aa r X i v : . [ h e p - t h ] F e b UTHEP-663
Comments on Takahashi-Tanimoto’s scalar solution
Nobuyuki Ishibashi ∗ Graduate School of Pure and Applied Sciences, University of Tsukuba,Tsukuba, Ibaraki 305-8571, Japan
Abstract
We study the identity-based solution of Witten’s cubic bosonic open string field theory constructedby Takahashi and Tanimoto, which is claimed to describe the tachyon vacuum. We argue that theobservables of the solution coincide with those of the tachyon vacuum using the method proposed byKishimoto and Takahashi. We also discuss how to treat the kinetic term of the string field theoryexpanded around it. ∗ e-mail: [email protected] Introduction
Since the discovery of the tachyon vacuum solution by Schnabl [1], various kinds of analytic solutions of theequation of motion of the cubic bosonic open string field theory [2] have been constructed (for reviews, see[3, 4, 5]). It is now possible to construct a solution corresponding to any known open string background [6].Most of the solutions found since [1] are so-called regular solutions which consist mainly of wedge stateswith non vanishing width with operator insertions. There exist some solutions which are not of this kind.An example is the solutionΨ TT = (cid:20)Z C left dξ πi (cid:0) e h a − (cid:1) j B ( ξ ) − Z C left dξ πi ( ∂h a ) e h a c ( ξ ) (cid:21) | I i , (1.1)given by Takahashi and Tanimoto [7], which is called the scalar solution. Here C left is a contour in the upperhalf plane depicted in Fig. 1, j B is the BRST current j B ( ξ ) = (cid:20) cT + bc∂c + 32 ∂ c (cid:21) ( ξ ) , (1.2) | I i is the identity string field and h a ( ξ ) is a function taken to be h a ( ξ ) = ln a (cid:18) ξ + 1 ξ (cid:19) ! , (1.3)for a ≥ − . Takahashi and Tanimoto claim that while the solution is a pure gauge solution for a > − , itis a tachyon vacuum solution for a = − . Figure 1: C left The solution (1.1) is expressed as an identity state with local operator insertions. The solutions of sucha form are called identity-based solutions. It is difficult to calculate observables like energy or Ellwoodinvariant of identity-based solutions. These quantities correspond to correlation functions of operators ona strip with vanishing width in the worldsheet theory and naive regularizations fail to yield definite values[8, 9, 10].On the other hand, the identity-based solutions have some advantages. In general, the string field actionexpanded around a classical solution Ψ cl can be given as S ′ [Ψ] = − g Z (cid:20)
12 Ψ Q ′ Ψ + 13 ΨΨΨ (cid:21) , (1.4)1here Q ′ A = QA + Ψ cl A − ( − | Ψ cl || A | A Ψ cl . In the case of regular solutions, Ψ cl involves wedge states with finite width and it will be very difficult to studythe string field theory action (1.4) with the kinetic operator Q ′ . However, if Ψ cl is an identity-based solution,the Q ′ can be expressed by local operators on the worldsheet. For example, if Ψ cl is the Takahashi-Tanimotosolution (1.1), the Q ′ becomes I dξ πi h e h a j B ( ξ ) − ( ∂h a ) e h a c ( ξ ) i . (1.5)With Q ′ being an operator like this, we expect it is relatively easy to deal with the string field theory action(1.4).Although the observables are not available, there are many evidences indicating that the Takahashi-Tanimoto solution (1.1) with a = − is a tachyon vacuum solution: • There are no physical open string excitations around the background corresponding to a = − . Thisfact has been shown by studying the BRST cohomology [11] or by constructing the homotopy operator[12]. • The open string amplitudes around the background can be shown to vanish [13]. • Solving the equation of motion in the background corresponding to a = − numerically, an unstablesolution which is supposed to correspond to the perturbative vacuum can be found [14, 15, 16].All these evidences imply that the solution corresponds to the tachyon vacuum. It should be interesting toexplore the string field theory around such a background and see whether or not the closed string amplitudescan be reproduced from it. Since the solution is an identity-based solution, the string field theory expandedaround the solution will have a tractable kinetic term.In this paper, we would like to study the Takahashi-Tanimoto solution (1.1) with a = − and the stringfield theory expanded around it. What we will do first is to evaluate the observables of the solution in arather indirect manner. In a recent paper [17], the authors consider the Erler-Schnabl solutions in the stringfield theory expanded around the identity-based marginal solutions found in [18, 7]. Since the Erler-Schnablsolutions will correspond to the tachyon vacuum, by calculating the observables of these solutions, they areable to evaluate the observables of the identity-based marginal solutions. We here apply this method to thescalar solution (1.1) with a = − and see what we can say about the observables of it. By doing so, we willget further evidences for the claim that the solution is a tachyon vacuum solution. In the latter half of thepaper, we will discuss the string field theory expanded around the solution. We will show how we shouldtreat the kinetic operator (1.5) in order for the solution to correspond to the tachyon vacuum.The organization of this paper is as follows. In section 2, we evaluate the observables of the Takahashi-Tanimoto solution by calculating those of the Erler-Schnabl solution in the string field theory expandedaround it. In section 3, we consider the string field theory around the Takahashi-Tanimoto solution anddiscuss how we should treat the kinetic operator. Section 4 is devoted to conclusions and discussions. Inappendix A, we discuss the method proposed recently by Maccaferri [19] to construct regular solutions gaugeequivalent to identity-based solutions. We explain what we can get by applying the method to the solution(1.1). In appendix B, we derive some identities concerning the operators U, U − which play important rolesin the main text. 2 ote added In the workshop “String field theory and related aspects VI, SFT2014” (July 28 -August 1, 2014, SISSAItaly), where this work is presented [20], we have learned that Kishimoto, Masuda and Takahashi work onthe same problem from a different point of view [21][22]. Their results have some overlap with those insection 2.While this paper was being typed, a paper [23] appeared on the arXiv, which also treat the sameproblem. There is some overlap with the contents of appendix A but the identity-based solution they dealwith is different from ours.
The Erler-Schnabl solution [24] Ψ ES = 11 + K ( c + Q ( Bc )) , (2.1)satisfies the equation of motion of the cubic string field theory. Here K, B, c are the string fields defined by B = Z + i ∞ − i ∞ dz πi b ( z ) | I i ,c = c ( z ) | z = | I i ,K = QB = Z + i ∞ − i ∞ dz πi T ( z ) | I i , and the product of them is the star product. z is the sliver frame coordinate which is expressed by the upperhalf plane coordinate ξ in (1.1) as z = 2 π arctan ξ .K, B, c and Q satisfy the so-called KBc algebra [25, 26] and one can show that Ψ ES is a solution by usingthe algebra. The Erler-Schnabl solution Ψ ES describes the tachyon vacuum. This fact can be shown bycalculating the observables or by showing that A = B
11 +
K , gives the homotopy operator for the background Ψ ES , i.e. QA + Ψ ES A + A Ψ ES = 1 [27]. The existence ofthe homotopy operator implies that there exist no physical open string states around the background Ψ ES .As was pointed out in [28], it is straightforward to construct the Erler-Schnabl solution in the string fieldtheory (1.4) expanded around an identity-based solution. Q ′ is a nilpotent operator and acts on string fieldsas a derivation. It is easy to see that Ψ ′ ES = 11 + K ′ ( c + Q ′ ( Bc )) . (2.2)3ith K ′ = Q ′ B , satisfies the equation of motion derived from the string field action (1.4), because the K ′ , B, c and Q ′ satisfythe same algebra as the KBc and Q do. Moreover, the homotopy operator for the solution Ψ ′ ES can beconstructed as A ′ = B
11 + K ′ . Therefore one can argue that the solution Ψ ′ ES describes the tachyon vacuum, provided K ′ is a regularquantity.Let us consider the Erler-Schnabl solution Ψ ′ ES in the string field theory expanded around the Takahashi-Tanimoto solution given in (1.1) with a = − . In this case, Q ′ is expressed by a contour integral I dz πi (cid:20) − sin πz cos πz j B ( z ) + 4 π cos πz c ( z ) (cid:21) , (2.3)in the sliver frame and K ′ becomes K ′ = K + J ,J ≡ Z + i ∞ − i ∞ dz πi (cid:20) − πz T ′ ( z ) + 4 π cos πz (cid:21) | I i ,T ′ ( z ) ≡ T matter ( z ) − b∂c ( z ) . (2.4) K ′ can be expressed as 11 + K ′ = Z ∞ dLe − L ( K ′ ) , in the usual way and we need to define e − LK ′ to make sense of such quantities. In this section, we expand e − LK ′ as e − LK ′ = e − L ( K + J ) = ∞ X n =0 ( − n lim δ → +0 Z ∞ δ dL · · · Z ∞ δ dL n +1 δ n +1 X i =1 L i − L ! e − L K Je − L K J · · · Je − L n +1 K . (2.5)and consider the right hand side as the definition of e − LK ′ . From the point of view of the worldsheet theory,we define e − LK ′ perturbatively treating J as perturbation. The perturbation corresponds to adding Z d z π (cid:20) − πz T ′ ( z ) + 4 π cos πz (cid:21) (2.6)to the worldsheet action. Since it is a chiral quantity integrated over the bulk worldsheet, we do not encounterany ultraviolet divergences [17] and the expression is well-defined . However, there is still a room for finiterenormalizations. A prescription for such renormalization is fixed by introducing a cut-off δ .Now let us consider the observables of the Erler-Schnabl solution Ψ ′ ES . The observables we consider arethe action and the Ellwood invariant [29, 30, 31]. The action of Ψ ′ ES in the string field theory (1.4) is equal Notice that the normalization of J is fixed by the equation of motion and there is no reason to expect that the higher orderterms in the expansion (2.5) are small in any sense. We will treat the operator K ′ without using such an expansion in section3.
4o the difference of the energy between the background corresponding to Ψ TT and that corresponding toΨ ′ ES . The Ellwood invariant of Ψ ′ ES becomes the difference of the 1-point function of a closed string vertexoperator V between these backgrounds. Thus they can be expressed as S [Ψ ′ ES ] = E TT − E Ψ ′ ES , (2.7)Tr V Ψ ′ ES = h V c i Ψ ′ ES − h V c i TT , (2.8)Here the Ellwood invariant Tr V Φ is given asTr V Φ = h I | V ( i, − i ) | Φ i , (2.9)where V ( i, − i ) = c ¯ cV m ( i, − i ) is a closed string vertex operator. E TT , E Ψ ′ ES , h V c i Ψ ′ ES , h V c i TT denote theenergy and the one-point function of each background respectively. In the following, we will show S [Ψ ′ ES ] =Tr V Ψ ′ ES = 0, which implies E Ψ ′ ES = E TT , (2.10) h V c i Ψ ′ ES = h V c i TT . (2.11)Since we assume that Ψ ′ ES corresponds to the tachyon vacuum, we can see that the observables E TT , h V c i TT of the identity-based solution Ψ TT coincide with those of the tachyon vacuum. Therefore showing S [Ψ ′ ES ] =Tr V Ψ ′ ES = 0 gives evidences for the claim that the Takahashi-Tanimoto solution Ψ TT describes the tachyonvacuum.In this section, we use this indirect way proposed in [17] to calculate the observables E TT , h V c i TT of theidentity-based solution Ψ TT . Recently there are somewhat more direct ways to calculate these quantities[28, 32] [19]. Especially Maccaferri [19] uses the so-called Zeze map [33] to construct regular solutions gaugeequivalent to identity-based ones and calculate the observables of the regular ones. Moreover, the calculationseventually reduce to those of the S [Ψ ′ ES ] , Tr V Ψ ′ ES . In appendix A, we explain how we can apply Maccaferri’smethod to the Takahashi-Tanimoto solution (1.1) with a = − . Ψ ′ ES Now let us calculate the observables S [Ψ ′ ES ] , Tr V Ψ ′ ES and show that both of them vanish . From theexpression (2.2), we obtain S [Ψ ′ ES ] = − g Tr (cid:20)
11 + K ′ c
11 + K ′ Q ′ c (cid:21) , Tr V Ψ ′ ES = Tr V (cid:20)
11 + K ′ c (cid:21) . (2.12)Therefore what we will prove are Tr V (cid:20)
11 + K ′ c (cid:21) = 0 , (2.13)Tr (cid:20)
11 + K ′ c
11 + K ′ Q ′ c (cid:21) = 0 . (2.14) Kishimoto, Masuda and Takahashi [22] generalize the method of [28, 32] to the case of the scalar solutions. Kishimoto, Masuda and Takahashi [22] obtain the same results using a different method, considering more general solutionsmade from K ′ Bc . e − ǫK Q ′ be − ǫK These can be proved by using the following identities: Q ′ (cid:18) π b (cid:19) = 1 , (2.15) Q ′ c = 0 . (2.16)Here 1 π b ≡ π b ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z = | I i = b ( ξ ) | ξ =1 | I i , and (2.15) suggests that the π b works as a homotopy operator of the BRST charge Q ′ .To be precise, one can show (2.15)(2.16) in the situation where we have some worldsheet around π b, c without any local operator insertions. Namely we should consider e − ǫK Q ′ (cid:18) π b (cid:19) e − ǫK = e − ǫK , (2.17) e − ǫK Q ′ ce − ǫK = 0 , (2.18)in which we attach e − ǫK ’s to generate worldsheet as is depicted in Fig. 2. . With the worldsheet, one canexpress the action of Q ′ by the contour integral (2.3) and get e − ǫK Q ′ (cid:18) π b (cid:19) e − ǫK = e − ǫK (cid:18)I dz πi (cid:20) − sin πz cos πz j B ( z ) + 4 π cos πz c ( z ) (cid:21) π b (0) (cid:19) e − ǫK = e − ǫK (cid:18)I dz πi (cid:20) − sin πz cos πz (cid:18) ∂ c ( z ) (cid:19) + 4 π cos πz c ( z ) (cid:21) π b (0) (cid:19) e − ǫK = e − ǫK . (2.19)(2.18) can be derived in the same way. In [19], such a prescription is used for the equation of motion. One can show that the Ψ TT in (1.1) satisfies the equationof motion in the same way. V h K ′ c i , we getTr V (cid:20)
11 + K ′ c (cid:21) = Tr V (cid:20) √ K ′ Q ′ (cid:18) π b (cid:19) √ K ′ c (cid:21) . Here we use the definition 1 √ K ′ = 1Γ (cid:0) (cid:1) Z ∞ dLL − e − L e − LK ′ , where e − LK ′ expressed as (2.5). With the cutoff δ , (2.15)(2.16) can be safely used because there are someworldsheets with no operator insertions around b, c . Thus we obtainTr V (cid:20)
11 + K ′ c (cid:21) = Tr V (cid:20) √ K ′ π b √ K ′ Q ′ c (cid:21) = 0 . Before closing this section, one comment is in order. Using Q ′ c = 0, one can see that from (2.2)Ψ ′ ES = c . Thus actually the Ψ ′ ES itself is an identity-based solution , although we do not have any trouble in calculatingthe right hand sides of (2.12). One can avoid this by replacing c by c y ≡ c (cid:18)
12 + iy (cid:19) | I i ( y = 0 , y ∈ R ) .K ′ , B, c y satisfy the KBc algebra and one can construct the Erler-Schnabl solutionΨ ′ ES ,y = 11 + K ′ ( c y + Q ′ ( Bc y )) , which is not identity-based, albeit it still includes an identity based piece. The observables to be calculatedbecome S (cid:2) Ψ ′ ES ,y (cid:3) = − g Tr (cid:20)
11 + K ′ c y
11 + K ′ Q ′ c y (cid:21) , Tr V Ψ ′ ES ,y = Tr V (cid:20)
11 + K ′ c y (cid:21) . (2.20)One can show that these quantities are actually independent of y . Indeed, using the KBc identity { B, c y } = 1 , the formulas given in [19] (eqs.(3.4),(3.10)-(3.19)) imply16 Tr (cid:20)
11 + K ′ c y
11 + K ′ Q ′ c y (cid:21) = 16 Tr (cid:20)
11 +
K c y
11 +
K Qc y (cid:21) −
16 Tr (cid:20)
11 + K Ψ TT
11 + K ′ Ψ TT
11 + K Ψ TT
11 + K ′ (cid:21) , Tr V (cid:20)
11 + K ′ c y (cid:21) = Tr V (cid:20)
11 +
K c y (cid:21) − Tr V (cid:20)
11 + K Ψ TT
11 + K ′ (cid:21) , One may be able to calculate the observables for such a solution following [34] or [23]. y . Therefore evaluating them at y = 0, we can see that theobservables (2.20) all vanish. The derivation in the previous section uses the perturbative definition (2.5) of e − LK ′ . Since Ψ TT is anidentity-based solution, the kinetic term Q ′ is given by an integral of local operators on the worldsheet andwe should be able to treat K ′ more directly. In this section, we will examine if we can derive the results insection 2 by doing so.In the calculation of the observables in the previous section, the following relations were essential: e − ǫK Q ′ (cid:18) π b (cid:19) e − ǫK = e − ǫK ,e − ǫK Q ′ ce − ǫK = 0 . These relations hold for the perturbative definition of e − LK ′ . In the treatment here, it will be more appro-priate to consider e − ǫK ′ Q ′ (cid:18) π b (cid:19) e − ǫK ′ = e − ǫK ′ , (3.1) e − ǫK ′ Q ′ ce − ǫK ′ = 0 , (3.2)where the e − ǫK ′ ’s are expected to provide worldsheet with no operator insertions.Actually, as we will see, the definition of e − LK ′ is very subtle and we need some regularization to definequantities involving it. There seem to be many ways to treat it, which should be related to the choice ofthe prescription of renormalization in the perturbative definition of e − LK ′ (2.5). Here we use the identities(3.1)(3.2) and their consequences (2.13)(2.14) as the guiding principle to find the definition of e − LK ′ so thatthe string field action (1.4) should describe the tachyon vacuum. The K ′ given in (2.4) involves T ′ ( z ) which is a twisted energy momentum tensor with central charge c = 24.Therefore we need to take care of the conformal anomaly on the worldsheet to deal with the correlationfunctions on surfaces generated by e − LK ′ and the calculations will become cumbersome. Here we would liketo use an alternative way of dealing with K ′ to do calculations.As was pointed out by Kishimoto and Takahashi [11], the kinetic operator Q ′ of the string field theoryexpanded around the solution (1.1) with a = − can be expressed as Q ′ = e − q (cid:18) − Q + c (cid:19) e q , (3.3)8here q = − I dξ πi ( − bc ) ( ξ ) ln (cid:18) − ξ (cid:19) , (3.4) Q k = I dξ πi ξ k j B ( ξ ) , (3.5) c k = I dξ πi ξ k − c ( ξ ) . (3.6)Using the mode expansion of the ghost number current − bc ( ξ ) = X n j n ξ − n − , the q is expressed as q = 2 ∞ X n =1 n j − n . bc -shift operation Eq. (3.3) can be rewritten by using the bc -shift operation [11] defined for k ∈ Z as c n → c ( k ) n = c n + k ,b n → b ( k ) n = b n − k , | i → | i ( k ) = b − k − b − k · · · b − | i k > c k +2 c k +3 · · · c | i k < , h | → ( k ) h | = h | c − c · · · c k − k > h | b b · · · b − k +1 k < , and φ → φ ( k ) = φ if φ involves only matter fields. A state | a i = φ − n · · · b − m · · · c − l · · · | i , in the Fock space is mapped to | a i ( k ) = φ ( k ) − n · · · b ( k ) − m · · · c ( k ) − l · · · | i ( k ) , under this operation. c ( k ) n , b ( k ) n , | i ( k ) , ( k ) h | satisfy n c ( k ) n , b ( h ) n o = δ n + m, , n c ( k ) n , c ( k ) m o = n b ( k ) n , b ( k ) m o = 0 ,b ( k ) n | i ( k ) = 0 ( n ≥ − ,c ( k ) n | i ( k ) = 0 ( n ≥ , ( k ) h | b ( k ) n = 0 ( n ≤ , ( k ) h | c ( k ) n = 0 ( n ≤ − , (3.7)9nd ( k ) h | c ( k ) − c ( k )0 c ( k )1 | i ( k ) = h | c − c c | i = 1 . (3.8)Since we can evaluate all the correlation functions of the bc system using the relations (3.7)(3.8), we can seethat for any states h a | , | b i in the Fock space, ( k ) h a | b i ( k ) = h a | b i . Under the bc -shift operation, the BRST charge is transformed as Q → Q ( k ) = Q k − k c k . Therefore (3.3) can be written as Q ′ = − e − q Q (2) e q . (3.9)It is convenient to introduce operators U k ( k ∈ Z ) which are defined so that U k | a i = | a i ( k ) , h a | U k = ( − k ) h a | .U k satisfies U k U − k | a i = | a i ,U k O U − k = O ( k ) , for any state | a i in the Fock space and any operator O . It turns out that U k can be expressed as U k = e − kσ , (3.10)where σ is the operator which appears in the bosonization formulas (B.5), (B.6). Indeed, e − kσ satisfies e − kσ c ( ξ ) e kσ = e − kσ exp " ∞ X n =1 n j − n ξ n e σ e j ln ξ exp " − ∞ X n =1 n j n ξ − n e kσ = ξ k c ( ξ ) ,e − kσ b ( ξ ) e kσ = e − kσ exp " − ∞ X n =1 n j − n ξ n e − σ e − j ln ξ exp " ∞ X n =1 n j n ξ − n e kσ = ξ − k b ( ξ ) ,e − kσ | i = b − k − b − k · · · b − | i = | i ( k ) k > c k +2 c k +3 · · · c | i = | i ( k ) k < , h | e − kσ = h | b b · · · b k +1 = ( − k ) h | k > h | c − c · · · c − k − = ( − k ) h | k < . From (3.10) and (cid:2) j , e − kσ (cid:3) = − ke − kσ ,
10e can see that U k carries ghost number − k .(3.9) can be written as Q ′ = − U QU − . (3.11)where U ≡ e − q U ,U − ≡ U − e q . (3.12)Notice that U, U − are of ghost number − , U and U − are inverse to each other, when these operatorsact on the states in the Fock space. However, when we are dealing with the states outside of the Fock space,such a statement may become subtle, as is discussed in appendix B. Another thing to be noticed is that theBPZ conjugates of U, U − do not coincide with either U or U − .Therefore, the Q ′ is related to the original kinetic operator Q by a similarity transformation (3.11), whichimplies that the solution Ψ TT is formally in the pure gauge form. By the similarity transformation, K ′ isturned into an operator made from T = { Q, b } and thus it is possible to evaluate quantities involving K ′ without dealing with the twisted energy momentum tensor T ′ . U, U − We need some identities satisfied by
U, U − to perform calculations using the relation (3.11). From thedefinition (3.12) we obtain U c ( ξ ) U − = (cid:0) ξ − (cid:1) ξ c ( ξ ) = − e h − ( ξ ) c ( ξ ) , (3.13) U − c ( ξ ) U = ξ ( ξ − c ( ξ ) = − e − h − ( ξ ) c ( ξ ) , (3.14) U b ( ξ ) U − = ξ ( ξ − b ( ξ ) = − e − h − ( ξ ) b ( ξ ) , (3.15) U − b ( ξ ) U = (cid:0) ξ − (cid:1) ξ b ( ξ ) = − e h − ( ξ ) c ( ξ ) . (3.16)It is also possible to derive how U, U − act on the states | i , | I i , h | , h I | : U | i = 116 ∂bb (1) ∂bb ( − c c | i , (3.17) U − | i = 116 ∂cc (1) ∂cc ( − b − b − | i , (3.18) h | U = h | b b , (3.19) h | U − = h | c − c . (3.20) U | I i = 132 ∂bb (1) | I i , (3.21) U − | I i = 2 ∂cc (1) | I i . (3.22)Moreover, one can show that h I | U and h I | U − can be set to zero in the situations where no ghost operatorsare inserted at ξ = ±
1. These properties are proved in appendix B.11ere let us comment on one thing concerning the operators
U, U − , which will be relevant to the subse-quent discussions. The pure gauge form (3.11) apparently contradicts the existence of the homotopy operator(2.15), as was pointed out in [12, 35]. Indeed, one can see from (3.11) that the representatives of the BRSTcohomology of Q ′ are given by the states of the form [11] U cV m (0) | i : gh − , (3.23) U ∂ccV m (0) | i : gh . (3.24)where V m is a primary field made from the matter fields with weight 1. Therefore one can conclude thatthere exist no physical open string excitations because they correspond to the states with ghost number 1.On the other hand, the existence of the homotopy operator b (1) implies that the states (3.23)(3.24) shouldbe written in a BRST exact form U cV m (0) | i = Q ′ b (1) U cV m (0) | i ,U ∂ccV m (0) | i = Q ′ b (1) U ∂ccV m (0) | i . Actually these do not hold. Indeed, using eqs.(3.14)(3.17), we obtain
U cV m (0) | i = 116 ∂bb (1) ∂bb ( − c − c c V m (0) | i , (3.25) U ∂ccV m (0) | i = 116 ∂bb (1) ∂bb ( − c − c − c c V m (0) | i , (3.26)and b (1) U cV (0) | i = b (1) U ∂ccV (0) | i = 0. The reason for this apparent contradiction is that the relation(2.15) holds only when there is some worldsheet around b (1) without any local operator insertions, as wementioned below eq.(2.18). Therefore, for the states (3.25)(3.26) which involve ∂bb (1), b (1) does not workas a homotopy operator of Q ′ . Now we would like to discuss how we can evaluate the observables (2.12) using the expression (3.11). In orderto facilitate the calculation using eq.(3.11), we rewrite everything in terms of the first-quantized operators,rather than string fields. Here let us introduce B + , L ′ + such that [36, 1] B + = I dξ πi (cid:0) ξ (cid:1) (cid:18) tan − ξ + tan − (cid:18) ξ (cid:19)(cid:19) b ( ξ )= π I dξ πi (cid:0) ξ (cid:1) ǫ (Re ξ ) b ( ξ ) , L ′ + ≡ (cid:8) Q ′ , B + (cid:9) . (3.27) L ′ + is the translation operator with respect to the sliver frame coordinate z for the left and right half of thestring. Therefore, the action of L ′ + on any state | φ i can be expressed by the string field K ′ as L ′ + | φ i = K ′ ∗ | φ i + | φ i ∗ K ′ . (3.28)12 ′ + can be used to express various quantities involving K ′ in our setup. For example, using (3.28) and (2.9),one can show that h I | e − L L ′ + c (1) V ( i, − i, ) e − L L ′ + | I i = Tr V h e − L K ′ ∗ c ∗ e − L K ′ ∗ | I i ∗ e − L K ′ ∗ e − L K ′ i = Tr V h e − LK ′ c i , (3.29)holds and the left hand side of eqs.(2.13) is expressed asTr V (cid:20)
11 + K ′ c (cid:21) = Z ∞ dLe − L h I | e − L L ′ + c (1) V ( i, − i, ) e − L L ′ + | I i . (3.30)In a similar way, one gets h I | e − L − L L ′ + c (1) e − L L ′ + Q ′ c (1) | I i = Tr h e − L − L K ′ ∗ c ∗ e − L K ′ ∗ Q ′ c ∗ e − L K ′ ∗ e − L − L K ′ i = Tr h e − L K ′ ce − L K ′ Q ′ c i . (3.31)We expect that e − L − L L ′ + is well-defined when L > L and this equation is valid only for L > L . When L > L , h I | e − L − L L ′ + c ( − e − L L ′ + Q ′ c (1) | I i can be used to express Tr h e − L K ′ ce − L K ′ Q ′ c i . Thereforethe left hand side of (2.14) is expressed asTr (cid:20)
11 + K ′ c
11 + K ′ Q ′ c (cid:21) = Z ∞ dL e − L Z ∞ L dL e − L × h I | e − L − L L ′ + c (1) e − L L ′ + Q ′ c (1) | I i + Z ∞ dL e − L Z L dL e − L × h I | e − L − L L ′ + c ( − e − L L ′ + Q ′ c (1) | I i , (3.32)Eqs. (3.1)(3.2) are also rewritten as e − ǫ L ′ + Q ′ b (1) | I i = e − ǫ L ′ + | I i , (3.33) e − ǫ L ′ + Q ′ c (1) | I i = 0 . (3.34)Let us check if one can prove (3.33)(3.34) by using the expression (3.11). Substituting (3.11) into the lefthand side of (3.33), we get − e − ǫ L ′ + U QU − b (1) | I i . In order to avoid the singularity which appears in moving the operator U − to the right, we shift the positionof b for regularization. Thus we consider −
14 lim ξ → e − ǫ L ′ + U QU − b ( ξ ) | I i = −
14 lim ξ → " (cid:0) ξ − (cid:1) ξ U e − ǫ ˜ L ′ + Qb ( ξ ) 2 ∂cc (1) | I i . (3.35)13here ˜ L ′ + = U − L ′ + U = (cid:26) Q, π I dξ πi (cid:0) ξ (cid:1) ǫ (Re ξ ) e h − ( ξ ) b ( ξ ) (cid:27) = π I dξ πi (cid:0) ξ (cid:1) ǫ (Re ξ ) e h − ( ξ ) T ( ξ ) , (3.36)Instead of K ′ or L ′ + , ˜ L ′ + is the fundamental translation operator to deal with in the subsequent calculation.Contrary to K ′ , ˜ L ′ + is made from T ( ξ ) and we do not have to worry about the conformal anomaly. If theoperator e − ǫ ˜ L ′ + should generate worldsheet around { Q, b ( ξ ) } in (3.35), we could express Q by a contourintegral and proceed further.The operator of the form (3.36) can be analyzed by the methods explained in [37]. Here it is convenientto go to the sliver frame and rewrite (3.36) as˜ L ′ + = Z i ∞− i ∞ dz πi e h ( + z ) T (cid:18)
12 + z (cid:19) + Z i ∞− i ∞ dz πi e h ( − + z ) T (cid:18) −
12 + z (cid:19) , where e h ( z ) = − cos πz sin πz . We introduce a new coordinate w such that ∂z∂w = e h ( z ) , which is integrated as w ( z ) = z − π sin πz cos πz . (3.37)Using these, ˜ L ′ + is expressed as ˜ L ′ + = "Z + i ∞ − i ∞ + Z − + i ∞− − i ∞ dz πi ∂w∂z T ( w ) , and ˜ L ′ + generates translations with respect to the coordinate w . The map w ( z ) (3.37) maps the region0 < Im z < ∞ to −∞ < Im w < ∞ and the region −∞ < Im z < −∞ < Im w < ∞ for Re z = ± and z = ± are singular points. z = ± are mapped to w = ±∞ and do not move under the translation generatedby ˜ L ′ + . Therefore the operator e − ǫ L ′ + acting on the identity state | I i generates the worldsheet of the formdepicted in Fig. 3. Hence e − ǫ L ′ + in (3.35) does not generate worldsheet around Qb ( ξ ) and we cannot proceedfrom (3.35). The correlation functions which appear on the right hand sides of eqs.(3.30)(3.32) correspondto cylinders of the form w ∼ w + L . Such a cylinder is mapped to two spheres whose coordinates are givenby e πiL w . Regularization
The operator e − ǫ ˜ L ′ + generates apparently singular surfaces, which should be defined as a limit of regularsurfaces. There are problems in performing calculations on such singular surfaces. We are not able to provethe homotopy relation (3.34) on such surfaces because no worldsheet is generated around the point on theboundary. We would like to define the string field theory so that it describes the tachyon vacuum. Therefore14igure 3: The worldsheet generated by e − ǫ ˜ L ′ + in contrast to the one generated by e − ǫK .Figure 4: The surface generated by e − L ˜ L ′ + a in contrast to the one generated by e − L ˜ L ′ + .what we need to do is to regularize the ˜ L ′ + , while preserving the relations (3.33)(3.34). The regularizationwe propose is to replace ˜ L ′ + by ˜ L ′ + a ≡ π I dξ πi (cid:0) ξ (cid:1) ǫ (Re ξ ) e h a ( ξ ) T ( ξ ) . (3.38) (cid:0) a > − (cid:1) with h a ( ξ ) given in (1.3). We define e − L ˜ L ′ + aslim a →− e − L ˜ L ′ + a . (3.39)For a > − , the surface generated by e − L ˜ L ′ + a is of the form depicted in Fig. 4 and we realize e − L ˜ L ′ + as asingular limit of e − L ˜ L a ′ + .With such a regularization, the right hand side of (3.35) becomes −
14 lim a →− lim ξ → " (cid:0) ξ − (cid:1) ξ U e − ǫ ˜ L ′ + a Qb ( ξ ) 2 ∂cc (1) | I i = U lim a →− e − ǫ ˜ L ′ + a ∂cc (1) | I i , U lim a →− e − ǫ ˜ L ′ + a ∂cc (1) | I i = U lim a →− e − ǫ ˜ L ′ + a U − | I i = e − ǫ L ′ + | I i , (3.40)and we eventually get (3.33). (3.34) can be proved in the same way: e − ǫ L ′ + Q ′ c (1) | I i = −
14 lim ξ → e − ǫ L ′ + U QU − c ( ξ ) | I i = −
14 lim a →− lim ξ → " ξ ( ξ − U e − ǫ ˜ L ′ + a Qc ( ξ ) 2 ∂cc (1) | I i = 0 . (3.41)One can immediately show that the terms on the right hand side of (3.32) vanish by using (3.41). In orderto show that the right hand side of (3.30) vanishes, we use (3.40) to get h I | e − L L ′ + c (1) V ( i, − i, ) e − L L ′ + | I i = lim a →− lim ξ → h I | e − L L ′ + c ( ξ ) V ( i, − i, ) U e − L ˜ L ′ + a ∂cc (1) | I i = lim a →− lim ξ → h I | U e − L ˜ L ′ + a (cid:0) ξ − (cid:1) ξ c ( ξ ) V ( i, − i, ) e − L ˜ L ′ + a ∂cc (1) | I i = 0 . (3.42)Here, with the regularization, h I | U is away from the other operators c ( ξ ) , ∂cc (1) and it can be set tozero. Thus we have shown how to regularize and define the operator e − L ˜ L ′ + so that we can derive(3.33)(3.34)(2.13)(2.14). These formulas imply that the string field theory describes the tachyon vacuum. In this paper, we have evaluated the observables of the Takahashi-Tanimoto’s scalar solution (1.1) with a = − , by studying the Erler-Schnabl solution in the string field theory expanded around it. The resultsare consistent with the claim that the solution corresponds to the tachyon vacuum. In the calculations, thestring field K ′ or its worldsheet operator counterpart plays crucial roles. In the latter half of this paper,we study the operator K ′ using the similarity transformation proposed by Kishimoto and Takahashi. Wediscuss how we should treat it in order to be consistent with the claim that the background is the tachyonvacuum.The relation (3.11) will be useful to evaluate various other quantities in the string field theory expandedaround Ψ TT . Since the solution is supposed to describe the tachyon vacuum, we expect all the amplitudesinvolving open string states to vanish. On the other hand, we may be able to calculate closed stringamplitudes using the string field theory [30, 38, 39]. In order to do such calculations, we should take Siegelgauge for example and construct the propagators. We will need some regularization like (3.39) to define thepropagator. We leave it as a future problem. 16he operator U, U − in (3.11) should be related to the boundary condition changing operators whichplay crucial roles in [6, 40]. Suppose that we formally divide the operators U, U − into the left and rightpiece U L , U R , (cid:0) U − (cid:1) L , (cid:0) U − (cid:1) R so that the operator U, U − acts on a string field A as U A = U L AU R ,U − A = (cid:0) U − (cid:1) L A (cid:0) U − (cid:1) R .U L , U R , (cid:0) U − (cid:1) L , (cid:0) U − (cid:1) R may be regarded as some kind of boundary condition changing operators and theidentities given in subsection 3.2 imply the OPE’s of them. It would be inspiring to study the Takahashi-Tanimoto background from the point of view of these operators. Acknowledgments
We are grateful to I. Kishimoto, C. Maccaferri, T. Masuda, and T. Takahashi for sharing their ideas on thistopic. We would like to acknowledge T. Erler and Y. Okawa for useful comments. We also would like tothank the organizers of the conference “String field theory and related aspects VI, SFT2014”, especiallyL. Bonora, for hospitality. This work was supported in part by Grant-in-Aid for Scientific Research (C)(25400242) from MEXT.
A Maccaferri’s method
In a recent paper [19], Maccaferri considered a special case of Zeze map [33], which maps an identity-basedsolution to a regular solution. In the case of the Takahashi-Tanimoto solution (1.1) with a = − , one obtainsΨ TT → Ψ reg . ≡ (cid:18) B − F ( K ) K Ψ TT (cid:19) ( Q + Ψ TT ) (cid:18) B − F ( K ) K Ψ TT (cid:19) − . (A.1)The Zeze map (A.1) is a gauge transformation and we can get a regular solution gauge equivalent to Ψ TT by choosing F ( K ) appropriately. A convenient choice is F ( K ) = k and we getΨ reg . = 11 + K Ψ TT
11 + K ′ − Q (cid:18)
11 + K Ψ TT
11 + K ′ (cid:19) , (A.2)which appears to be a regular solution. From the expression (A.2), it is straightforward to calculate theenergy and the Ellwood invariant and one obtains [19] S [Ψ reg . ] = − g Tr (cid:20)
11 +
K c
11 +
K Qc (cid:21) + 16 g Tr (cid:20)
11 + K ′ c
11 + K ′ Q ′ c (cid:21) , Tr V Ψ reg . = Tr V (cid:20)
11 +
K c (cid:21) − Tr V (cid:20)
11 + K ′ c (cid:21) . (A.3)The right hand sides of eq.(A.3) can be written as S [Ψ reg . ] = S [Ψ ES ] − S [Ψ ′ ES ] , Tr V Ψ reg . = Tr V Ψ ES − Tr V Ψ ′ ES , Since
U, U − involve operators like U , U − , we are not so sure if we could do such a decomposition. ES , Ψ ′ ES are the Erler-Schnabl solutions given in (2.1)(2.2). Thus the observables of Ψ reg . are obtainedfrom those of the Erler-Schnabl solution Ψ ′ ES . Using S [Ψ ′ ES ] = Tr V Ψ ′ ES = 0 derived in section 2, we can seethat the observables of Ψ reg . coincide with those of the tachyon vacuum solution Ψ ES . Singularities
Actually, the calculation of the observables above suffers from singularities discussed by Maccaferri [19]. Incalculating the action, one typically encounters quantities of the form h c ( z ) c∂c (0) i C L = − (cid:18) Lπ (cid:19) sin πzL , (A.4)where h·i C L denotes the correlation function on a semi-infinite cylinder with circumference L . (A.4) divergesin the limit Im z → ±∞ for small enough L > L → z = 0. Since the Takahashi-Tanimoto solution (1.1) involves an integral of the ghost c up to Im z = ±∞ , we have trouble in calculatingthe action .Therefore we need to find a good regularization to calculate the action . In [19], a solution with F ( K ) = F ǫ ( K ) = e − ǫK − ǫ ) K , (A.5)(0 ≤ ǫ ≤
1) in (A.1) is considered as a regularization. Let Ψ ǫ denote the Ψ reg . with this choice of F ( K ). Itis easy to see that Ψ ǫ = 11 + K ǫ (cid:18) Ψ TT − Ψ TT B ǫ
11 + K ′ ǫ Ψ TT (cid:19) , where c ǫ = c KBG ǫ ( K ) c ,B ǫ = B G ǫ ( K ) K ,K ǫ = QB ǫ = G ǫ ( K ) ,J ǫ = { B ǫ , Ψ TT } ,K ′ ǫ = K ǫ + J ǫ , (A.6)and 11 + K ǫ = 11 + G ǫ ( K ) = e − ǫK − ǫ ) K .
The Ψ ǫ consists of wedge states of width not smaller than ǫ with operator insertions and we can avoid theabove-mentioned divergences taking ǫ > . K ǫ , B ǫ , c ǫ in (A.6) satisfy the KBc algebra [41, 42, 34] and it is straightforward to show that the observ-ables for the solution Ψ ǫ coincide with the shift in those of the modified Erler-Schnabl solutions [19]Ψ ES ,ǫ = 11 + K ǫ ( c ǫ + Q ( B ǫ c ǫ )) , (A.7)Ψ ′ ES ,ǫ = 11 + K ′ ǫ ( c ǫ + Q ′ ( B ǫ c ǫ )) , (A.8) We do not encounter such divergences in the calculation of the Ellwood invariant or the overlap of Ψ reg . with Fock spacestates. In [23], the author modifies the form of the solution (1.1) as was presented in [19] and avoids the singularity. S [Ψ ǫ ] = S [Ψ ES ,ǫ ] − S (cid:2) Ψ ′ ES ,ǫ (cid:3) = − g Tr (cid:20)
11 + K ǫ c ǫ
11 + K ǫ Qc ǫ (cid:21) + 16 g Tr (cid:20)
11 + K ′ ǫ c ǫ
11 + K ′ ǫ Q ′ c ǫ (cid:21) . Tr V Ψ ǫ = Tr V Ψ ES ,ǫ − Tr V Ψ ′ ES ,ǫ = Tr V
11 + K ǫ c ǫ − Tr V
11 + K ′ ǫ c ǫ , (A.9)Now we can use (A.9) to calculate the observables. As is pointed in [19], although Ψ ǫ itself may involvesingularities for small ǫ , Ψ ′ ES ,ǫ is regular for all 0 ≤ ǫ ≤
1. Moreover one can show ∂∂ǫ Ψ ES ,ǫ = Q Λ + Ψ ES ,ǫ Λ − ΛΨ ES ,ǫ ,∂∂ǫ Ψ ′ ES ,ǫ = Q ′ Λ ′ + Ψ ′ ES ,ǫ Λ ′ − Λ ′ Ψ ′ ES ,ǫ , where Λ = B ǫ
11 + K ǫ ∂∂ǫ Ψ ES ,ǫ , Λ ′ = B ǫ
11 + K ′ ǫ ∂∂ǫ Ψ ′ ES ,ǫ . Since the observables Tr V Ψ ES ,ǫ , Tr V Ψ ′ ES ,ǫ , S [Ψ ES ,ǫ ] , S (cid:2) Ψ ′ ES ,ǫ (cid:3) are gauge invariant quantities, they are in-dependent of ǫ provided the gauge parameters Λ , Λ ′ are regular string fields. Thus we can evaluate themchoosing ǫ for which the calculation is easy. The most convenient choice is ǫ = 0 and we getTr V Ψ ǫ = Tr V Ψ ES − Tr V Ψ ′ ES , (A.10) S [Ψ ǫ ] = S [Ψ ES ] − S [Ψ ′ ES ] . (A.11)From (2.13)(2.14), we can see that the observables of Ψ ǫ coincide with those of the tachyon vacuum solutionΨ ES .Thus, by using the Maccaferri’s method, it is possible to construct regular solutions gauge equivalent toΨ TT , calculate the observables of them and show that they coincide with those of the tachyon vacuum. In asense, this gives a more direct derivation of the observables of the identity-based solutions than the one givenin section 2. On the other hand, since the gauge transformation (A.1) transforms an identity-based solutioninto a regular solution, the transformation itself might be somewhat singular. Therefore if the observables(A.3) can be identified with those of Ψ TT may be debatable.Before closing this appendix, one comment is in order. The string field theory expanded around theTakahashi-Tanimoto solution possesses a classical solution − Ψ TT corresponding to the perturbative vacuum.Although the solution itself is an identity-based solution, one can construct a solution gauge equivalent to it −
11 + K ′ Ψ TT
11 + K + Q ′ (cid:18)
11 + K ′ Ψ TT
11 + K (cid:19) , by Maccaferri’s method. The observables can be calculated at least formally and they coincide with thoseof the perturbative vacuum. 19 Properties of
U, U − U, U − act on the states | i , h | , | I i , h I | .Let us first prove the following identities: U | i = 116 ∂bb (1) ∂bb ( − c c | i , (B.1) U − | i = 116 ∂cc (1) ∂cc ( − b − b − | i , (B.2) h | U = h | b b , (B.3) h | U − = h | c − c . (B.4)Since q = 2 P ∞ n =1 1 n j − n , e ± q | i = exp " ± ∞ X n =1 n j − n | i . On the other hand, we have the bosonization formula c ( ξ ) = exp " ∞ X n =1 n j − n ξ n e σ e j ln ξ exp " − ∞ X n =1 n j n ξ − n , (B.5) b ( ξ ) = exp " − ∞ X n =1 n j − n ξ n e − σ e − j ln ξ exp " ∞ X n =1 n j n ξ − n , (B.6)where σ is the canonical conjugate of j satisfying[ j , σ ] = 1 . Eqs.(B.5)(B.6) imply ∂bb (1) ∂bb ( − c − c − c c | i = 16 e − q | i ,∂cc (1) ∂cc ( − b − b − b − b − | i = 16 e q | i . From these, we get U | i = e − q U | i = e − q lim ε → ∂bb ( ε ) | i = lim ε → (cid:18) − ε (cid:19) − ∂bb ( ε ) 116 ∂bb (1) ∂bb ( − c − c − c c | i = 116 ∂bb (1) ∂bb ( − c c | i ,U − | i = U − e q | i = U − ∂cc (1) ∂cc ( − b − b − b − b − | i = 116 ∂cc (1) ∂cc ( − b − b − | i . (B.3)(B.4) are obtained from h | e ± q = h | . U, U − act on | I i . We will show U | I i = 132 ∂bb (1) | I i , (B.7) U − | I i = 2 ∂cc (1) | I i . (B.8)These are shown by using the defining relation [43, 8, 36] of h I |h I | φ (0) | i = h f ◦ φ (0) i UHP , (B.9)where f ( ξ ) = 2 ξ − ξ , and h·i UHP denotes the correlation function on the upper half plane. In order to derive (B.8), for example,what we should do is to calculate h | φ (0) U | I i , and show that it is equal to h | φ (0) 2 ∂cc (1) | I i for any φ (0). Since U only changes the ghost part of h I | ,we only have to deal with the case where φ (0) is made from ghost operators. Therefore what we shouldcalculate are the quantities of the form h | Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) U | I i . (B.10)Using eqs.(B.4), (3.13), (3.15), we obtain h | Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) U − | I i = h | c − c Y i (cid:0) ξ i − (cid:1) ξ i c ( ξ i ) ! Y j ξ ′ j (cid:0) ξ ′ j − (cid:1) b (cid:0) ξ ′ j (cid:1)! | I i = h I | Y i (cid:0) ξ i − (cid:1) ξ i I ◦ c ( ξ i ) ! Y j ξ ′ j (cid:0) ξ ′ j − (cid:1) I ◦ b (cid:0) ξ ′ j (cid:1)! c c | i = h | Y i (cid:18) f ( ξ i ) (cid:19) f ◦ I ◦ c ( ξ i ) ! Y j f (cid:0) ξ ′ j (cid:1) ! f ◦ I ◦ b (cid:0) ξ ′ j (cid:1) c c | i = 2 h | U − Y i f ◦ I ◦ c ( ξ i ) Y j f ◦ I ◦ b (cid:0) ξ ′ j (cid:1) | i = 2 h | c − c Y i f ◦ I ◦ c ( ξ i ) Y j f ◦ I ◦ b (cid:0) ξ ′ j (cid:1) | i = 2 h | f ◦ I ◦ ( ∂cc ) (1) Y i f ◦ I ◦ c ( ξ i ) Y j f ◦ I ◦ b (cid:0) ξ ′ j (cid:1) | i = h | Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) ∂cc (1) | I i , (B.11)where I : ξ → − ξ , is the inversion map. Eq.(B.11) implies U − | I i = 2 ∂cc (1) | I i . Eq.(B.7) can be shown in the same way.21lthough the state | I i is not included in the Fock space, the operators U, U − are inverse to one another,when they are acting on it. Indeed, U (cid:0) U − | I i (cid:1) = 2 U ∂cc (1) | I i = 2 U lim ξ → ∂cc ( ξ ) | I i = 2 U lim ξ → (cid:0) ξ − (cid:1) ξ ! ∂cc ( ξ ) 132 ∂bb (1) | I i , = | I i , (B.12)and we can also get U − ( U | I i ) = | I i in the same way.Now let us consider the action of U, U − on h I | . In order to get h I | U , we need to calculate h I | U Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) | i . (B.13)Using (3.14)(3.16)(B.1), it is straightforward to get U Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) | i = 116 Y i (cid:0) ξ i − (cid:1) ξ i c ( ξ i ) ! Y j ξ ′ j (cid:0) ξ ′ j − (cid:1) b (cid:0) ξ ′ j (cid:1)! ∂bb (1) ∂bb ( − c c | i . Now using (B.9), we obtain h I | U Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) | i = h | f ◦ ( ∂bb ) (1) f ◦ ( ∂bb ) ( − Y i (cid:0) ξ i − (cid:1) ξ i f ◦ c ( ξ i ) ! Y j ξ ′ j (cid:0) ξ ′ j − (cid:1) f ◦ b (cid:0) ξ ′ j (cid:1)! f ◦ ( ∂cc ) (0) | i . (B.14)Since f ◦ ( ∂bb ) ( ±
1) = lim ε → (cid:18) ∂f∂ξ (cid:19) ∂bb (cid:18) ξ − ξ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ± ε = lim ε → ε − ∂bb (cid:18) − ε (cid:19) ∼ b b , acting on h | , h I | U Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) | i = 0 , (B.15)provided none of ξ i , ξ ′ j coincides with ±
1. We can also derive, for example, h I | U ∂cc ( ± Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) | i = 32 h I | Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) | i , (B.16)if none of ξ i , ξ ′ j coincides with ±
1. Therefore we can set h I | U to zero in the case where there are no ghostoperator insertions at ξ = ±
1. One can show that h I | U − can be set to zero in such situations, in the same22ay. However h I | U ∂cc ( ±
1) and h I | U − ∂bb ( ±
1) are not zero identically. We do not know how to express h I | U and h I | U − with such properties in a closed form.The vanishing of h I | U, h I | U − in some situations does not mean that the operators U, U − are notinvertible. For example, if one considers correlation function of the form( h I | U ) U − Y i c ( ξ i ) Y j b (cid:0) ξ ′ j (cid:1) | i , (B.17)with ξ i = ± , ξ ′ j = ±
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