficient to have a rough idea depicted in Fig. 1.

The star product is noncommutative,

A ∗B �= B ∗ A , (2.29)

and there is a useful analogy with the matrix multiplication. Let us formally associatethe left half of the string field with the left index of the matrix and the right half ofthe string field with the right index of the matrix:

A←→ Aij . (2.30)∗) With our definitions of the BPZ inner product and the star product, The open string coupling

constant gT here is related to the coupling constant go and g′o in Ref. 3) as

gT = g′o

r2

α′ =go

α′ .

at Akadem



ownloaded from


1008 Y. Okawa

Then the star product A ∗B corresponds to the matrix multiplication,

A ∗B ←→ (AB)ij = AikBkj , (2.31)

and the BPZ inner product 〈A,B 〉 corresponds to the trace of AB:

〈A,B 〉 ←→ trAB = AijBji . (2.32)

We will give a precise definition of the star product later, but the gauge invarianceof the action follows only from the properties (2.27) and the relations

〈A,B ∗ C 〉 = 〈A ∗B,C 〉 ,QB(A ∗B) = QBA ∗B + (−1)AA ∗QBB . (2.33)

The first relation corresponds to the associativity of the matrix multiplication,

trA(BC) = tr (AB)C = AijBjkCki , (2.34)

and the second relation is the derivation property of the BRST operator with respectto the star product, which corresponds to the derivation property of the ordinaryderivative with respect to the matrix multiplication:

d

dx

(Aij(x)Bjk(x)

)=dAij(x)dx

Bjk(x) +Aij(x)dBjk(x)dx

. (2.35)

The equation of motion derived from the action is

QBΨ + Ψ ∗ Ψ = 0 . (2.36)

The nonlinearly extended gauge transformation is

δΛΨ = QBΛ+ Ψ ∗ Λ− Λ ∗ Ψ , (2.37)

and the action is invariant under this gauge transformation:

δΛS = 0 . (2.38)

The invariance can be shown only using the following relations:

〈A,B 〉 = (−1)AB〈B,A 〉 , 〈QBA,B 〉 = −(−1)A 〈A,QBB 〉 , Q2B = 0 ,

〈A,B ∗ C 〉 = 〈A ∗B,C 〉 , QB(A ∗B) = QBA ∗B + (−1)AA ∗QBB . (2.39)

The proof is algebraically the same as that for the Chern-Simons theory in threedimensions, and the action is often called the Chern-Simons-like action.

We will define the BRST operator QB, the BPZ inner product, and the starproduct for general boundary CFT, and the relations (2.39) hold for any boundaryCFT with the central charge c = 26 in the matter sector. We can therefore constructa gauge-invariant action of open string field theory for any given boundary CFT withc = 26 in the matter sector.

at Akadem



ownloaded from



Fig. 2. The strip coordinates.

Fig. 3. The upper half-plane. The shaded region of Fig. 2 in the strip coordinate w is mapped to

the shaded region of this figure in the coordinate z by the conformal transformation z = ew.

2.3. CFT description

The string field of open string field theory is a state in a boundary CFT intwo dimensions. In the strip coordinates (τ, σ) with the range 0 ≤ σ ≤ π and−∞ < τ < ∞, the state is defined on a segment with the same value of τ such asτ = 0. See Fig. 2. CFT is invariant under conformal transformations:

z = x+ iy → z′ = x′ + iy′ = f(z) . (2.40)

The complex strip coordinate w = τ + iσ can be mapped to the coordinate z of theupper half-plane (UHP) by the conformal transformation given by

z = ew . (2.41)

The coordinate z of the upper half-plane is useful because we can use the state-operator correspondence to represent states in the Fock space. The segment withτ = 0 in the strip coordinates is mapped to the semi-circle at |z| = 1 in the upperhalf-plane so that the state is defined on this semi-circle. Since z → 0 as τ → −∞,specifying an initial condition at τ = −∞ in the strip coordinates can be translatedinto an operator insertion at z = 0 in the upper half-plane. We denote the operatorat the origin of the coordinate z corresponding to the state |ϕ〉 in the Fock space byϕ(0):

|ϕ〉 ←→ ϕ(0) . (2.42)

Namely, the state |ϕ〉 can be represented as the state we obtain at the semi-circle|z| = 1 in the upper half-plane by path integral over the region |z| < 1 in the upperhalf-plane with an insertion of ϕ(0). See Fig. 3.

The matter sector of the boundary CFT depends on the background we choseto formulate string field theory. For the open bosonic string in 26-dimensional flatspacetime, the matter sector consists of 26 scalar fields Xμ(z, z) with the equationsof motion given by

∂∂Xμ(z, z) = 0 , (2.43)

at Akadem



ownloaded from


1010 Y. Okawa

where

∂ =∂

∂z, ∂ =

∂

∂z. (2.44)

The ghost sector of the boundary CFT consists of the b ghost and the c ghost whoseequations of motion are

∂b(z) = 0 , ∂b(z) = 0 , ∂c(z) = 0 , ∂c(z) = 0 . (2.45)

On the upper half-plane of z, the boundary conditions imposed on the real axis are

∂Xμ(z) = ∂Xμ(z) , b(z) = b(z) , c(z) = c(z) at �z = 0 . (2.46)

The solutions to the equations of motion satisfying the boundary conditions can beexpanded as follows:

Xμ(z, z) = xμ − iα′pμ ln |z|2 + i

√α′

2

∞∑n=−∞

n �=0

αμn

n

( 1zn

+1zn

),

b(z) =∞∑

n=−∞

bnzn+2

, b(z) =∞∑

n=−∞

bnzn+2

,

c(z) =∞∑

n=−∞

cnzn−1

, c(z) =∞∑

n=−∞

cnzn−1

. (2.47)

We use the doubling trick and define ∂Xμ(z), b(z), and c(z) in the region �z < 0 by

∂Xμ(z) ≡ ∂Xμ(z′) , b(z) ≡ b(z′) , c(z) ≡ c(z′) for �z < 0 , z′ = z . (2.48)

If we write z = x + iy and z′ = x′ + iy′, we can translate �z < 0 and z′ = z intoy < 0 and x′ = x, y′ = −y. Therefore, for example, ∂Xμ(x, y) at (x, y) = (2,−3)is given by ∂Xμ(x′, y′) at (x′, y′) = (2, 3). In terms of the complex coordinates,it is slightly confusing as ∂Xμ(z) at z = x + iy = 2 − 3i is given by ∂Xμ(z′) atz′ = x′ − iy′ = 2− 3i. Using the doubling trick, the modes αμ

n, bn, and cn are givenby

αμn = i

√2α′

∮dz

2πizn ∂Xμ(z) , αμ

0 =√

2α′ pμ ,

bn =∮

dz

2πizn+1 b(z) , cn =

∮dz

2πizn−2 c(z) , (2.49)

where the contour of these integrals runs along the unit circle counterclockwise.In CFT, operator products can be expanded in a complete set of local operators:

Oi(z)Oj(w) =∑

k

ckij(z − w)Ok(w) . (2.50)

at Akadem



ownloaded from



The expansions of the operator products ∂Xμ(z) ∂Xν(w) and b(z) c(w) are given by

∂Xμ(z) ∂Xν(w) = −α′

2ημν 1

(z − w)2+ : ∂Xμ(z) ∂Xν(w) :

= −α′

2ημν 1

(z − w)2+

∞∑k=0

1k!

(z − w)k : ∂k+1Xμ ∂Xν : (w) ,

b(z) c(w) =1

z − w + : b(z) c(w) : =1

z − w +∞∑

k=0

1k!

(z − w)k : (∂kb)c : (w) ,

(2.51)where conformal normal ordering of O is denoted by : O :. The singular part of theoperator product expansion (OPE) is often written as

∂Xμ(z) ∂Xν(w) ∼ −α′

2ημν 1

(z − w)2, b(z) c(w) ∼ 1

z − w . (2.52)

Let us use operator product expansions to identify the state |1〉 correspondingto the unit operator 1 in the state-operator correspondence. The state |1〉 is definedat the semi-circle |z| = 1 in the upper half-plane by path integral over the region|z| < 1 in the upper half-plane without any insertion at the origin. Now considerthe states αμ

n |1〉, bn |1〉, and cn |1〉. Since no operators are inserted at the origin, wefind ∮

dz

2πizn ∂Xμ(z) = 0 for n ≥ 0 ,∮

dz

2πizn+1 b(z) = 0 for n ≥ −1 ,∮

dz

2πizn−2 c(z) = 0 for n ≥ 2 . (2.53)

These are translated into

αμn |1〉 = 0 for n ≥ 0 ,bμn |1〉 = 0 for n ≥ −1 ,cμn |1〉 = 0 for n ≥ 2 (2.54)

so that we identify |1〉 with | 0; 0 〉 ≡ |0〉:|1〉 = | 0; 0 〉 ≡ |0〉 ←→ 1 . (2.55)

We use this identification to fix the normalization of the state-operator correspon-dence. Let us next consider the operator corresponding to the state c1|0〉. Sincec1|0〉 = c1|1〉 and ∮

dz

2πi1zc(z) = c(0) , (2.56)

we find that c(0) is the corresponding operator:

c1|0〉 ←→ c(0) . (2.57)

at Akadem



ownloaded from


1012 Y. Okawa

As an exercise, let us calculate b−1c1|0〉 using the state-operator correspondence.Since the state c1|0〉 corresponds to the operator c(0), the operator corresponding tob−1c1|0〉 can be calculated as follows:∮

dz

2πib(z) c(0) =

∮dz

2πi1z

= 1 , (2.58)

where we used the OPEb(z) c(0) ∼ 1

z. (2.59)

Therefore, we find thatb−1c1|0〉 = |0〉 , (2.60)

which reproduces the result using the anticommutation relation:

b−1c1|0〉 = {b−1, c1}|0〉 = |0〉 . (2.61)

The action of CFT is invariant under appropriate conformal transformationsof operators. Consider a conformal transformation from the coordinate z to thecoordinate z′ given by z′ = f(z). We denote the operator in the coordinate z′

transformed from an operator ϕ(z) in the coordinate z by f ◦ ϕ(z):

ϕ(z)→ f ◦ ϕ(z) . (2.62)

Using this notation, the conformal invariance can be stated as

〈ϕ1(z1) ϕ2(z2) . . . ϕn(zn) 〉Σ = 〈 f ◦ ϕ1(z1) f ◦ ϕ2(z2) . . . f ◦ ϕn(zn) 〉f◦Σ , (2.63)

where we denoted by f ◦Σ the surface mapped from the surface Σ under the confor-mal transformation f(z). A primary field of weight h is an operator which transformsas follows:

O(z)→ f ◦ O(z) =(df(z)

dz

)hO(f(z)) . (2.64)

The operators ∂Xμ(z), b(z), and c(z) are primary fields. Their weights are 1, 2, and−1, respectively:

∂Xμ(z)→ f ◦ ∂Xμ(z) =df(z)dz

∂Xμ(f(z)) ,

b(z)→ f ◦ b(z) =(df(z)

dz

)2b(f(z)) ,

c(z)→ f ◦ c(z) =(df(z)

dz

)−1c(f(z)) . (2.65)

The operator ∂c(z) is not a primary field. Its transformation can be derived fromthat of c(z) by taking a derivative with respect to z:

∂c(z)→ f ◦ ∂c(z) = ∂c(f(z))− f ′′(z)f ′(z)2

c(f(z)) , (2.66)

where

f ′(z) =df(z)dz

, f ′′(z) =d2f(z)dz2

. (2.67)

at Akadem



ownloaded from



Conformal transformations are generated by the energy-momentum tensor:

T (z) ≡ Tzz(z) , T (z) ≡ Tzz(z) , Tzz = 0 . (2.68)

The conservation of the energy-momentum tensor can be stated as

∂T (z) = 0 , ∂T (z) = 0 , (2.69)

and the energy-momentum tensor satisfies the following boundary condition on theupper half-plane:

T (z) = T (z) at �z = 0 . (2.70)

The energy-momentum tensor can be expanded as

T (z) =∞∑

n=−∞

Ln

zn+2, T (z) =

∞∑n=−∞

Ln

zn+2, (2.71)

and the modes Ln are called the Virasoro generators. We use the doubling trick forthe energy-momentum tensor as well:

T (z) ≡ T (z′) for �z < 0 , z′ = z . (2.72)

Then the Virasoro generators are given by

Ln =∮

dz

2πizn+1 T (z) , (2.73)

where the contour of the integral runs along the unit circle counterclockwise. Forthe open bosonic string in 26-dimensional flat spacetime, the matter part of theenergy-momentum tensor T (m) is given by

T (m)(z) = − 1α′ : ∂Xμ ∂Xμ : (z) , (2.74)

and the ghost part of the energy-momentum tensor T (bc) is

T (bc)(z) = : (∂b)c : (z)− 2 ∂(: bc :)(z) . (2.75)

The OPE of the energy-momentum tensor and a primary field O of weight h is

T (z)O(w) ∼ h

(z − w)2O(w) +

1z − w ∂O(w) , (2.76)

and the OPE of the energy-momentum tensor with itself takes the form

T (z)T (w) ∼ c

21

(z − w)4+

2(z − w)2

T (w) +1

z − w ∂T (w) , (2.77)

where c is a constant called the central charge. The central charge of the scalar fieldXμ is 1 for each direction, and the central charge of the bc ghost CFT is −26. Thetotal central charge must vanish for a consistent string background. In this case, the

at Akadem



ownloaded from


1014 Y. Okawa

total energy-momentum tensor is a primary field of weight 2. For a 26-dimensionalspacetime, the total central charge indeed vanishes.

The BRST operator QB is an integral of the BRST current jB(z):

QB =∮

dz

2πijB(z) (2.78)

withjB(z) = cT (m)(z)+ : bc∂c : (z) +

32∂2c(z) . (2.79)

We can show using various OPE’s that

Q2B = 0 (2.80)

when the central charge of the matter sector is 26. We can also confirm that theBRST current is a primary field of weight 1 by calculating the OPE of the BRSTcurrent with the energy-momentum tensor. Under the conformal transformationz′ = f(z), the BRST operator, which is the zero mode of jB(z), is therefore invariant:∮

dz

2πijB(z)→

∮dz′

2πijB(z′) , (2.81)

where the contour should be mapped according to z′ = f(z). This is an importantproperty of an integral of a primary field of weight 1. Another important propertyof the BRST operator is that it transforms the b ghost to the energy-momentumtensor:

QB · b(w) ≡∮

dz

2πijB(z) b(w) = T (w) , (2.82)

where the contour of the integral encircles the point w counterclockwise. The energy-momentum tensor is invariant under the BRST transformation:

QB · T (w) = 0 , (2.83)

which is consistent with the nilpotency: Q2B = 0. These relations can be translated

into the following anticommutation and commutation relations:

{QB, bn} = Ln , [QB, Ln ] = 0 . (2.84)

2.4. BPZ inner product

We are now ready to provide the general definition of the BPZ inner product〈ϕ1, ϕ2 〉 of the states ϕ1 and ϕ2 in the Fock space. We prepare the states ϕ1 andϕ2 in the upper half-plane using the state-operator correspondence. The BPZ innerproduct 〈ϕ1, ϕ2 〉 is defined by the following two-point function:

〈ϕ1, ϕ2 〉 = 〈 I ◦ ϕ1(0)ϕ2(0) 〉UHP , (2.85)

where the conformal transformation I(ξ) is given by

I(ξ) = −1ξ. (2.86)

at Akadem



ownloaded from



Fig. 4. The definition (2.85) of the BPZ inner product. The half unit disk 1 is mapped to the outer

region by the conformal transformation I(ξ), and it is combined with the other half unit disk 2

to form an upper half-plane.

See Fig. 4. In this representation, however, the origin ξ = 0 is mapped to the pointat infinity by I(ξ) so that the insertion point of the operator I ◦ ϕ1(0) is outsidethe coordinate patch. We can make a conformal transformation such that bothoperators are inserted at finite points. For example, we can perform the conformaltransformation h(z) given by

h(z) = −z + 1z − 1

, (2.87)

which maps the upper half-plane to itself. Then the BPZ inner product is expressedas follows:

〈ϕ1, ϕ2 〉 = 〈h ◦ I ◦ ϕ1(0) h ◦ ϕ2(0) 〉UHP . (2.88)

Sinceh ◦ I(ξ) = h(I(ξ)) =

ξ − 1ξ + 1

, (2.89)

the operator h◦I ◦ϕ1(0) is inserted at −1, while h◦ϕ2(0) is inserted at 1. See Fig. 5.Let us now prove the important property of the BPZ inner product

〈ϕ2, ϕ1 〉 = (−1)ϕ1ϕ2〈ϕ1, ϕ2 〉 . (2.90)

Since I(I(ξ)) = ξ, we have

〈ϕ2, ϕ1 〉 = 〈 I ◦ ϕ2(0) ϕ1(0) 〉UHP = 〈 I ◦ I ◦ ϕ2(0) I ◦ ϕ1(0) 〉UHP

= 〈ϕ2(0) I ◦ ϕ1(0) 〉UHP = (−1)ϕ1ϕ2〈 I ◦ ϕ1(0) ϕ2(0) 〉UHP

= (−1)ϕ1ϕ2〈ϕ2, ϕ1 〉 . (2.91)

For a proof of (2.90) based on the expression (2.88), we can use the conformaltransformation

h ◦ I ◦ h−1(z) = −1z, (2.92)

at Akadem



ownloaded from


1016 Y. Okawa

Fig. 5. The representation (2.88) of the BPZ inner product. The half unit disk 1 is mapped to

the left half of the upper half-plane, and the half unit disk 2 is mapped to the right half of the

upper half-plane. The two operators h ◦ I ◦ ϕ1(0) and h ◦ ϕ2(0) in (2.88) are both within this

coordinate patch.

which coincides with I(z) and maps the upper half-plane to itself, to find

〈ϕ2, ϕ1 〉 = 〈h ◦ I ◦ ϕ2(0) h ◦ ϕ1(0) 〉UHP

= 〈h ◦ I ◦ h−1 ◦ h ◦ I ◦ ϕ2(0) h ◦ I ◦ h−1 ◦ h ◦ ϕ1(0) 〉UHP

= 〈h ◦ ϕ2(0) h ◦ I ◦ ϕ1(0) 〉UHP = (−1)ϕ1ϕ2〈h ◦ I ◦ ϕ1(0) h ◦ ϕ2(0) 〉UHP

= (−1)ϕ1ϕ2〈ϕ2, ϕ1 〉 . (2.93)

We explained that the degrees of freedom of string field theory is a set of infinitespacetime component fields, and these component fields are incorporated into thestring field as coefficients in front of complete states of the boundary CFT. How canwe extract these component fields from the string field? We explained the stringfield using the analogy with the matrix field of the SU(2) gauge fields:

Aμ(x) =12

3∑a=1

Aaμ(x)σa . (2.94)

The component fields Aaμ(x) with a = 1, 2, 3 can be extracted from the matrix field

Aμ(x) by multiplying it with σa and taking the trace:

Aaμ(x) = trσaAμ(x) . (2.95)

As can be understood from the analogy between trAB for matrices and 〈A,B 〉 forstring fields, component fields of the string field can be extracted from BPZ innerproducts of the string field with states in the Fock space. Since the dynamicaldegrees of freedom are these component fields, a string field Ψ can be specified by

at Akadem



ownloaded from



giving 〈ϕ, Ψ 〉 for all ϕ in the Fock space. In what follows we use ϕ to denote ageneric state in the Fock space and ϕ(0) to denote its corresponding operator in thestate-operator correspondence, and we will often define a string field Ψ by giving〈ϕ, Ψ 〉. If we need more states in the Fock space, we use ϕ1, ϕ2, ϕ3, . . . and denotetheir corresponding operators in the state-operator correspondence by ϕ1(0), ϕ2(0),ϕ3(0), . . . .

It would be helpful to illustrate the definition of the BPZ inner product with anexample. Consider 〈T,QBT 〉 with T = c1|0〉. The operator corresponding to T isc(0). The operator corresponding to QBT can be calculated as follows. Since

QB · c(w) =∮

dz

2πijB(z) c(w) =

∮dz

2πi: bc∂c : (z) c(w)

=∮

dz

2πi1

z − w c∂c(z) = c∂c(w) , (2.96)

where the contour of the integral encircles the point w counterclockwise, the operatorcorresponding to QBT is c∂c(0). The BPZ inner product 〈T,QBT 〉 can be expressedas

〈T,QBT 〉 = 〈h ◦ I ◦ c(0) h ◦ c∂c(0) 〉UHP . (2.97)

There are two ways to proceed from here. The first one is to calculate the conformaltransformations of c(0) and c∂c(0). The operator c(z) is a primary field of weight −1.How about the operator c∂c(z)? Instead of calculating the conformal transformationof c∂c(0) directly, let us take the second way. Since the BRST operator is invariantunder conformal transformations, we have the following general relation:

f ◦ (QB · ϕ(ξ)) = f ◦[ ∮

dz

2πijB(z)ϕ(ξ)

]=

∮dz′

2πijB(z′) f ◦ ϕ(ξ) (2.98)

with z′ = f(z). Namely, the BRST transformation and the conformal transformationcommute:

f ◦ (QB · ϕ(ξ)) = QB · (f ◦ ϕ(ξ)) . (2.99)

Applying this to the BPZ inner product, we find

〈T,QBT 〉 = 〈h◦I◦c(0) h◦(QB ·c(0)) 〉UHP = 〈h◦I◦c(0)QB ·(h◦c(0)) 〉UHP . (2.100)

Sinceh ◦ c(0) =

12c(1) , h ◦ I ◦ c(0) =

12c(−1) (2.101)

and thusQB · (h ◦ c(0)) =

12QB · c(1) =

12c∂c(1) , (2.102)

the BPZ inner product is given by

〈T,QBT 〉 =14〈 c(−1) c∂c(1) 〉UHP . (2.103)

Actually, the relation (2.99) implies that the BRST transformation of a primaryfield of weight h is a primary field of weight h. Therefore, the operator c∂c(z),

at Akadem



ownloaded from


1018 Y. Okawa

which is the BRST transformation of c(z), is a primary field of weight −1. We canthus reproduce the expression (2.103) by calculating the conformal transformationof c∂c(0) directly. We normalize the correlation function as

〈 c(z1) c(z2) c(z3) 〉UHP, density = (z1 − z2)(z1 − z3)(z2 − z3) . (2.104)

Here and in what follows we use the subscript density to denote quantities dividedby the spacetime volume factor. The BPZ inner product (2.103) is

〈T,QBT 〉density = −1 . (2.105)

Based on the definition (2.85) of the BPZ inner product, let us derive the for-mulas of the BPZ conjugation for the operators αμ

n, bn, and cn in (2.22). Consider aprimary field O(z) of weight h. It is expanded as

O(z) =∞∑

n=−∞

On

zn+h, (2.106)

and the mode On is given by

On =∮

dz

2πizn+h−1O(z) , (2.107)

where the contour of the integral runs along the unit circle counterclockwise. Letus assume, for simplicity, that O(z) is Grassmann even, and consider the BPZ innerproduct 〈ϕ1,Onϕ2 〉. Under the conformal transformation

z′ = −1z, (2.108)

the mode On transforms as

On =∮

dz

2πizn+h−1O(z)→ −

∮dz′

2πidz

dz′(− 1z′

)n+h−1 ( dzdz′

)−hO(z′)

= (−1)n+h

∮dz′

2πiz′ −n+h−1O(z′) = (−1)n+hO−n , (2.109)

where the first minus sign came from reversing the direction of the contour integral.We therefore find

〈ϕ1,Onϕ2 〉 = (−1)n+h〈O−nϕ1, ϕ2 〉 , (2.110)

orO�

n = (−1)n+hO−n . (2.111)

Since αμn, bn, and cn are modes of primary fields of weights 1, 2, and −1, respectively,

we reproduce the formulas in (2.22). Furthermore, the BRST operator is the zeromode of the BRST current, which is a primary field of weight 1, so that we confirmthat the BRST operator is BPZ odd: Q�

B = −QB.

at Akadem



ownloaded from



2.5. Star product

Let us next provide the definition of the star product 〈ϕ1, ϕ2 ∗ϕ3 〉 for arbitrarystates ϕ1, ϕ2, and ϕ3 in the Fock space. As in the case of the BPZ inner product, weprepare the states ϕ1, ϕ2, and ϕ3 in the upper half-plane using the state-operatorcorrespondence. Then the inner product 〈ϕ1, ϕ2 ∗ ϕ3 〉 is defined by

〈ϕ1, ϕ2 ∗ ϕ3 〉 = 〈 f1 ◦ ϕ1(0) f2 ◦ ϕ2(0) f3 ◦ ϕ3(0) 〉UHP , (2.112)

where

f1(ξ) = tan[

23

(arctan ξ − π

2

) ], f2(ξ) = tan

(23

arctan ξ),

f3(ξ) = tan[

23

(arctan ξ +

π

2

) ]. (2.113)

See Fig. 6.Let us prove the important properties (2.33) based on the definition (2.112).

The first one〈ϕ1 ∗ ϕ2, ϕ3 〉 = 〈ϕ1, ϕ2 ∗ ϕ3 〉 (2.114)

can be shown using the conformal transformation

f(z) = tan(

arctan z +π

3

)=

z +√

31−√3 z

, (2.115)

which maps the upper half-plane to itself. We can confirm that

f ◦ f1(ξ) = f2(ξ) , f ◦ f2(ξ) = f3(ξ) , f ◦ f3(ξ) = f1(ξ) . (2.116)

Fig. 6. The definition (2.112) of the star product. The three half unit disks 1, 2, and 3 are mapped

to the left, center, and right regions, respectively, to form an upper half-plane.

at Akadem



ownloaded from


1020 Y. Okawa

We therefore find

〈ϕ1, ϕ2 ∗ ϕ3 〉 = 〈 f ◦ f1 ◦ ϕ1(0) f ◦ f2 ◦ ϕ2(0) f ◦ f3 ◦ ϕ3(0) 〉UHP

= 〈 f2 ◦ ϕ1(0) f3 ◦ ϕ2(0) f1 ◦ ϕ3(0) 〉UHP

= (−1)(ϕ1+ϕ2) ϕ3〈 f1 ◦ ϕ3(0) f2 ◦ ϕ1(0) f3 ◦ ϕ2(0) 〉UHP

= (−1)(ϕ1+ϕ2) ϕ3〈ϕ3, ϕ1 ∗ ϕ2 〉 = 〈ϕ1 ∗ ϕ2, ϕ3 〉 . (2.117)

The derivation property of the BRST operator with respect to the star productalso follows from the definition easily. Consider the inner product 〈QBϕ1, ϕ2 ∗ ϕ3 〉.Since Q�

B = −QB , we have

〈QBϕ1, ϕ2 ∗ ϕ3 〉 = −(−1)ϕ1〈ϕ1, QB(ϕ2 ∗ ϕ3) 〉 . (2.118)

On the other hand, it follows from the definition of the star produce that

〈QBϕ1, ϕ2 ∗ ϕ3 〉 = 〈 f1 ◦ (QB · ϕ1(0)) f2 ◦ ϕ2(0) f3 ◦ ϕ3(0) 〉UHP

= 〈QB · (f1 ◦ ϕ1(0)) f2 ◦ ϕ2(0) f3 ◦ ϕ3(0) 〉UHP . (2.119)

The contour of the integral for the BRST operator can be deformed as in Fig. 7.Changing the direction of the contours, we find

〈QB · (f1 ◦ ϕ1(0)) f2 ◦ ϕ2(0) f3 ◦ ϕ3(0) 〉UHP

= −(−1)ϕ1〈 f1 ◦ ϕ1(0) QB · (f2 ◦ ϕ2(0)) f3 ◦ ϕ3(0) 〉UHP

− (−1)ϕ1(−1)ϕ2〈 f1 ◦ ϕ1(0) f2 ◦ ϕ2(0) QB · (f3 ◦ ϕ3(0)) 〉UHP

= −(−1)ϕ1〈 f1 ◦ ϕ1(0) f2 ◦ (QB · ϕ2(0)) f3 ◦ ϕ3(0) 〉UHP

− (−1)ϕ1(−1)ϕ2〈 f1 ◦ ϕ1(0) f2 ◦ ϕ2(0) f3 ◦ (QB · ϕ3(0)) 〉UHP

= −(−1)ϕ1〈ϕ1, (QBϕ2) ∗ ϕ3 〉 − (−1)ϕ1(−1)ϕ2〈ϕ1, ϕ2 ∗ (QBϕ3) 〉 . (2.120)

We have thus derived

QB(ϕ2 ∗ ϕ3) = (QBϕ2) ∗ ϕ3 + (−1)ϕ2ϕ2 ∗ (QBϕ3) . (2.121)

It would again be helpful to illustrate the definition (2.112) of the star productby an example. Let us calculate 〈T, T ∗ T 〉 with T = c1|0〉. Since

f1 ◦ c(0) =38c(−√

3 ) , f2 ◦ c(0) =32c(0) , f3 ◦ c(0) =

38c(√

3 ) , (2.122)

Fig. 7. The deformation of the contour used in (2.120). The contour in the left figure can be

deformed to the sum of the two contours in the right figure.

at Akadem



ownloaded from



we find〈T, T ∗ T 〉 =

27128〈 c(−

√3 ) c(0) c(

√3 ) 〉UHP . (2.123)

Using (2.104), it is evaluated as

〈T, T ∗ T 〉density = −81√

364

. (2.124)

It should be noted that there is another convention for the star product. It isgiven by

〈ϕ1, ϕ2 ∗ ϕ3 〉 = 〈 f3 ◦ ϕ3(0) f2 ◦ ϕ2(0) f1 ◦ ϕ1(0) 〉UHP . (different convention)(2.125)

Namely, the operators in the correlation function are ordered in the opposite way.While both conventions (2.112) and (2.125) are consistent, let us explain why wechoose the convention (2.112). As we have seen, the three operators f1 ◦ ϕ1(0),f2 ◦ ϕ2(0), and f3 ◦ ϕ3(0) are inserted at −√3, 0, and

√3, respectively. In the

standard way of writing the real axis, the operators are therefore located along thereal axis in the order f1 ◦ ϕ1(0) f2 ◦ ϕ2(0) f3 ◦ ϕ3(0). In the convention (2.112),the operators are ordered in the same way inside the correlation function. We thinkthat using the convention (2.112) helps us reduce sign mistakes when we handleGrassmann-odd operators.

§3. Tachyon condensation

3.1. Approximate solutions in level truncation

Consider an ordinary field theory of a scalar field φ with its potential V (φ) shownin Fig. 8. The static configuration φ = 0 is a solution to the equation of motion,but it is an unstable background of φ, and there is a tachyonic mode around thebackground. In such cases, we look for a solution φ = φ∗ to the equation of motionsatisfying

dV (φ)dφ

∣∣∣∣φ=φ∗

= 0 withd2V (φ)dφ2

∣∣∣∣φ=φ∗

> 0 , (3.1)

Fig. 8. An example of V (φ).

at Akadem



ownloaded from


1022 Y. Okawa

and we expand φ around the background φ∗ as φ = φ∗ + δφ.In string theory, open strings are excitations on solitonic extended objects called

D-branes. The background we have considered corresponds to the D25-brane, andthere is a tachyonic mode around the background. Sen conjectured that the in-stability associated with this open string tachyon corresponds to the decay of theD-brane and there exists a background where the unstable D-brane disappeared.This conjecture makes a quantitative prediction that the difference of the energydensity between the two backgrounds is given by the energy density of the D-brane.However, string theory is defined only perturbatively, and we can only investigatean infinitesimal neighborhood around a background. We are not able to draw a po-tential like the one in Fig. 8, as it is an off-shell quantity. On the other hand, we canin principle discuss the relationship between backgrounds using string field theory.When we formulate open string field theory around the background with the unsta-ble D-brane, we expect a classical solution to the equation of motion correspondingto the background without the D-brane. The problem of finding this solution is usu-ally referred to as tachyon condensation, and it is conceptually the same as findinga nontrivial solution in the scalar field theory with the potential shown in Fig. 8. Inthe case of string field theory, however, there are infinitely many fields and findinga nontrivial solution is technically complicated. The expected background withoutthe D-brane is often called tachyon vacuum, and the corresponding solution of openstring field theory is called tachyon vacuum solution.

Sen and Zwiebach tested the quantitative conjecture using open string fieldtheory.5) They used the approximation scheme called level truncation. In leveltruncation, component fields of the string field are truncated to a finite number offields. At level 0, the zero mode of the tachyon t is the only degree of freedom:

Ψ = t c1|0〉 . (3.2)

The tachyon potential is given by

V (t) =1

α′ 3g2T

[12t2 〈T,QBT 〉+ 1

3t3 〈T, T ∗ T 〉

]density

, (3.3)

where T = c1|0〉. We have already calculated these terms in (2.105) and (2.124), andso the potential at level 0 is

V (t) =1

α′ 3g2T

(−1

2t2 − 27

√3

64t3

). (3.4)

The D25-brane tension T25 is given by∗)

T25 =1

2π2α′ 3g2T

. (3.5)

The tachyon potential normalized by the D25-brane tension is thus

V (t)T25

= 2π2

(−1

2t2 − 27

√3

64t3

). (3.6)

∗) The expression of T25 in terms of gT can be found, for example, in Appendix A of Ref. 6).

at Akadem



ownloaded from



This potential has a critical point at

t = t∗ = − 6481√

3, (3.7)

and the value of the potential at the critical point is

V (t∗)T25

= −4096π2

59049� −0.68 . (3.8)

This is about 68% of the value predicted by Sen’s conjecture.At level 2, the string field is truncated to

Ψ = t c1|0〉+ u c−1|0〉+ v L(m)−2 c1|0〉 , (3.9)

where t, u, and v are dynamical degrees of freedom. The potential as a function oft, u, and v is given by

V (t, u, v)T25

= 2π2

(−1

2t2 − 27

√3

64t3 − 1

2u2 +

12v2

−3364

√3 t2u+

1564

√39 t2v − 19

64√

3tu2 − 581

192√

3tv2 +

5596

√133tuv

− 164√

3u3 +

209515184

√39v3 +

951728

√133u2v − 6391

5184√

3uv2

). (3.10)

The equations of motion,

∂V (t, u, v)∂t

= 0 ,∂V (t, u, v)

∂u= 0 ,

∂V (t, u, v)∂v

= 0 , (3.11)

can be solved numerically, and we find a critical point at

t = t∗ � −0.544 , u = u∗ � −0.190 , v = v∗ � −0.202 . (3.12)

The value of the potential at the critical point is

V (t∗, u∗, v∗)T25

� −0.959 . (3.13)

This is about 96% of the value predicted by Sen’s conjecture.The calculation up to level 18 was performed by Gaiotto and Rastelli7) and the

results up to level 12 are summarized in Table I.∗) As can be seen from the table,the value of the potential at the critical point normalized by the D25-brane tensionis very close to the value −1 predicted by Sen’s conjecture. We can think of theresult of this analysis as evidence for Sen’s conjecture, but we can also think of itas evidence that open string field theory can describe nonperturbative phenomenasuch as tachyon condensation.

While the improvement of the numerical solution is impressive, it is still anapproximate solution. As we mentioned in the introduction, an analytic solutionwas constructed by Schnabl,1) and it is for this problem of tachyon condensation.In order to understand his construction, we need to develop the CFT description ofstring field theory further.

∗) The current world record is level 26.8)

at Akadem



ownloaded from


1024 Y. Okawa

Table I. The value of the potential at the critical point.

level V/T25

4 −0.987822

6 −0.995177

8 −0.997930

10 −0.999183

12 −0.999822

3.2. Sliver frame

The coordinate of the upper half-plane is useful and we can use the state-operatorcorrespondence to represent a state. To describe the star product, there is a moreconvenient coordinate called the sliver frame introduced by Rastelli and Zwiebach.9)

The coordinate z of the sliver frame is related to the coordinate ξ of the upperhalf-plane by

z = f(ξ) =2π

arctan ξ . (3.14)

The right half of the open string eiθ with 0 ≤ θ ≤ π/2 in the upper half-plane ismapped to the semi-infinite line with �z = 1/2 and �z ≥ 0 in the sliver frame. Theleft half of the open string eiθ with π/2 ≤ θ ≤ π in the upper half-plane is mapped tothe semi-infinite line with �z = −1/2, �z ≥ 0 in the sliver frame. The region |ξ| ≤ 1with �ξ ≥ 0 is mapped to the semi-infinite strip −1/2 ≤ �z ≤ 1/2, �z ≥ 0. SeeFig. 9. Since either of the right half or the left half of the open string is mapped to asemi-infinite line, the gluing of half open strings can be done simply by translation.

Let us recalculate 〈T,QBT 〉 and 〈T, T ∗ T 〉 with T = c1|0〉 in the sliver frame.The vacuum state |0〉 is represented in the sliver frame by the region −1/2 ≤ �z ≤1/2, �z ≥ 0 without any operator insertions. Since

f ◦ c(0) =1

f ′(0)c(f(0)) =

π

2c(0) , (3.15)

the state T = c1|0〉 is represented in the same region with an insertion of π2 c(0).

Fig. 9. The sliver frame. The half unit disk of the upper half-plane of ξ in the left figure is

mapped to the semi-infinite strip in the sliver frame with the coordinate z by the conformal

transformation (3.14).

at Akadem



ownloaded from



Fig. 10. The BPZ inner product in the sliver frame. The left semi-infinite line and the right semi-

infinite line are identified by translation. BPZ inner products are represented by two-point

functions on this semi-infinite cylinder.

Fig. 11. The star product in the sliver frame. The left semi-infinite line and the right semi-infinite

line are identified by translation. Star products are represented by three-point functions on this

semi-infinite cylinder.

The state QBT is also represented in the same region with an insertion of π2 c∂c(0).

In the BPZ inner product 〈T,QBT 〉, the region for QBT can be translated to theregion 1/2 ≤ �z ≤ 3/2, �z ≥ 0 with the insertion π

2 c∂c(1), and the semi-infiniteline �z = −1/2, �z ≥ 0 of the state T and the semi-infinite line �z = 3/2, �z ≥ 0 ofthe state QBT are identified by translation. The resulting surface is a semi-infinitecylinder. See Fig. 10. Similarly, the inner product 〈T, T ∗ T 〉 can be representedby the region −1/2 ≤ � ≤ 5/2, �z ≥ 0 with the semi-infinite line �z = −1/2,�z ≥ 0 and the semi-infinite line �z = 5/2, �z ≥ 0 are identified by translation,and the operators π

2 c(0), π2 c(1) and π

2 c(2) are inserted. See Fig. 11. We denoteby Cn the semi-infinite cylinder constructed from the upper half-plane of z by the

at Akadem



ownloaded from


1026 Y. Okawa

identification z ∼ z+n, and we represent it in the region −1/2 ≤ � ≤ n−1/2 unlessotherwise specified. The formula for correlation functions on Cn can be derived fromthe formula (2.104) for the upper half-plane. The surface Cn can be mapped to theupper half-plane by

z′ = gn(z) = f−1(2zn

)= tan

π z

n, (3.16)

so we have

〈 c(z1) c(z2) c(z3) 〉Cn = 〈 gn ◦ c(z1) gn ◦ c(z2) gn ◦ c(z3) 〉gn◦Cn

=1

g′n(z1) g′n(z2) g′n(z3)〈 c(gn(z1)) c(gn(z2)) c(gn(z3)) 〉UHP . (3.17)

From this we obtain the formula for correlation functions on Cn given by

〈 c(z1) c(z2) c(z3) 〉Cn, density =(nπ

)3sin

π(z1 − z2)n

sinπ(z1 − z3)

nsin

π(z2 − z3)n

.

(3.18)In the case of 〈T,QBT 〉, we use the formula with n = 2 and find

〈T,QBT 〉density = −1 . (3.19)

In the case of 〈T, T ∗ T 〉, we use the formula with n = 3 and find

〈T, T ∗ T 〉density = −81√

364

. (3.20)

We have thus reproduced the previous results (2.105) and (2.124).

3.3. Wedge state

Let us now introduce a class of states called wedge states, which play an impor-tant role in the development of analytic methods in open string field theory. Firstconsider the star product |0〉 ∗ |0〉. As we explained before, the star product |0〉 ∗ |0〉is specified by 〈ϕ, 0 ∗ 0 〉 for an arbitrary state ϕ in the Fock space, where we repre-sented |0〉 ∗ |0〉 by 0 ∗ 0 in the BPZ inner product. Using the sliver frame, the innerproduct 〈ϕ, 0 ∗ 0 〉 is given by a one-point function of f ◦ ϕ(0) on the semi-infinitecylinder C3:

〈ϕ, 0 ∗ 0 〉 = 〈 f ◦ ϕ(0) 〉C3 . (3.21)

Similarly, the star product |0〉 ∗ |0〉 ∗ . . . ∗ |0〉 from n vacuum states is specified by〈ϕ, 0 ∗ 0 ∗ . . . ∗ 0 〉 for an arbitrary state ϕ in the Fock space, where we represented|0〉 ∗ |0〉 ∗ . . . ∗ |0〉 by 0 ∗ 0 ∗ . . . ∗ 0 in the BPZ inner product. The inner product〈ϕ, 0 ∗ 0 ∗ . . . ∗ 0 〉 is given by a one-point function of f ◦ ϕ(0) on the semi-infinitecylinder Cn+1:

〈ϕ, 0 ∗ 0 ∗ . . . ∗ 0︸︷︷︸n

〉 = 〈 f ◦ ϕ(0) 〉Cn+1 . (3.22)

Let us generalize these star products. The wedge state Wα for non-negative realnumber α is defined by

〈ϕ,Wα 〉 = 〈 f ◦ ϕ(0) 〉Cα+1 (3.23)

at Akadem



ownloaded from



for an arbitrary state ϕ in the Fock space. In this notation, the vacuum state |0〉 andthe star product |0〉 ∗ |0〉 are W1 and W2, respectively. Let us consider star productsof wedge states. The star product Wα ∗Wβ is specified by 〈ϕ,Wα ∗Wβ 〉, which isevaluated as follows:

〈ϕ,Wα ∗Wβ 〉 = 〈 f ◦ ϕ(0) 〉Cα+β+1= 〈ϕ,Wα+β 〉 . (3.24)

We therefore findWα ∗Wβ = Wα+β . (3.25)

Namely, wedge states are closed under the star multiplication. We also considerwedge states with operator insertions. We call this class of states wedge-based states.Wedge-based states are closed under the star multiplication and closed also underthe BRST transformation. Therefore, the space of wedge-based states is a good placeto look for solutions to the equation of motion: QBΨ + Ψ ∗ Ψ = 0.

It is interesting to consider the limit α→∞ of the wedge state Wα. The innerproduct 〈ϕ,Wα 〉 in the limit α →∞ is given by the one-point function of f ◦ ϕ(0)in the upper half-plane and is thus well defined and finite. The state obtained in thelimit α → ∞ of Wα is called the sliver state and denoted by W∞. If we formallytake the limit α→∞ and β →∞ in (3.25), we obtain

W∞ ∗W∞ = W∞ . (formal) (3.26)

Namely, the sliver state W∞ is formally a star-algebra projector. We keep sayingthat the sliver limit is formal because in general the sliver limit is not uniquelydetermined when multiple sliver states are involved. For example, the inner product〈ϕ1,W∞ ∗ϕ2 ∗W∞ 〉 for states ϕ1 and ϕ2 in the Fock space should be understood asthe limit α→∞ and β →∞ of 〈ϕ1,Wα ∗ ϕ2 ∗Wβ 〉, but the result depends on howwe send α and β to infinity. Similarly, the inner product 〈ϕ1,W∞ ∗ϕ2 ∗W∞ ∗W∞ 〉,which is expected to coincide with 〈ϕ1,W∞ ∗ ϕ2 ∗W∞ 〉 from the relation (3.26), isalso ambiguous and whether this coincides with 〈ϕ1,W∞ ∗ ϕ2 ∗W∞ 〉 depends onhow we take the sliver limit.

3.4. Derivatives of the wedge state

Let us next consider the derivative of the wedge state Wα with respect to theparameter α:

W ′α =

dWα

dα= lim

β→0

Wα+β −Wα

β. (3.27)

Since 〈ϕ,Wα+β 〉 and 〈ϕ,Wα 〉 are represented on different surfaces, it is convenientto perform the conformal transformation g(z) given by

g(z) =α+ 1

α+ β + 1z (3.28)

such that the surface for 〈ϕ,Wα+β 〉 is independent of β:

〈ϕ,Wα+β 〉 = 〈 f ◦ ϕ(0) 〉Cα+β+1= 〈 g ◦ f ◦ ϕ(0) 〉Cα+1 . (3.29)

at Akadem



ownloaded from


1028 Y. Okawa

Sinceg(z) =

α+ 1α+ β + 1

z = z − β

α+ 1z +O(β2) , (3.30)

the transformation with infinitesimal β is generated by the energy-momentum tensoras follows:

〈ϕ,Wα+β 〉 = 〈 g ◦ f ◦ ϕ(0) 〉Cα+1

= 〈 f ◦ ϕ(0) 〉Cα+1 −β

α+ 1〈∮

dz

2πiz T (z) f ◦ ϕ(0) 〉Cα+1 +O(β2) , (3.31)

where the contour of the integral encircles the origin counterclockwise. The derivativeof the wedge state W ′

α is thus given by

〈ϕ,W ′α 〉 = − 1

α+ 1〈∮

dz

2πiz T (z) f ◦ ϕ(0) 〉Cα+1 . (3.32)

Let us now deform the contour of the integral. We denote by Vα the vertical contouralong the infinite line with �z = α in the complex z plane with the direction fromi∞ to −i∞. Namely, it runs downwards. The contour of the integral in (3.32) canbe deformed as follows:∮

dz

2πiz T (z) =

∫−Vε

dz

2πiz T (z) +

∫V−ε

dz

2πiz T (z)

=∫−Vα+1/2−ε

dz+2πi

z+ T (z+) +∫

V−1/2+ε

dz−2πi

z− T (z−) , (3.33)

where ε is an infinitesimal positive real number. Let us further deform the contourV−1/2+ε in the coordinate z− to V−1/2−ε, which is identified with Vα+1/2−ε in thecoordinate z+ by the conformal transformation z+ = z− + α+ 1. We therefore have∫

−Vα+1/2−ε

dz+2πi

z+ T (z+) +∫

V−1/2+ε

dz−2πi

z− T (z−)

=∫−Vα+1/2−ε

dz+2πi

z+ T (z+) +∫

Vα+1/2−ε

dz+2πi

(z+ − α− 1)T (z+)

= −(α+ 1)∫

Vα+1/2−ε

dz+2πi

T (z+) . (3.34)

The upshot is

〈ϕ,W ′α 〉 = 〈 f ◦ ϕ(0)

∫Vγ

dz

2πiT (z) 〉Cα+1 , (3.35)

where γ can be any value in the range 1/2 < γ < α + 1/2. We have learned thattaking a derivative with respect to α corresponds to an insertion of the line integralof the energy-momentum tensor given by

L ≡∫

Vγ

dz

2πiT (z) (3.36)

at Akadem



ownloaded from



with γ in the region of the wedge state. We write

〈ϕ,W ′α 〉 = 〈 f ◦ ϕ(0)L 〉Cα+1 (3.37)

with the understanding that the location of the line integral L is to the right of theoperator f ◦ϕ(0), although in this case we could bring L to the left of f ◦ϕ(0) usingthe identification z ∼ z + α+ 1. This can be generalized to higher derivatives:

〈ϕ,W (n)α 〉 = 〈 f ◦ ϕ(0)

∫Vγ1

dz

2πiT (z)

∫Vγ2

dz

2πiT (z) . . .

∫Vγn

dz

2πiT (z) 〉Cα+1 , (3.38)

where γi can be any value in the range 1/2 < γi < α + 1/2. It is not difficult toconfirm that insertions of the line integral commute using the OPE of the energy-momentum tensor with itself. We therefore write (3.38) as follows:

〈ϕ,W (n)α 〉 = 〈 f ◦ ϕ(0) LL . . . L︸︷︷︸

n

〉Cα+1 = 〈 f ◦ ϕ(0)Ln 〉Cα+1 . (3.39)

We can also write Wα+β as

〈ϕ,Wα+β 〉 =∞∑

n=0

βn

n!〈 f ◦ ϕ(0)

∫Vγ1

dz

2πiT (z)

∫Vγ2

dz

2πiT (z) . . .

∫Vγn

dz

2πiT (z) 〉Cα+1

=∞∑

n=0

βn

n!〈 f ◦ ϕ(0) LL . . . L︸︷︷︸

n

〉Cα+1 =∞∑

n=0

βn

n!〈 f ◦ ϕ(0)Ln 〉Cα+1 . (3.40)

The correspondence between taking a derivative with respect to α and an inser-tion of the line integral of the energy-momentum tensor is exactly what we expected.In the strip coordinate, the operator e−tL0 generates a rectangular surface. Takinga derivative with respect to t corresponds to an insertion of −L0:

∂t e−tL0 = −L0 e

−tL0 . (3.41)

The line integral L is locally the same as −L0 in the strip coordinate. There is,however, an important difference. The operator −L0 is a line integral of the energy-momentum tensor from one boundary to the other boundary. The operator L is aline integral of the energy-momentum tensor from the open string midpoint to theboundary before using the doubling trick. In the coordinate ξ of the upper half-plane,the integral is written as∫

dz

2πiT (z)→

∫dξ

2πi

(dzdξ

)−1T (ξ) =

π

2

∫dξ

2πi(ξ2 + 1)T (ξ) . (3.42)

Namely, the operator L is a half integral ofπ

2(L1 + L−1) ≡ π

2K1 . (3.43)

When an operator O is defined by an integral along the unit circle counterclockwise,

O =∮

dξ

2πiφ(ξ) , (3.44)

at Akadem



ownloaded from


1030 Y. Okawa

where φ(ξ) can be a field or a field multiplied by a function of ξ, we define the righthalf integral OR and the left half integral OL by

OR =∫

CR

dξ

2πiφ(ξ) , OL =

∫CL

dξ

2πiφ(ξ) , (3.45)

where the contour CR runs along the right half of the unit circle from −i to icounterclockwise and the contour CL runs along the left half of the unit circle fromi to −i counterclockwise.∗) Using this notation, the state W ′

1 can be written as

W ′1 =

π

2KL

1 |0〉 (3.46)

or asW ′

1 = −π2KR

1 |0〉 . (3.47)

3.5. Eigenstates of the operator L

The space of wedge-based states is closed under star multiplication and underthe BRST transformation so that it is a good subspace of string fields to look forsolutions to the equation of motion. Schnabl1) found a much smaller subspace closedunder star multiplication and under the BRST transformation and introduced abasis of the space based on eigenvalues of the operator L, which is the generator ofdilatation in the sliver space:∗∗)

L =∮

dz

2πiz T (z) =

∮dξ

2πif(ξ)f ′(ξ)

T (ξ) = L0 +23L2 − 2

15L4 + . . . . (3.48)

The vacuum state |0〉 is an eigenstate of L, and its eigenvalue is 0:

L |0〉 = 0 . (3.49)

Next consider the vacuum state |0〉 with an insertion of L, which is W ′1. Since∫

Vγ

dw

2πi

∮dz

2πiz T (z)T (w) =

∫Vγ

dw

2πiT (w) , (3.50)

where the contour of the integral over z encircles the point w counterclockwise, thestate W ′

1 is an eigenstate of L, and its eigenvalue is 1:

LW ′1 = W ′

1 . (3.51)

Similarly, we find that W (n)1 is an eigenstate of L with its eigenvalue n:

LW(n)1 = nW

(n)1 . (3.52)

Schnabl also considered insertions of the line integral B, which is obtained fromL by replacing the energy-momentum tensor with the b ghost:

B ≡∫

Vγ

dz

2πib(z) . (3.53)

∗) In Ref. 1) Schnabl’s OL is our OR, and Schnabl’s OR is our OL.∗∗) The operator L here is denoted by L0 in Ref. 1).

at Akadem



ownloaded from



The vacuum state |0〉 with an insertion of B whose contraction with ϕ given by

〈 f ◦ ϕ(0)B 〉C2 = 〈 f ◦ ϕ(0)∫

Vγ

dz

2πib(z) 〉C2 , (3.54)

where 1/2 < γ < 3/2, is an eigenstate of L with its eigenvalue 1 because∫Vγ

dw

2πi

∮dz

2πiz T (z) b(w) =

∫Vγ

dw

2πib(w) , (3.55)

where the contour of the integral over z encircles the point w counterclockwise. Ifwe define the line integral B1 by

B1 ≡ b1 + b−1 =∮

dξ

2πi(ξ2 + 1) b(ξ) , (3.56)

the vacuum state |0〉 with an insertion of B can be written using the half integralBL

1 asπ

2BL

1 |0〉 (3.57)

or using BR1 as

−π2BR

1 |0〉 . (3.58)

We can confirm from the OPE of the energy-momentum tensor and the b ghost that∫Vγ

dz

2πiT (z)

∫Vγ+ε

dw

2πib(w) =

∫Vγ−ε

dw

2πib(w)

∫Vγ

dz

2πiT (z) , (3.59)

where ε > 0. We writeLB = BL (3.60)

with the understanding that the contour of the right integral is to the right of thecontour of the left integral. We also write it as the commutation relation:

[B,L ] = 0 . (3.61)

Another important property is that∫Vγ1

dz12πi

b(z1)∫

Vγ2

dz22πi

b(z2) = 0 (3.62)

when there are no operators in the region γ1 < �z < γ2. We write

B2 = 0 . (3.63)

Finally, consider local insertions of the c ghost and its derivatives on the bound-ary. For example, the state defined by the contraction with ϕ given by

〈 f ◦ ϕ(0) c∂2c∂5c(1) 〉C2 (3.64)

at Akadem



ownloaded from


1032 Y. Okawa

is an eigenstate of L with its eigenvalue 2+5−3 = 4, where the positive contribution2 + 5 comes from the number of derivatives and the negative contribution −3 comesfrom the number of c ghosts.

We have found that a set of L eigenstates in the form

〈 f ◦ ϕ(0)∏

i

∂nic(1)Lm 〉C2 (3.65)

or〈 f ◦ ϕ(0)

∏i

∂nic(1)Lm B 〉C2 . (3.66)

This is the basis of the states introduced in Ref. 1).∗) States in this class are closedunder the BRST transformation and under the star multiplication. First, the BRSToperator acts on the ingredients of the states as follows:

QB ·∏

i

∂nic(1) =∏j

∂n′jc(1) , QB · B = L , QB · L = 0 . (3.67)

Thus the set of states is closed under the BRST transformation. Note also that theeigenvalue of L does not change under the BRST transformation because [QB, L ] =0. Let us next consider the star product of two states in the class (3.66):

〈 f ◦ ϕ(0)∏

i

∂nic(1)Lm B∏j

∂kjc(2)Ll B 〉C3 . (3.68)

We first move the line integrals Lm B in the region 1 < �z < 2 to the right of theoperators at z = 2. For the line integral B, we have∫

Vz−ε

dw

2πib(w) c(z) =

∮dw

2πib(w) c(z)− c(z)

∫Vz+ε

dw

2πib(w) , (3.69)

where ε > 0 and the contour of the integral of the first term on the right-hand sideencircles the point z counterclockwise. We therefore find∫

Vz−ε

dw

2πib(w) c(z) = 1− c(z)

∫Vz+ε

dw

2πib(w) , (3.70)

which can be written asB c(z) = 1− c(z)B (3.71)

or as the anticommutation relation in the form

{B, c(z) } = 1 . (3.72)

Taking derivatives with respect to z, we also find

B ∂nc(z) + ∂nc(z)B = 0 for n ≥ 1 . (3.73)∗) To be more precise, states in this set do not in general satisfy the reality condition of the

string field,10) and the set of states introduced in Ref. 1) is a refined set of states such that the reality

condition is satisfied. In what follows we neglect the reality condition to simplify the presentation.

at Akadem



ownloaded from



For the line integral L, we have∫Vz−ε

dw

2πiT (w)ϕ(z) =

∮dw

2πiT (w)ϕ(z) + ϕ(z)

∫Vz+ε

dw

2πiT (w) , (3.74)

where ϕ(z) is an arbitrary operator and the contour of the integral of the first termon the right-hand side encircles the point z counterclockwise. We therefore find∫

Vz−ε

dw

2πiT (w)ϕ(z) = ∂ϕ(z) + ϕ(z)

∫Vz+ε

dw

2πiT (w) , (3.75)

which can be written asLϕ(z) = ∂ϕ(z) + ϕ(z)L (3.76)

or as the commutation relation given by

[L, ϕ(z) ] = ∂ϕ(z) . (3.77)

After moving the line integrals, we have states which can be expanded in the statesof the form

〈 f ◦ ϕ(0)∏

i

∂nic(1)∏j

∂mjc(2)Lk B 〉C3 . (3.78)

Note that [B,L ] = 0 and B2 = 0 so that there are no states with two insertions ofB. Similarly, the star product of two states in the class (3.65) can be transformedto the form expanded in the states of the form

〈 f ◦ ϕ(0)∏

i

∂nic(1)∏j

∂mjc(2)Lk 〉C3 , (3.79)

and the star product of a state in the class (3.65) and a state in the class (3.66) canbe transformed to the form expanded in the states of the forms (3.78) and (3.79).We can then expand the c ghosts around z = 1 to obtain states of the form

〈 f ◦ ϕ(0)∏

i

∂nic(1)Lk 〉C3 (3.80)

or of the form〈 f ◦ ϕ(0)

∏i

∂nic(1)Lk B 〉C3 . (3.81)

Finally, we use the formula (3.40) to express the states based on the surface C3

in terms of states based on the surface C2, which are states in the classes (3.65)and (3.66). We have thus shown that the set of states in the classes (3.65) and (3.66)is closed under the star multiplication.

Furthermore, the basis of the states (3.65) and (3.66) has the following property.Let us denote the set of states in the basis by {φi} and the eigenvalue of L for φi byhi. We have shown that the star product of two states φ1 and φ2 in the basis can beexpanded as

φ1 ∗ φ2 =∑

i

ci φi , (3.82)

at Akadem



ownloaded from


1034 Y. Okawa

where ci’s are nonvanishing coefficients. By carefully investigating each step of theproof, we find that the eigenvalue hi of a state which appears on the right-hand sidecannot be less than the sum of the eigenvalues h1 and h2:

hi ≥ h1 + h2 . (3.83)

This allows us to solve the equation of motion recursively.In the sector of ghost number 1, the lowest eigenvalue of L is −1, and the

corresponding eigenstate φ(−1) can be expressed by its contraction with ϕ given by

〈ϕ, φ(−1) 〉 = 〈 f ◦ ϕ(0) c(1) 〉C2 . (3.84)

Namely,

φ(−1) =2πc1|0〉 . (3.85)

There are three states in the basis with eigenvalue 0, which we call φ(0)1 , φ(0)

2 , andφ

(0)3 . Their inner products with ϕ are given by

〈ϕ, φ(0)1 〉 = 〈 f ◦ ϕ(0) ∂c(1) 〉C2 , 〈ϕ, φ(0)

2 〉 = 〈 f ◦ ϕ(0) c(1)L 〉C2 ,

〈ϕ, φ(0)3 〉 = 〈 f ◦ ϕ(0) c∂c(1)B 〉C2 . (3.86)

3.6. Schnabl gauge

While we can now start looking for solutions in this subspace of string fields,Schnabl further restricted the space of states by imposing a gauge condition. It isan analog of the Siegel gauge condition b0 Ψ = 0 in the sliver frame and is given by

B Ψ = 0 , (3.87)

where∗)

B =∮

dz

2πiz b(z) =

∮dξ

2πif(ξ)f ′(ξ)

b(ξ) = b0 +23b2 − 2

15b4 + . . . . (3.88)

This gauge condition is called the Schnabl gauge condition. The eigenstate φ(−1)

with eigenvalue −1 satisfies the Schnabl gauge condition:

Bφ(−1) = 0 . (3.89)

While it is obvious from the mode expansion of B and φ(−1) ∝ c1|0〉, it can be shownusing the CFT expression as follows:

〈ϕ,Bφ(−1) 〉 = 〈 f ◦ ϕ(0)B · c(1) 〉C2 = 〈 f ◦ ϕ(0)∮

dz

2πi(z − 1) b(z) c(1) 〉C2 = 0 .

(3.90)Let us next calculate Bφ(0)

1 , Bφ(0)2 , and Bφ(0)

3 . The first one is

〈ϕ,Bφ(0)1 〉 = 〈 f ◦ ϕ(0)

∮dz

2πi(z − 1) b(z) ∂c(1) 〉C2 = 〈 f ◦ ϕ(0) 〉C2 . (3.91)

∗) The operator B here is denoted by B0 in Ref. 1).

at Akadem



ownloaded from



Namely, Bφ(0)1 = |0〉. For Bφ(0)

2 , we find

〈ϕ,Bφ(0)2 〉= 〈 f ◦ ϕ(0)

∮dz

2πi(z − 1) b(z) c(1)L 〉C2

= 〈 f ◦ ϕ(0)[ ∫

V1/2

dz

2πi(z − 1) b(z) c(1)L + c(1)L

∫V3/2

dz

2πi(z − 1) b(z)

]〉C2

= 〈 f ◦ ϕ(0)[ ∫

V1−ε

dz

2πi(z − 1) b(z) c(1) + c(1)

∫V1+ε

dz

2πi(z − 1) b(z)

]L 〉C2

−〈 f ◦ ϕ(0) c(1)B 〉C2

= −〈 f ◦ ϕ(0) c(1)B 〉C2 , (3.92)

where we have used

L∫

Vγ+ε

dz

2πi(z − 1) b(z) =

∫Vγ

dw

2πiT (w)

∫Vγ+ε

dz

2πi(z − 1) b(z)

=∫

Vγ−ε

dz

2πi(z − 1) b(z)

∫Vγ

dw

2πiT (w)−

∫Vγ

dw

2πi

∮dz

2πi(z − 1) b(z)T (w)

=∫

Vγ−ε

dz

2πi(z − 1) b(z)L− B . (3.93)

For Bφ(0)3 , we find

〈ϕ,Bφ(0)3 〉 = 〈 f ◦ ϕ(0)

∮dz

2πi(z − 1) b(z) c∂c(1)B 〉C2

= 〈 f ◦ ϕ(0)[ ∫

V1/2

dz

2πi(z − 1) b(z) c∂c(1)B + c∂c(1)B

∫V3/2

dz

2πi(z − 1) b(z)

]〉C2

= 〈 f ◦ ϕ(0)[ ∫

V1−ε

dz

2πi(z − 1) b(z) c∂c(1) − c∂c(1)

∫V1+ε

dz

2πi(z − 1) b(z)

]B 〉C2

= − 〈 f ◦ ϕ(0) c(1)B 〉C2 . (3.94)

Therefore, the linear combination φ(0)2 − φ(0)

3 satisfies the Schnabl gauge condition:

B (φ(0)2 − φ(0)

3 ) = 0 . (3.95)

In fact, the states φ(−1) and φ(0)2 − φ(0)

3 can be written as

φ(−1) = ψ0 , φ(0)2 − φ(0)

3 = ψ′0 (3.96)

using ψn labeled by a real parameter n defined by

〈ϕ, ψn 〉 = 〈 f ◦ ϕ(0) c(1)B c(n+ 1) 〉Cn+2

= 〈 f ◦ ϕ(0) c(1)∫

Vγ

dz

2πib(z) c(n+ 1) 〉Cn+2 (3.97)

at Akadem



ownloaded from


1036 Y. Okawa

with 1 < γ < n+ 1, and its derivative ψ′n with respect to n:

ψ′n ≡

dψn

dn. (3.98)

The string field ψn is an important ingredient of the analytic solution by Schnabl.We will show shortly that ψn satisfies the Schnabl gauge condition for any n, but

let us first prove the relations in (3.96). The string field ψ0 should be understood asthe limit n → 0 of ψn. In the limit, the three operators c(1)B c(n + 1) collide. Intaking the limit, we first move the line integral B to the right of c(n+ 1) using theanticommutation relation (3.72). The inner product 〈ϕ, ψn 〉 can then be written as

〈ϕ, ψn 〉 = 〈 f ◦ ϕ(0) c(1) 〉Cn+2 − 〈 f ◦ ϕ(0) c(1) c(n+ 1)B 〉Cn+2 . (3.99)

It is now straightforward to take the limit n→ 0. The second term on the right-handside vanishes in the limit because two c ghosts collide. We thus have

〈ϕ, ψ0 〉 = limn→0〈ϕ, ψn 〉 = 〈 f ◦ ϕ(0) c(1) 〉C2 , (3.100)

orψ0 = lim

n→0ψn =

2πc1|0〉 . (3.101)

We have thus shown the first relation in (3.96).To prove the second relation in (3.96), let us express 〈ϕ, ψ′

n 〉 in the CFT lan-guage. This can be done in a way analogous to that we did for 〈ϕ,W ′

α 〉, and thefinal result is

〈ϕ, ψ′n 〉 = 〈 f ◦ ϕ(0) c(1)LB c(n+ 1) 〉Cn+2 . (3.102)

In the limit n → 0, the four operators c(1)LB c(n + 1) collide. We first move B tothe right of c(n+ 1) to obtain

〈ϕ, ψ′n 〉 = 〈 f ◦ ϕ(0) c(1)L 〉Cn+2 − 〈 f ◦ ϕ(0) c(1)L c(n+ 1)B 〉Cn+2 . (3.103)

We then use the commutation relation (3.77) to find

〈ϕ, ψ′n 〉 = 〈 f ◦ ϕ(0) c(1)L 〉Cn+2 − 〈 f ◦ ϕ(0) c(1) ∂c(n+ 1)B 〉Cn+2

− 〈 f ◦ ϕ(0) c(1) c(n+ 1)LB 〉Cn+2 . (3.104)

The third term on the right-hand side vanishes in the limit n → 0 because two cghosts collide. We thus have

〈ϕ, ψ′0 〉 = lim

n→0〈ϕ, ψ′

n 〉 = 〈 f ◦ ϕ(0) c(1)L 〉C2 − 〈 f ◦ ϕ(0) c∂c(1)B 〉C2 . (3.105)

The second relation in (3.96) has now been shown.Let us move to the proof that ψn satisfies the Schnabl gauge condition:

B ψn = 0 (3.106)

for any n. It then follows that

B ψ0 = 0 , B ψ′0 = 0 , . . . , B ψ

(n)0 = 0 , . . . . (3.107)

at Akadem



ownloaded from



What we need to prove is that 〈ϕ,B ψn 〉 = 0. We know how the operator B actson a state ϕ in the Fock space. In the sliver frame, the state Bϕ corresponds to theoperator ∮

dz

2πiz b(z) f ◦ ϕ(0) , (3.108)

where the contour of the integral encircles the origin counterclockwise. On the otherhand, we do not know how B acts on general wedge-based states. We therefore usethe relation

〈ϕ,B ψn 〉 = −〈B�ϕ, ψn 〉 (3.109)

and consider B�ϕ. Here the minus sign on the right-hand side comes from theanticommutation of B� and ϕ because ϕ has to carry ghost number 3 for the innerproduct to be nonvanishing and thus ϕ is Grassmann odd. The BPZ conjugation isdefined in the upper half-plane of ξ using the conformal map

ξ′ = I(ξ) = −1ξ. (3.110)

In the sliver frame z, we have to use the map

z′ = f(ξ′) = f ◦ I(ξ) = f ◦ I ◦ f−1(z) . (3.111)

The operator B in the sliver frame z can be decomposed into two pieces:

B = −∫

V1/2

dz

2πiz b(z) +

∫V−1/2

dz

2πiz b(z) . (3.112)

For the first piece, the coordinates z′ and z are related as z′ = z − 1. The BPZconjugation of the first piece is thus given by

−∫

V1/2

dz

2πiz b(z)→ −

∫V−1/2

dz′

2πi(z′ + 1) b(z′) . (3.113)

For the second piece, the coordinates z′ and z are related as z′ = z + 1. The BPZconjugation of the second piece is thus given by∫

V−1/2

dz

2πiz b(z)→

∫V1/2

dz′

2πi(z′ − 1) b(z′) . (3.114)

Combining the two pieces, we obtain the following expression for the inner product〈B�ϕ, ψn 〉:

〈B�ϕ, ψn 〉 = −〈 f ◦ ϕ(0)∫

V1/2

dz

2πi(z − 1) b(z) c(1)B c(n+ 1) 〉Cn+2

−〈∫

V−1/2

dz

2πi(z + 1) b(z) f ◦ ϕ(0) c(1)B c(n+ 1) 〉Cn+2

= −〈 f ◦ ϕ(0)∫

V1/2

dz

2πi(z − 1) b(z) c(1)B c(n+ 1) 〉Cn+2

−〈 f ◦ ϕ(0) c(1)B c(n+ 1)∫

Vn+3/2

dz

2πi(z − n− 1) b(z) 〉Cn+2 , (3.115)

at Akadem



ownloaded from


1038 Y. Okawa

where the minus sign of the term with the line integral along V1/2 is from the an-ticommutation of b(z) and f ◦ ϕ(0) and we used the identification z ∼ z + n + 2 ofthe surface Cn+2. An important point of this expression is that the integrand of theline integral along V1/2 has a zero at the location of the c ghost insertion c(1) andthe integrand of the line integral along Vn+3/2 has a zero at the location of the cghost insertion c(n+ 1). Therefore, the pole of the OPE between the b and c ghostsis canceled for each case and we can simply anticommute the line integral and the cghost to find

〈B�ϕ, ψn 〉 = 〈 f ◦ ϕ(0) c(1)∫

V1+n/2

dz

2πi(z − 1) b(z)B c(n+ 1) 〉Cn+2

+〈 f ◦ ϕ(0) c(1)B∫

V1+n/2

dz

2πi(z − n− 1) b(z) c(n+ 1) 〉Cn+2 . (3.116)

Now the part of the line integral at �z = 1+n/2 proportional to B vanishes becauseB2 = 0,

〈B�ϕ, ψn 〉 = 〈 f ◦ ϕ(0) c(1)∫

V1+n/2

dz

2πiz b(z)B c(n+ 1) 〉Cn+2

+ 〈 f ◦ ϕ(0) c(1)B∫

V1+n/2

dz

2πiz b(z) c(n+ 1) 〉Cn+2 , (3.117)

and the remaining part of the line integral at �z = 1 +n/2 anticommutes with B tofind

〈B�ϕ, ψn 〉 = 0 . (3.118)

We have thus shown that ψn satisfies the Schnabl gauge condition for any n.

3.7. Schnabl’s analytic solution

The space of string fields where we look for solutions to the equation of motionhas been narrowed down to the form

α0 ψ0 + α1 ψ′0 + α2 ψ

′′0 + α3 ψ

(3)0 + . . .+ αn ψ

(n)0 + . . . , (3.119)

and we know that the coefficients αi can be determined recursively. Schnabl calcu-lated them to find

Ψ = − ψ0 +12ψ′

0 −16ψ′′

0

2!+

130ψ

(4)0

4!− 1

42ψ

(6)0

6!+

130ψ

(8)0

8!

− 566ψ

(10)0

10!+

6912730

ψ(12)0

12!− 7

6ψ

(14)0

14!+ . . . . (3.120)

What would be the general form of the coefficient αi? Schnabl inferred that thecoefficient αi can be written using the Bernoulli number Bn as follows:

Ψ = −∞∑

n=0

Bn

n!ψ

(n)0 , (3.121)

at Akadem



ownloaded from



where the Bernoulli number Bn is defined by

x

ex − 1=

∞∑n=0

Bn xn

n!. (3.122)

We will later prove that Ψ in the form (3.121) formally satisfies the equation ofmotion, but let us continue the route Schnabl followed in Ref. 1). Schnabl used theEuler-Maclaurin sum formula

∞∑k=0

Bk

k!

[f (k)(b)− f (k)(a)

]=

b−1∑n=a

f ′(n) , (3.123)

which is∞∑

k=0

Bk

k!

[ψ

(k)N+1 − ψ(k)

0

]=

N∑n=0

ψ′n (3.124)

for f(n) = ψn, a = 0, and b = N + 1, to turn the solution to the form

Ψ = −∞∑

k=0

Bk

k!ψ

(k)0 =

N∑n=0

ψ′n −

∞∑k=0

Bk

k!ψ

(k)N+1 . (3.125)

Then Schnabl defined the solution in the limit given by

Ψ = limN→∞

[N∑

n=0

ψ′n − ψN

]. (3.126)

Here derivatives ψ(k)N+1 with k > 0 are dropped because they vanish in the limit

N → ∞. In fact, the string field ψN , which is retained, also vanishes in the limitN →∞ in the following sense:

limN→∞

〈ϕ, ψN 〉 = 0 (3.127)

for any ϕ in the Fock space. However, the string field ψN gives a finite contributionin the limit N →∞ in the calculation of the energy of the solution, while derivativesψ

(k)N with k > 0 do not. The term with ψN is often called the phantom piece. It

would be important to understand why the inner product 〈ϕ, ψN 〉 vanishes in thelimit N →∞. The inner product 〈ϕ, ψN 〉 is given by

〈ϕ, ψN 〉 = 〈 f ◦ ϕ(0) c(1)B c(N + 1) 〉CN+2= 〈 c(−1)f ◦ ϕ(0) c(1)B 〉CN+2

, (3.128)

where we used the identification z ∼ z+N +2 of the surface CN+2 to bring c(N +1)to c(−1) so that the surface CN+2 of the last expression of the inner product shouldbe understood to be represented, for example, in the region −(N + 2)/2 ≤ �z ≤(N+2)/2. The operators c(−1)f◦ϕ(0) c(1) carrying ghost number 4 can be expandedaround the origin, and the expansion generically takes the form

c(−1)f ◦ ϕ(0) c(1) ∝ c∂c∂2c∂3c(0) + . . . , (3.129)

at Akadem



ownloaded from


1040 Y. Okawa

where c∂c∂2c∂3c is the operator carrying the lowest weight 2 in the sector of ghostnumber 4 and the dots represent operators whose weight is larger than 2. The Ndependence of the leading term is given by

〈 c∂c∂2c∂3c(0)B 〉CN+2=

1(N + 2)3

〈 c∂c∂2c∂3c(0)B 〉C1 , (3.130)

where we used the conformal transformation

z′ =1

N + 2z . (3.131)

We therefore find that〈ϕ, ψN 〉 ∼ O

( 1N3

)(3.132)

for a generic state ϕ. If the leading term c∂c∂2c∂3c(0) happens to be absent in theOPE of c(−1)f ◦ϕ(0) c(1), the inner product vanishes faster than 1/N3. We also seethat derivatives ψ(k)

N vanish faster than ψN :

〈ϕ, ψ(k)N 〉 ∼ O

( 1Nk+3

)(3.133)

for a generic state ϕ.Since the phantom piece ψN formally vanishes in the limit N →∞, we can write

the solution as

Ψ =∞∑

n=0

ψ′n. (formal) (3.134)

We can prove that Ψ in this form, again formally, satisfies the equation of motionfrom

QBψ′0 = 0 , QBψ

′n+1 = −

n∑m=0

ψ′m ∗ ψ′

n−m . (3.135)

These relations do not involve any limit and they hold without any subtlety.Let us prove these relations. The first relation states that ψ′

0 is BRST closed. Itis convenient to use the expression (3.105). The BRST transformation of ψ′

0 is givenby

〈ϕ,QBψ′0 〉 = 〈 f ◦ ϕ(0) (QB · c(1))L 〉C2 − 〈 f ◦ ϕ(0) c(1)QB · L 〉C2

− 〈 f ◦ ϕ(0) (QB · c∂c(1))B 〉C2 − 〈 f ◦ ϕ(0) c∂c(1)QB · B 〉C2 .(3.136)

Since

QB · c(z) = c∂c(z) , QB · c∂c(z) = 0 , QB · B = L , QB · L = 0 , (3.137)

we have

〈ϕ,QBψ′0 〉 = 〈 f ◦ ϕ(0) c∂c(1)L 〉C2 − 〈 f ◦ ϕ(0) c∂c(1)L 〉C2 = 0 . (3.138)

In fact, we can write

〈ϕ, ψ′0 〉 = − 〈 f ◦ ϕ(0) QB · ( c(1)B ) 〉C2 . (3.139)

at Akadem



ownloaded from



The string field ψ′0 is thus BRST exact. Since

QB · ( c(1)B ) = QB · ( 1− B c(1) ) = −QB · (B c(1) ) , (3.140)

the inner product 〈ϕ, ψ′0 〉 can also be written as

〈ϕ, ψ′0 〉 = 〈 f ◦ ϕ(0) QB · (B c(1) ) 〉C2 . (3.141)

In terms of the half integrals BR1 and BL

1 of B1 in (3.56), ψ′0 can thus be written as

ψ′0 = −QBB

R1 c1|0〉 (3.142)

or asψ′

0 = QBBL1 c1|0〉 . (3.143)

Let us next prove the second relation in (3.135). The BRST transformation ofψ′

n+1 is〈ϕ,QBψ

′n+1 〉 = 〈 f ◦ ϕ(0) c∂c(1)LB c(n+ 2) 〉Cn+3

− 〈 f ◦ ϕ(0) c(1)L2 c(n+ 2) 〉Cn+3

+ 〈 f ◦ ϕ(0) c(1)LB c∂c(n+ 2) 〉Cn+3 . (3.144)

The star product ψ′m ∗ ψ′

n−m with 0 < m < n is

〈ϕ, ψ′m ∗ ψ′

n−m 〉 = 〈 f ◦ ϕ(0) c(1)LB c(m+ 1) c(m+ 2)BL c(n+ 2) 〉Cn+1 . (3.145)

Let us calculate the part B c(m+ 1) c(m+ 2)B in the middle. When there is one cghost between two line integrals of B, we find

B c(z)B = B . (3.146)

This follows from {B, c(z) } = 1 and B2 = 0:

B c(z)B = B [ 1− B c(z) ] = B . (3.147)

When there are two c ghosts between two line integrals of B, we find

B c(z1) c(z2)B = B c(z1) [ 1− B c(z2) ]= B c(z1)− B c(z1)B c(z2) = B c(z1)− B c(z2) . (3.148)

Using this formula, the inner product 〈ϕ, ψ′m ∗ ψ′

n−m 〉 can be written as

〈ϕ, ψ′m ∗ ψ′

n−m 〉 = 〈 f ◦ ϕ(0) c(1)LB c(m+ 1)L c(n+ 2) 〉Cn+1

− 〈 f ◦ ϕ(0) c(1)LB c(m+ 2)L c(n+ 2) 〉Cn+1 . (3.149)

Note the simple dependence on m. The relation also holds in the limit m → 0 andin the limit m→ n. Using the results we obtained in deriving (3.105), we have

〈ϕ, ψ′0 ∗ ψ′

n 〉 = 〈 f ◦ ϕ(0) c(1)L2 c(n+ 2) 〉Cn+1

− 〈 f ◦ ϕ(0) c∂c(1)BL c(n+ 2) 〉Cn+1

− 〈 f ◦ ϕ(0) c(1)LB c(2)L c(n+ 2) 〉Cn+1 ,

〈ϕ, ψ′n ∗ ψ′

0 〉 = 〈 f ◦ ϕ(0) c(1)LB c(n+ 1)L c(n+ 2) 〉Cn+1

− 〈 f ◦ ϕ(0) c(1)LB c∂c(n+ 2) 〉Cn+1 . (3.150)

at Akadem



ownloaded from


1042 Y. Okawa

Therefore, we findn∑

m=0

〈ϕ, ψ′m ∗ ψ′

n−m 〉 = 〈 f ◦ ϕ(0) c(1)L2 c(n+ 2) 〉Cn+1

−〈 f ◦ ϕ(0) c∂c(1)BL c(n+ 2) 〉Cn+1

−〈 f ◦ ϕ(0) c(1)LB c∂c(n+ 2) 〉Cn+1 . (3.151)

We have thus shown the second relation in (3.135).We have actually shown that

QB Ψλ + Ψλ ∗ Ψλ = 0 (3.152)

for

Ψλ =∞∑

n=0

λn+1 ψ′n (3.153)

with formally any λ. We have a one-parameter family of solutions labeled by λ. Itis believed, however, that Ψλ is a pure gauge for |λ| < 1. Let us first rewrite Ψλ ina pure-gauge form. We will mostly omit the star symbol in the rest of this section.We have already shown that ψ′

0 is BRST exact:

ψ′0 = QB Φ , (3.154)

where Φ is defined by〈ϕ,Φ 〉 = 〈 f ◦ ϕ(0)B c(1) 〉C2 . (3.155)

Star products Φn are given by

〈ϕ,Φn 〉 = 〈 f ◦ ϕ(0)B c(1)B c(2) . . . B c(n) 〉Cn+1 = 〈 f ◦ ϕ(0)B c(n) 〉Cn+1 , (3.156)

and (QBΦ)Φn are

〈ϕ, (QBΦ)Φn 〉 = limε→0〈 f ◦ ϕ(0) c(1)LB c(1 + ε)B c(n) 〉Cn+1

= 〈 f ◦ ϕ(0) c(1)LB c(n) 〉Cn+1 . (3.157)

We findψ′

n = (QBΦ)Φn . (3.158)

The solution Ψλ can be written in terms of Φ. We have

Ψλ =∞∑

n=0

λn+1 ψ′n =

∞∑n=0

λn+1 (QBΦ)Φn = λ (QBΦ)1

1− λΦ . (3.159)

This is a pure-gauge form

e−Λ (QBeΛ ) = −(QBe

−Λ ) eΛ = λ (QBΦ)1

1− λΦ (3.160)

under the identificationeΛ =

11− λΦ . (3.161)

at Akadem



ownloaded from



3.8. KBc subalgebra

What we have derived so far can be summarized in a set of simple algebraicrelations of three string fields K, B, and c, which is often called KBc subalgebra.11)

Let us start with the wedge state Wα. As we have seen in (3.40), the wedge state isgenerated by the line integral L. We therefore write the wedge state Wα as

Wα = eαK , (3.162)

where the string field K is the wedge state W0 of vanishing width with an insertionof the line integral L:

〈ϕ,K 〉 = 〈 f ◦ ϕ(0)L 〉C1 . (3.163)

In particular, the vacuum state |0〉 is written as

|0〉 = eK . (3.164)

The algebra of the wedge state is now manifest:

eαKeβK = e(α+β)K . (3.165)

The zero-momentum tachyon state ψ0 can be represented as

ψ0 = eK/2ceK/2 , (3.166)

where the string field c is the wedge stateW0 of vanishing width with a local insertionof a c ghost on the boundary:

〈ϕ, c 〉 = 〈 f ◦ ϕ(0) c(1

2

)〉C1 . (3.167)

The string field ψn is written as

ψn = eK/2cBenKceK/2 , (3.168)

where the string field B is the wedge state W0 of vanishing width with an insertionof the line integral B:

〈ϕ,B 〉 = 〈 f ◦ ϕ(0)B 〉C1 . (3.169)

The string field ψ′n is then

ψ′n = eK/2cBKenKceK/2 . (3.170)

The string fields K, B, and c satisfy the following relations:

[K,B ] = 0 , {B, c } = 1 , c2 = 0 , B2 = 0 ,QBB = K , QBK = 0 , QBc = cKc . (3.171)

This set of relations is calledKBc subalgebra. The commutation relation [K,B ] = 0follows from (3.61), and the anticommutation relation {B, c } = 1 comes from (3.72).The relation c2 = 0 represents the property of the c ghost that c(z) c(z+ε) vanishes inthe limit ε→ 0, and the relation B2 = 0 is from (3.63). The BRST transformations

at Akadem



ownloaded from


1044 Y. Okawa

of B and K simply follow from QB · B and QB · L, respectively. As we calculatedin (2.96), the BRST transformation of c(z) is c∂c(z), and ∂c(z) can be realized asL c(z)− c(z)L, which is a special case of the relation (3.77). We therefore find thatQBc = c [K, c ] = cKc− c2K = cKc.

Let us prove that Ψλ satisfies the equation of motion only using KBc subalgebra.The string field Ψλ can be written as

Ψλ = λ eK/2cBK

1− λ eKceK/2 = f(K) c

BK

1− f(K)2c f(K) (3.172)

withf(K) =

√λ eK/2 . (3.173)

The BRST transformation of Ψλ is

QBΨλ = QB

[f(K) c

BK

1− f(K)2c f(K)

]= f(K) cKc

BK

1− f(K)2c f(K)

− f(K) cK2

1− f(K)2c f(K) + f(K) c

BK

1− f(K)2cKc f(K) . (3.174)

The product Ψ2λ is

Ψ2λ = f(K) c

BK

1− f(K)2c f(K)2 c

BK

1− f(K)2c f(K)

= f(K) cK

1− f(K)2Bc f(K)2 cB

K

1− f(K)2c f(K) . (3.175)

Since

Bc f(K)2 cB = Bc f(K)2 −Bc f(K)2Bc = Bc f(K)2 −BcB f(K)2 c

= Bc f(K)2 −B f(K)2 c = Bc f(K)2 − f(K)2Bc

= [Bc, f(K)2 ] = −[Bc, 1− f(K)2 ] , (3.176)

we find

Ψ2λ = −f(K) c

K

1− f(K)2[Bc, 1− f(K)2 ]

K

1− f(K)2c f(K)

= −f(K) cK

1− f(K)2BcKc f(K) + f(K) cKBc

K

1− f(K)2c f(K)

= −f(K) cK

1− f(K)2BcKc f(K) + f(K) cK(1− cB)

K

1− f(K)2c f(K)

= −f(K) cBK

1− f(K)2cKc f(K)

+ f(K) cK2

1− f(K)2c f(K)− f(K) cKc

BK

1− f(K)2c f(K) . (3.177)

at Akadem



ownloaded from



We have thus shown that QBΨλ + Ψ2λ = 0.

Note that we have never used the explicit form of f(K) in the proof. We haveactually shown that Ψ given by

Ψ = f(K) cBK

1− f(K)2c f(K) (3.178)

satisfies the equation of motion QBΨ + Ψ2 = 0 formally for any f(K). This stringfield Ψ can also be written as a pure-gauge form. The string field Φ defined by (3.155)can be generalized to

Φ = f(K)Bc f(K) . (3.179)

Star products Φn are

Φn = f(K)Bc f(K)2Bc f(K)2 . . . Bc f(K) = f(K)2n−1Bc f(K) , (3.180)

and (QBΦ)Φn are

(QBΦ)Φn = QB [ f(K)Bc f(K) ] f(K)2(n−1)+1Bc f(K)

= f(K) cKBc f(K)2nBc f(K) = f(K) cBK f(K)2n c f(K) . (3.181)

Therefore, Ψ in (3.178) can be written as the following pure-gauge form:

Ψ = f(K) cBK

1− f(K)2c f(K) = (QBΦ)

11− Φ (3.182)

with Φ defined in (3.179).Let us come back to Ψλ in (3.172) with f(K) =

√λ eK/2. An important difference

between the case |λ| < 1 and the case λ = 1 can be seen from this form of Ψλ. If weexpand the factor K/(1− f(K)2) in the middle of Ψλ in powers of K, we find

K

1− f(K)2=

K

1− λ eK= O(K) for |λ| < 1 . (3.183)

When λ = 1, the order of K changes:

K

1− eK= O(K0) . (3.184)

The expansion of the factor K/(1 − λ eK) in K corresponds to the expansion withrespect to the eigenvalue of L, and when λ = 1, we find that the solution acquiresthe zero-momentum tachyon term ψ0. In fact, since

K

1− eK= −

∞∑n=0

BnKn

n!, (3.185)

we recover the tachyon vacuum solution (3.121) in terms of the Bernoulli numberBn:

Ψλ=1 = eK/2cBK

1− eKceK/2 = −

∞∑n=0

Bn

n!eK/2cBKnceK/2 = −

∞∑n=0

Bn

n!ψ

(n)0 . (3.186)

at Akadem



ownloaded from


1046 Y. Okawa

On the other hand, the factor 1/(1 − eK) can also be expanded in eK . Then weformally have

K

1− eK=

∞∑n=0

KenK , (3.187)

but it turned out that there is an issue of convergence with this sum. If we truncatethe sum up to the term n = N , we find

N∑n=0

KenK =K(1− e(N+1)K)

1− eK, (3.188)

so K/(1− eK) can be written as

K

1− eK=

N∑n=0

KenK +K

1− eKe(N+1)K . (3.189)

In the limit N → ∞, component fields of the string field eNK are finite and theydefine the sliver state W∞, as we discussed before. This means that component fieldsof KneNK vanish for any positive integer n in the limit N → ∞. We thus defineK/(1− eK) as follows:

limN→∞

[N∑

n=0

KenK − eNK

]. (3.190)

This corresponds to the form of the tachyon vacuum solution (3.126) with the phan-tom piece.

It is sometimes convenient to introduce the trace notation

trAB ≡ 〈A,B 〉density (3.191)

for BPZ inner products following the analogy we mentioned before. BPZ innerproducts of two states which consist of K, B, and c can be reduced using the KBcsubalgebra to a superposition of traces of the form tr cet1Kcet2Kcet3K and of the formtr cet1Kcet2Kcet3Kcet4KB. In fact, the former tr cet1Kcet2Kcet3K is a special case ofthe latter tr cet1Kcet2Kcet3Kcet4KB with t4 = 0. In terms of correlation functions,they can be expressed as

tr cet1Kcet2Kcet3K = 〈 c(0) c(t1) c(t1 + t2) 〉Cn, density with n = t1 + t2 + t3 ,

(3.192)and

tr cet1Kcet2Kcet3Kcet4KB = 〈 c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3)B 〉Cn, density

with n = t1 + t2 + t3 + t4 . (3.193)

Using (3.18) we find

tr cet1Kcet2Kcet3K = −(nπ

)3sin

π t1n

sinπ t2n

sinπ t3n

with n = t1 + t2 + t3 .

(3.194)

at Akadem



ownloaded from



To calculate the correlation function in (3.193), let us consider the correlation func-tion given by

〈∫

V−ε

dz

2πiz b(z) c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3) 〉Cn (3.195)

with

n = t1 + t2 + t3 + t4 . (3.196)

We evaluate this in two ways. First, the b-ghost integral can be moved to the rightas follows:

〈∫

V−ε

dz

2πiz b(z) c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3) 〉Cn

= 〈[ ∮

dz

2πiz b(z) c(0)

]c(t1) c(t1 + t2) c(t1 + t2 + t3) 〉Cn

− 〈 c(0)∫

Vε

dz

2πiz b(z) c(t1) c(t1 + t2) c(t1 + t2 + t3) 〉Cn

= 〈[ ∮

dz

2πiz b(z) c(0)

]c(t1) c(t1 + t2) c(t1 + t2 + t3) 〉Cn

− 〈 c(0)[∮

dz

2πiz b(z) c(t1)

]c(t1 + t2) c(t1 + t2 + t3) 〉Cn

+ 〈 c(0) c(t1)∫

Vt1+ε

dz

2πiz b(z) c(t1 + t2) c(t1 + t2 + t3) 〉Cn . (3.197)

We further continue the procedure to obtain

〈∫

V−ε

dz


= 〈[ ∮

dz

2πiz b(z) c(0)

]c(t1) c(t1 + t2) c(t1 + t2 + t3) 〉Cn

− 〈 c(0)[∮

dz

2πiz b(z) c(t1)

]c(t1 + t2) c(t1 + t2 + t3) 〉Cn

+ 〈 c(0) c(t1)[ ∮

dz

2πiz b(z) c(t1 + t2)

]c(t1 + t2 + t3) 〉Cn

− 〈 c(0) c(t1) c(t1 + t2)[ ∮

dz

2πiz b(z) c(t1 + t2 + t3)

]〉Cn

+ 〈 c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3)∫

Vt1+t2+t3+ε

dz

2πiz b(z) 〉Cn . (3.198)

at Akadem



ownloaded from


1048 Y. Okawa

We therefore find

〈∫

V−ε

dz


= − t1 〈 c(0) c(t1 + t2) c(t1 + t2 + t3) 〉Cn

+ (t1 + t2) 〈 c(0) c(t1) c(t1 + t2 + t3) 〉Cn

− (t1 + t2 + t3) 〈 c(0) c(t1) c(t1 + t2) 〉Cn

+ 〈 c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3)∫

Vt1+t2+t3+ε

dz

2πiz b(z) 〉Cn . (3.199)

Second, using the identification z ∼ z+n of the surface Cn, the correlation functioncan be transformed to

〈∫

V−ε

dz


= 〈 c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3)∫

Vt1+t2+t3+t4−ε

dz

2πi(z − n) b(z) 〉Cn

= 〈 c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3)∫

Vt1+t2+t3+ε

dz

2πiz b(z) 〉Cn

− n 〈 c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3)B 〉Cn . (3.200)

We therefore find that

〈 c(0) c(t1) c(t1 + t2) c(t1 + t2 + t3)B 〉Cn

=t1n〈 c(0) c(t1 + t2) c(t1 + t2 + t3) 〉Cn

− t1 + t2n〈 c(0) c(t1) c(t1 + t2 + t3) 〉Cn

+t1 + t2 + t3

n〈 c(0) c(t1) c(t1 + t2) 〉Cn . (3.201)

The trace in (3.193) is thus given by

tr cet1Kcet2Kcet3Kcet4KB

= − t1n

(n

π

)3

sinπ(t1 + t2)

nsin

πt3n

sinπt4n

+t1 + t2n

(n

π

)3

sinπt1n

sinπ(t2 + t3)

nsin

πt4n

− t1 + t2 + t3n

(n

π

)3

sinπt1n

sinπt2n

sinπ(t3 + t4)

n(3.202)

withn = t1 + t2 + t3 + t4 . (3.203)

We can in principle calculate any trace of string fields made of K, B, and cusing the formulas (3.194) and (3.202). The kinetic energy of the tachyon vacuum

at Akadem



ownloaded from



solution in the form (3.126) with the phantom piece was calculated in Ref. 1). Thecalculation is technically complicated so that we only present the result, and we willcalculate the energy of a simpler solution for tachyon vacuum later. We have

K2 = limN→∞

N∑n=0

N∑m=0

〈ψ′n, QBψ

′m 〉density =

12− 1π2

,

K1 = limN→∞

N∑m=0

〈ψN , QBψ′m 〉density =

12

+1π2

,

K0 = limN→∞

〈ψN , QBψN 〉density =12

+1π2

, (3.204)

so that the kinetic term of the solution is evaluated as

〈Ψ,QBΨ 〉density = K2 − 2K1 +K0 = − 3π2

. (3.205)

The cubic term was evaluated in Refs. 11) and 12). It turned out that the phantompiece ψN is necessary for the equation of motion contracted with the solution

〈Ψ,QBΨ 〉+ 〈Ψ, Ψ2 〉 = 0 (3.206)

to be satisfied. The energy density of the solution normalized by the D25-branetension T25 is thus

1T25

1α′ 3g2

T

[12〈Ψ,QBΨ 〉density +

13〈Ψ, Ψ ∗ Ψ 〉density

]= 2π2

[12〈Ψ,QBΨ 〉density +

13〈Ψ, Ψ ∗ Ψ 〉density

]=π2

3〈Ψ,QBΨ 〉density = −1 .

(3.207)The value of the energy density predicted by Sen’s conjecture is thus reproducedanalytically.

3.9. Phantomless solution

Later Erler and Schnabl found a phantomless solution for the tachyon vacuum.13)

They chose the function f(K) to be

f(K) =1√

1−K . (3.208)

SinceK

1− f(K)2= K − 1 (3.209)

for this choice of f(K), the solution Ψ is

Ψ =1√

1−K cB(K − 1)c1√

1−K . (3.210)

If Ψ satisfies the equation of motion, Ψ = U(K)−1 Ψ U(K) also satisfies the equationof motion because the BRST transformation of any function of K vanishes. This

at Akadem



ownloaded from


1050 Y. Okawa

can also be considered as a gauge transformation analogous to a homogeneous gaugetransformation in ordinary gauge theory. Choosing U(K) = 1/

√1−K, we can

simplify the solution as

Ψ = U(K)−1 Ψ U(K) = cB(K − 1)c1

1−K . (3.211)

While Ψ satisfies the reality condition, Ψ does not. The factors 1/√

1−K and1/(1−K) should be understood as being defined using the Laplace transformation:

1√1−K =

1√π

∫ ∞

0dte−t

√tetK ,

11−K =

∫ ∞

0dt e−(1−K) t =

∫ ∞

0dt e−t etK .

(3.212)Then these can be interpreted as superpositions of wedge states.

The calculation of the energy is much simpler than that of the solution in Schnablgauge. The potential V (Ψ) normalized by the D-brane tension is

V (Ψ) ≡ V (Ψ)T25

= 2π2 tr[

12Ψ QBΨ +

13Ψ3

]. (3.213)

Obviously, V (Ψ) = V (Ψ). The solution Ψ can be written as

Ψ = cB(K − 1)c1

1−K = [QB(Bc) ]1

1−K − c1

1−K . (3.214)

The BRST transformation of Ψ is

QBΨ = −(QBc)1

1−K = −cKc 11−K , (3.215)

and Ψ QBΨ can be written as

Ψ QBΨ = −[QB(Bc) ]1

1−K (QBc)1

1−K + c1

1−K cKc1

1−K= −QB

[Bc

11−K (QBc)

11−K

]+ c

11−K cKc

11−K . (3.216)

The first term is BRST exact and thus does not contribute to the energy. Thenormalized potential is

V (Ψ) =π2

3tr Ψ QBΨ =

π2

3tr c

11−K cKc

11−K . (3.217)

Since

tr cet1KcKcet2K = −( t1 + t2

π

)2sin

π t1t1 + t2

sinπ t2t1 + t2

= −( t1 + t2

π

)2sin2 π t1

t1 + t2,

(3.218)we have

V (Ψ) = −π2

3

∫ ∞

0dt1

∫ ∞

0dt2 e

−t1 e−t2( t1 + t2

π

)2sin2 π t1

t1 + t2. (3.219)

at Akadem



ownloaded from



Changing the variables as

u = t1 + t2 , v =t1

t1 + t2, (3.220)

we evaluate the normalized potential as follows:

V (Ψ) = −13

∫ ∞

0duu3 e−u

∫ 1

0dv sin2 π v = −1

3Γ (4)× 1

2= −1 . (3.221)

We have thus reproduced the value predicted by Sen’s conjecture.

§4. Analytic solutions for marginal deformations

4.1. Marginal deformations with regular operator products

Using string field theory, we can potentially discuss various backgrounds of stringtheory. In the current construction of string field theory, however, we need to chooseone consistent background to formulate the theory, and other backgrounds are ex-pected to be described by classical solutions.

In the case of the open string, a consistent background corresponds to a bound-ary CFT with c = 26. Deformations of the boundary condition which preservethe conformal invariance are called marginal deformations, and we expect a one-parameter family of solutions to the equation of motion of open string field theoryfor each marginal deformation.

The deformation of the boundary CFT

SBCFT → SBCFT + λ

∫∂Σ

dt V (t) , (4.1)

where we denoted the boundary of the world-sheet by ∂Σ, is marginal to O(λ) if Vis a primary field of weight 1. For example, the operator V (t) given by

V (t) =i√2α′ ∂tX

μ(t) (4.2)

corresponds to turning on a constant mode of the gauge field on the D-brane, andthe operator V (t) given by

V (t) =i√2α′ ∂⊥X

α(t) , (4.3)

where ∂⊥ is the derivative normal to the boundary, corresponds to a transversecoordinate of the D-brane. A more nontrivial example is

V (t) =√

2 : cosXμ(t)√α′ : , (4.4)

which deforms the Neumann boundary condition to the Dirichlet boundary condi-tion. For example, the D25-brane is deformed into a periodic array of D24-branesat a particular value of the deformation parameter λ.

at Akadem



ownloaded from


1052 Y. Okawa

For any primary field V of weight 1 in the matter sector, the operator cV isBRST closed:

QB · cV (t) =∮

dz

2πijB(z) cV (t) = 0 , (4.5)

where the contour of the integral encircles the point t counterclockwise. Therefore,the string field Ψ (1) given by

〈ϕ, Ψ (1) 〉 = 〈 f ◦ ϕ(0) cV (1) 〉C2 (4.6)

satisfies the linearized equation of motion:

QB Ψ(1) = 0 . (4.7)

When the deformation is exactly marginal, we expect a solution of the form

Ψ =∞∑

n=1

λn Ψ (n) (4.8)

to the equation of motion QB Ψ + Ψ ∗ Ψ = 0 . Since the equation of motion has tobe satisfied for any λ, we have the following equations of motion for Ψ (n):

QB Ψ(1) = 0 ,

QB Ψ(2) = −Ψ (1) ∗ Ψ (1) ,

...

QB Ψ(n) = −

n−1∑m=1

Ψ (m) ∗ Ψ (n−m) . (4.9)

Formally, the equation of motion for Ψ (2) can be solved by

Ψ (2) = − b0L0

[Ψ (1) ∗ Ψ (1)

]. (4.10)

Combined actions of b0/L0 and the star multiplication, however, are in general com-plicated.

When operator products of the marginal operator

V (t1)V (t2) . . . V (tn) (4.11)

are regular, analytic solutions were constructed in Refs. 14) and 15) . The basic ideais to replace b0/L0 by the corresponding operator B/L in the sliver frame:

Ψ (2) = −BL

[Ψ (1) ∗ Ψ (1)

]. (4.12)

We can write this as

Ψ (2) = −∫ ∞

0dT B e−TL

[Ψ (1) ∗ Ψ (1)

], (4.13)

at Akadem



ownloaded from



although there is some subtlety to be discussed later. After a few nontrivial steps,Ψ (2) can be expressed as

〈ϕ, Ψ (2) 〉 =∫ 1

0dt 〈 f ◦ ϕ(0) cV (1)B cV (1 + t) 〉C2+t . (4.14)

Instead of explaining the derivation of this expression, let us confirm that Ψ (2) satis-fies the equation of motion, which is in fact much easier. The BRST transformationof Ψ (2) is given by

〈ϕ,QB Ψ(2) 〉 = −

∫ 1

0dt 〈 f ◦ ϕ(0) cV (1)L cV (1 + t) 〉C2+t

= −∫ 1

0dt

∂

∂t〈 f ◦ ϕ(0) cV (1) cV (1 + t) 〉C2+t . (4.15)

There are two surface terms. At t = 1, we find

− 〈 f ◦ ϕ(0) cV (1) cV (2) 〉C3 = − 〈ϕ, Ψ (1) ∗ Ψ (1) 〉 . (4.16)

The other surface term at t = 0 vanishes when

limε→0

cV (0) cV (ε) = 0 . (4.17)

Therefore, iflimε→0

V (0)V (ε) (4.18)

is finite or vanishing, Ψ (2) is finite and satisfies the equation of motion QB Ψ(2) =

− Ψ (1) ∗ Ψ (1). It is also easy to see that Ψ (2) satisfies the Schnabl gauge conditionbecause it takes the form of an integral of a state whose ghost part is ψt.

Once we understand how Ψ (2) in the form of (4.14) satisfies the equation ofmotion, it is easy to construct Ψ (n) satisfying the equation of motion. It is given by

〈ϕ, Ψ (n) 〉 =∫ 1

0dt1

∫ 1

0dt2 . . .

∫ 1

0dtn−1 〈 f ◦ ϕ(0) cV (1)B cV (1 + t1)

× B cV (1 + t1 + t2) . . .B cV (1 + t1 + t2 + . . .+ tn−1) 〉C2+t1+t2+...+tn−1.

(4.19)Introducing the length parameters

�i ≡i∑

k=1

tk , (4.20)

the solution can be written more compactly as

〈ϕ, Ψ (n) 〉 =∫ 1

0dt1

∫ 1

0dt2 . . .

∫ 1

0dtn−1 〈 f ◦ φ(0) cV (1)

n−1∏i=1

[B cV (1 + �i)

]〉C2+�n−1

.

(4.21)

at Akadem



ownloaded from


1054 Y. Okawa

Let us now prove that the equation of motion is satisfied for Ψ (n). It is straightfor-ward to generalize the calculation of 〈φ,QBΨ

(2) 〉 to that of 〈φ,QBΨ(n) 〉 . Since the

operator cV is BRST closed, the BRST operator acts only on the insertions of B:

〈ϕ,QBΨ(n) 〉 = −

n−1∑j=1

∫ 1

0dt1

∫ 1

0dt2 . . .

∫ 1

0dtn−1 〈 f ◦ φ(0) cV (1)

j−1∏i=1

[B cV (1 + �i)

]

×L cV (1 + �j)n−1∏

k=j+1

[B cV (1 + �k)

]〉C2+�n−1

. (4.22)

An insertion of L between cV (1 + �j−1) and cV (1 + �j) corresponds to taking aderivative with respect to tj . When operator products of V are regular, we have

〈ϕ,QBΨ(n) 〉 = −

n−1∑j=1

∫ 1

0dt1

∫ 1

0dt2 . . .

∫ 1

0dtn−1

∂

∂tj〈 f ◦ φ(0) cV (1)

×j−1∏i=1

[B cV (1 + �i)

]cV (1 + �j)

n−1∏k=j+1

[B cV (1 + �k)

]〉C2+�n−1

= −n−1∑j=1

∫ 1

0dt1 . . .

∫ 1

0dtj−1

∫ 1

0dtj+1 . . .

∫ 1

0dtn−1 〈 f ◦ φ(0) cV (1)

×j−1∏i=1

[B cV (1 + �i)

]cV (1 + �j)

n−1∏k=j+1

[B cV (1 + �k)

]〉C2+�n−1

∣∣∣∣tj=1

= −n−1∑j=1

〈ϕ, Ψ (j) ∗ Ψ (n−j) 〉 . (4.23)

The equation of motion is thus satisfied.An important example of marginal operators which satisfy the regularity condi-

tion isV (t) = : exp

[ 1√α′ X

0(t)]

: . (4.24)

This describes a time-dependent decay process of an unstable D-brane, which isusually referred to as rolling tachyon. Actually, the deformation by

λ

∫∂Σ

dt : exp[ 1√

α′ X0(t)

]: (4.25)

does not generate a one-parameter family of physically distinct solutions becausechanging the absolute value of λ without changing the sign can be absorbed by ashift in the origin of the time coordinate. However, the construction of the solutioncan be applied to this case without any problem. It is an exact time-dependentsolution incorporating all α′ corrections. By the way, it is difficult to deal with time-dependent solutions in string theory, but in string field theory it is conceptually the

at Akadem



ownloaded from



same problem as that in ordinary field theory. This can be considered as one of theadvantages of string field theory.

While the rolling tachyon is an interesting example of V (t) with regular operatorproducts, the marginal operator in general has singular operator products. Since itis a primary field of weight 1, its typical OPE is

V (t1)V (t2) ∼ 1(t1 − t2)2 . (4.26)

In this case, the construction we described has to be modified. One possible approachwas presented in Ref. 15). First, the integral of ti is regularized as follows:∫ 1

0dti →

∫ 1

εdti . (4.27)

In (4.13), this corresponds to regularizing the integral for large T . Namely, therepresentation of B/L used in (4.13) is not well defined for V (t) with the OPE (4.26).The equation of motion is no longer satisfied because of the regularization, but wecan add counterterms such that the equation of motion is satisfied and the solutionis finite in the limit ε → 0. In Ref. 15) Ψ (2) and Ψ (3) were constructed in this way.It was also found that there exists an obstruction in constructing a solution whenthe deformation is not exactly marginal. It is also interesting to note that whilethe solution when operator products of the marginal operator are regular satisfiesthe Schnabl gauge condition, the solution for the singular case does not satisfy theSchnabl gauge condition. In fact, there does not seem to exist a solution in Schnablgauge.

4.2. Solutions for general marginal deformations

The solution Ψ (n) for marginal deformations with regular operator products inthe preceding subsection consists of n unintegrated vertex operators, n− 1 integralsof moduli, and n − 1 corresponding b-ghost insertions. In attempting to generalizethe solution for the singular case, one source of difficulties came from the moduliintegrals. The solution Ψ (n) is represented on wedge states of various lengths, whichmakes the regularization complicated. A different approach to the construction ofsolutions for marginal deformations with singular operator products was presentedin Ref. 16). The new solution Ψ (n) consists of one unintegrated vertex operator andn − 1 integrated vertex operators. The moduli integral is absent and the solutionΨ (n) is represented on a single surface Cn+1.

The unintegrated vertex operator cV is BRST closed:

QB · cV (t) = 0 . (4.28)

The integrated vertex operator

V (a, b) =∫ b

adt V (t) (4.29)

is BRST closed up to boundary terms:

QB · V (a, b) = QB ·∫ b

adt V (t) =

∫ b

adt ∂t

[cV (t)

]= cV (b)− cV (a) . (4.30)

at Akadem



ownloaded from


1056 Y. Okawa

It follows only from these basic properties of unintegrated and integrated vertexoperators that Ψ (2)

L given by

〈ϕ, Ψ (2)L 〉 = 〈 f ◦ ϕ(0)

∫ 2

1dt cV (1)V (t) 〉C3 (4.31)

satisfies the equation of motion. In fact, the BRST transformation of Ψ (2)L can be

calculated from

QB ·∫ 2

1dt cV (1)V (t) = −

∫ 2

1dt cV (1) ∂t

[cV (t)

]= − cV (1) cV (2) (4.32)

when the operator product V (1)V (t) is regular in the limit t → 1, and we confirmthat QBΨ

(2)L = − Ψ (1) ∗ Ψ (1). It is also easy to verify that Ψ (3)

L given by

〈ϕ, Ψ (3)L 〉 = 〈 f ◦ ϕ(0)

12cV (1)

[∫ 3

1dt V (t)

]2

〉C4

− 〈 f ◦ ϕ(0)12cV (1)

[ ∫ 3

2dt V (t)

]2

〉C4 (4.33)

satisfies the equation of motion QBΨ(3)L = −Ψ (1) ∗ Ψ (2)

L − Ψ (2)L ∗ Ψ (1) when operator

products of the marginal operator are regular. The solution Ψ (3)L can also be written

as

〈ϕ, Ψ (3)L 〉 = 〈 f ◦ ϕ(0)

∫ 2

1dt1

∫ 3

t1

dt2 cV (1)V (t1)V (t2) 〉C4 , (4.34)

and this form of Ψ (3)L generalizes to Ψ (n)

L given by

〈ϕ, Ψ (n)L 〉 = 〈 f ◦ ϕ(0)

∫ 2

1dt1

∫ 3

t1

dt2

∫ 4

t2

dt3 . . .

∫ n

tn−2

dtn−1 cV (1)

× V (t1)V (t2) . . . V (tn−1)〉Cn+1 . (4.35)

The integration region is complicated, but it can be decomposed and the solutioncan be constructed only from operators of the forms V (a, b)m and cV (a)V (a, b)m

with appropriate values of the parameters a, b, and m. These are the operatorswhich appear when we expand eλV (a,b) and λ cV (a) eλV (a,b) in λ as

eλV (a,b) =∞∑

m=0

λm

m!V (a, b)m , λ cV (a) eλV (a,b) =

∞∑m=0

λm+1

m!cV (a)V (a, b)m , (4.36)

and we only need the relations

QB · eλV (a,b) = eλV (a,b) λ cV (b)− λ cV (a) eλV (a,b) ,

QB ·[λ cV (a) eλV (a,b)

]= − λ cV (a) eλV (a,b) λ cV (b) (4.37)

in proving that Ψ (n)L satisfies the equation of motion.

at Akadem



ownloaded from



These are the relations we expect when the operator eλV (a,b) implements thechange of the boundary condition for the region between a and b. The effect ofthe BRST transformation localizes to the points where the boundary condition ischanged, and it can be interpreted in terms of boundary condition changing op-erators. When operator products of the marginal operator V (t) are singular, theoperator eλV (a,b) has to be properly renormalized, but there should be a renormal-ized operator [ eλV (a,b) ]r which implements the change of the boundary condition forthe region between a and b, and there should be relations which generalize (4.37)and can be interpreted in terms of boundary condition changing operators. Thenanalytic solutions can be constructed systematically from [ eλV (a,b) ]r.16)

§5. Future directions

When the first analytic solution was constructed by Schnabl,1) it was not clearwhether it is limited to the particular problem of tachyon condensation or it can bemore universal. Later analytic solutions for marginal deformations were constructed,and they provided a perspective that the development is more universal. Further-more, the development does not depend on the bosonic nature of the problem, andthere is no immediate obstruction to the extension to the superstring. In fact, an-alytic solutions for marginal deformations were extended to open superstring fieldtheory. The development is not even restricted to the construction of classical solu-tions. As can be seen from the construction of the analytic solutions for marginaldeformations, we have better control over the combination of the star multiplicationand the propagator in Schnabl gauge. It is therefore promising to calculate scat-tering amplitudes analytically in Schnabl gauge. Four-point amplitudes in Schnablgauge at the tree level were discussed in detail,17) and the generalization to one-loopamplitudes has also been explored.18)

We have seen that open string field theory is able to describe tachyon conden-sation. How far can we explore nonperturbative phenomena using open string fieldtheory?

(i) Can open string field theory describe other backgrounds?

The current formulation of open string field theory requires a choice of oneboundary CFT. The question is whether open string field theory formulated aroundthe chosen boundary CFT describes backgrounds corresponding to other boundaryCFT’s. For backgrounds connected by marginal deformations, analytic solutionshave been constructed to all orders in the deformation parameter. Whether or notthe expansion has a finite radius of convergence would depend on a choice of theoriginal boundary CFT and details of marginal deformations, and little has beenunderstood so far.

In Siegel gauge, numerical solutions in level truncation have been constructed,where the deformation parameter is treated nonperturbatively.19) It turned out thatthe branch of the solutions truncates at a finite distance from the origin. While thisis not an immediate problem, the correspondence between the space of boundary

at Akadem



ownloaded from


1058 Y. Okawa

CFT’s and the space of solutions in open string field theory, if we assume that itexists, may not be so simple. It may also be related to the issue that the gaugecondition may not be imposed globally in the configuration space of string fields.

For backgrounds which are not continuously connected to the original back-ground, the situation is more complicated. Analytic solutions for relevant deforma-tions were proposed in Ref. 20), but later it was pointed out21) that the equationof motion was not completely satisfied, while the energy and the coupling to closedstrings were correctly reproduced. Since the violation of the equation of motionseems to be related to the assumption of nontrivial cohomology,21) this may not bea technical problem, and it might be related to a more fundamental issue of whatthe space of string fields is. On the other hand, it would still be possible that theproblem comes from the particular ansatz and that we can find solutions in differentgauges. In fact, numerical solutions for such relevant deformations were constructedin Siegel gauge.

(ii) Can open string field theory describe creation of D-branes?

We often say that D-branes are kind of solitonic objects in string theory, andopen strings correspond to collective excitations on the soliton. In this perspective,the tachyon vacuum solution in open string field theory is remarkable because itcorresponds to the description of the annihilation of the soliton in terms of its col-lective excitations. Then how about creation of the soliton? Namely, starting fromopen string field theory formulated around a background with one D-brane, can weconstruct solutions for multiple D-branes? Such multi-brane solutions were proposedby Murata and Schnabl,22),23) and it was demonstrated that the energy and the cou-pling to closed strings can be reproduced correctly. The solutions, however, containsome singularities and it is intensively discussed how to treat such singularities.

(iii) Can open string field theory describe closed strings?

If we succeed in quantizing open string field theory consistently and calculatingon-shell loop amplitudes, there should be poles corresponding to closed strings. Thenfrom the standard argument using unitarity there should be external closed stringsas well in perturbation theory. Therefore, the question of whether open string fieldtheory is able to describe closed strings is closely related to the question of whetheropen string field theory can be quantized consistently.

Imagine a closed string field theory on a background with one space-like di-mension compactified on a circle. Truncation to closed string fields with vanishingwinding number is classically consistent. For example, in perturbation theory of on-shell scattering amplitudes, closed strings with nonvanishing winding number neverappear in tree-level Feynman diagrams when all external closed strings carry vanish-ing winding number. Such truncation, however, is not consistent in quantum theory.In loop amplitudes, pair creation of closed strings with nonvanishing winding num-ber is possible, and we will not be able to forbid such processes consistently. Weknow in string perturbation theory that the inconsistency is related to the viola-

at Akadem



ownloaded from



tion of modular invariance for such truncation. In the context of string field theory,the inconsistency would manifest itself as anomaly of the gauge invariance uponquantization.

Classically, open string field theory is a consistent gauge theory. The questionis whether we can preserve the gauge invariance upon quantization without addingindependent degrees of freedom such as closed string fields. This question has beendiscussed in open bosonic string field theory, but the quantization of the bosonictheory is necessarily formal because of the tachyons in the open string sector and inthe closed string sector, and no definite conclusion has been obtained. We shouldconsider quantization of open superstring field theory where tachyons are absent.In particular, as we mentioned earlier, the recent analytic developments can be ex-tended to the superstring, and we may be able to carry out analytic calculations inSchnabl gauge at least at one loop. We should seriously consider quantization ofopen superstring field theory now!

With the analytic methods that have been developed so far, we are now ableto explore these directions of research more concretely than before. We hope thatfurther developments of string field theory, together with those of various otherapproaches, will help us uncover the nature of the nonperturbative theory underlyingstring theory.

Acknowledgements

This work was supported in part by Grants-in-Aid for Scientific Research (B)No. 20340048 and (C) No. 24540254 from the Japan Society for the Promotion ofScience (JSPS). The work was also supported in part by JSPS, the Academy ofSciences of the Czech Republic (ASCR), and the MSMT contract No. LH11106under the Research Cooperative Program between Japan and the Czech Republic.

References

1) M. Schnabl, Adv. Theor. Math. Phys. 10 (2006), 433, hep-th/0511286.2) E. Fuchs and M. Kroyter, Phys. Rep. 502 (2011), 89, arXiv:0807.4722.3) J. Polchinski, String theory, Vol. 1: An introduction to the bosonic string (Cambridge

University Press, 1998).4) E. Witten, Nucl. Phys. B 268 (1986), 253.5) A. Sen and B. Zwiebach, J. High Energy Phys. 03 (2000), 002, hep-th/9912249.6) Y. Okawa, J. High Energy Phys. 07 (2002), 003, hep-th/0204012.7) D. Gaiotto and L. Rastelli, J. High Energy Phys. 08 (2003), 048, hep-th/0211012.8) I. Kishimoto, Prog. Theor. Phys. Suppl. No. 188 (2011), 155.9) L. Rastelli and B. Zwiebach, J. High Energy Phys. 09 (2001), 038, hep-th/0006240.

10) M. R. Gaberdiel and B. Zwiebach, Nucl. Phys. B 505 (1907), 569, hep-th/9705038.11) Y. Okawa, J. High Energy Phys. 04 (2006), 055, hep-th/0603159.12) E. Fuchs and M. Kroyter, J. High Energy Phys. 05 (2006), 006, hep-th/0603195.13) T. Erler and M. Schnabl, J. High Energy Phys. 10 (2009), 066, arXiv:0906.0979.14) M. Schnabl, Phys. Lett. B 654 (2007), 194, hep-th/0701248.15) M. Kiermaier, Y. Okawa, L. Rastelli and B. Zwiebach, J. High Energy Phys. 01 (2008),

028, hep-th/0701249.16) M. Kiermaier and Y. Okawa, J. High Energy Phys. 11 (2009), 041, arXiv:0707.4472.17) L. Rastelli and B. Zwiebach, J. High Energy Phys. 01 (2008), 018, arXiv:0708.2591.

at Akadem



ownloaded from


1060 Y. Okawa

18) M. Kiermaier and B. Zwiebach, J. High Energy Phys. 07 (2008), 063, arXiv:0805.3701.19) A. Sen and B. Zwiebach, J. High Energy Phys. 10 (2000), 009, hep-th/0007153.20) L. Bonora, C. Maccaferri and D. D. Tolla, J. High Energy Phys. 11 (2011), 107,

arXiv:1009.4158.21) T. Erler and C. Maccaferri, J. High Energy Phys. 11 (2011), 092, arXiv:1105.6057.22) M. Murata and M. Schnabl, Prog. Theor. Phys. Suppl. No. 188 (2011), 50, arXiv:1103.1382.23) M. Murata and M. Schnabl, J. High Energy Phys. 07 (2012), 063, arXiv:1112.0591.

at Akadem



ownloaded from


Prog. Theor. Phys. 2012 Okawa 1001 60

Documents

perturbation theory

free theory

quantum eld theory

string theoryis

string theorywill

eld theory of strings

conformal field theory

aconsistent quantum