The phantom term in open string field theory

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JHEP06(2012)084Published for SISSA by SpringerReceived: February 14, 2012Accepted: May 18, 2012Published: June 14, 2012The phantom term in open string field theoryTheodore Erlera and Carlo Maccaferria,baInstitute of Physics of the ASCR, v.v.i.,Na Slovance 2, 182 21 Prague 8, Czech RepublicbINFN Sezione di Torino,Via Pietro Giuria 1, I-10125 Torino, ItalyE-mail: tchovi@gmail.com, maccafer@gmail.comAbstract: We show that given any two classical solutions in open string field theory anda singular gauge transformation relating them, it is possible to write the second solutionas a gauge transformation of the first plus a singular, projector-like state which describesthe shift in the open string background between the two solutions. This is the phantomterm. We give some applications in the computation of gauge invariant observables.Keywords: Tachyon Condensation, String Field TheoryArXiv ePrint: 1201.5122c SISSA 2012 doi:10.1007/JHEP06(2012)084JHEP06(2012)084Contents1 Introduction 12 The phantom term 23 Relation to Schnabls phantom term 44 Example 1: Identity-like marginals 55 Example 2: Energy for Schnabls solution 76 Example 3: Tadpole shift between two marginals 101 IntroductionPerhaps the most mysterious aspect of Schnabls analytic solution for tachyon conden-sation [1] is the so-called phantom term a singular and formally vanishing term in thesolution which appears to be solely responsible for the disappearance of the D-brane. Muchwork has since shed light on the term, either specifically in the context of Schnabls solu-tion [13], or in some generalizations [49], but so far there has been limited understandingof why the phantom term should be present.In this paper we show that the phantom term is a consequence of a particular andgeneric property of string field theory solutions: Given any two solutions 1 and 2, it isalways possible to find a ghost number zero string field U satisfying(Q+1)U = U2. (1.1)This is called a left gauge transformation from 1 to 2 [10]. Under some conditions,1the existence of U implies that 2 can be expressed as a gauge transformation of 1 plusa singular projector-like state which encapsulates the shift in the open string backgroundbetween 1 and 2. This is the phantom term. The phantom term is proportional to a staralgebra projector called the boundary condition changing projector, which is conjecturedto describe a surface of stretched string connecting two BCFTs [10]. One consequence ofthis description is that phantom terms, in general, do not vanish in the Fock space, as isthe case for Schnabls solution.This paper can be viewed as a companion to reference [10], to which we refer thereader for more detailed discussion of singular gauge transformations and boundary condi-tion changing projectors. Our main goal is to show how the phantom term can be used to1Specifically, U should have the property that it defines a boundary condition changing projector, whichmeans that as a formal linear operator its image and kernel should be linearly independent and span thewhole space [10]. 1 JHEP06(2012)084calculate physical observables, even for solutions where the existence of a phantom termwas not previously suspected. We give three examples: The closed string tadpole ampli-tude [11] for identity-like marginal deformations [12]; the energy for Schnabls solution [1];and the shift in the closed string tadpole amplitude between two Schnabl-gauge marginalsolutions [13, 14]. The last two computations reproduce results which have been obtained inother ways [1, 15], but our approach brings a different perspective and some simplifications.This description of the phantom term will be useful for the study of future solutions.2 The phantom termTo start, lets review some concepts and terminology from [10]. Given a pair of classicalsolutions 1 and 2, a ghost number zero state U satisfying(Q+1)U = U2 (2.1)is called a left gauge transformation from 1 to 2. If U is invertible, then 1 and 2 aregauge equivalent solutions. However U does not need to be invertible. In this case, we saythat the left gauge transformation is a singular gauge transformation.It is always possible to relate any pair of solutions by a left gauge transformation.Given a ghost number 1 field b, we can construct a left gauge transformation from 1 to2 explicitly with the formula:U = Qb+1b+ b2= Q12b. (2.2)Here, Q12 is the shifted kinetic operator for a stretched string between the solutions 1and 2. Equation (2.2) is not necessarily the most general left gauge transformation from1 to 2. This depends on whether Q12 has cohomology at ghost number zero [10].Even when U is not invertible, it often happens that U + is invertible for any suffi-ciently small > 0. However this raises a question: If U + is invertible, how is it possiblethat 1 and 2 are not automatically gauge equivalent in the 0 limit? The answer tothis question is contained in the following equation:2 =1+ U[Q+1](+ U) 1()++ U(2 1) 12(), (2.3)Computing the right hand side using (2.1), this equation is easily seen to be an identity.The first term 1() is a gauge transformation of 1, and the second term 12() is aremainder. Apparently, if 1 and 2 are not gauge equivalent, the remainder must benontrivial in the 0 limit:lim0+12() = X(2 1). (2.4) 2 JHEP06(2012)084This is the phantom term. Here we must assume that the 0 limit of +U in some senseconverges, defining the star algebra projectorX = lim0++ U. (2.5)called the boundary condition changing projector [10]. The boundary condition changingprojector is a subtle object, and is responsible for some of the mystery of the phantomterm. Based on formal arguments and examples, it was argued in [10] that the boundarycondition changing projector represents a surface of stretched string connecting the BCFTsof 2 and 1.The identity (2.3) offers a potentially useful method for computing gauge invariantquantities from classical solutions. The basic idea is that, with a judicious choice of U , thepure gauge term can be arranged to absorb all of the gauge-redundant artifacts of thesolution, leaving the phantom term to to describe the shift in the open string backgroundin a transparent manner. In later examples we will compute the on-shell action and theclosed string tadpole [11]:S[] = 16Tr[Q], A(V) = TrV []. (2.6)where is a tachyon vacuum solution and A(V) is the disk amplitude for a single on-shellclosed string V = ccVmatter to be emitted from the BCFT of .2 The change in the actionor tadpole between solutions 1 and 2 can be computed using the phantom term:S[2] S[1] = 16Tr[12()Q12()] 13Tr[12()Q2](2.7)A2(V)A1(V) = TrV[12()]. (2.8)These equations are exact for any , though they will be most useful in the 0 limit.It would be interesting to see whether the phantom term can also be used to compute theboundary state [17].The phantom term defined here is useful only if U+ is invertible3 and the 0 limitsensibly defines a boundary condition changing projector which simplifies the calculationof observables. The latter might not occur if the boundary condition changing projectordoes not exist (a possibility explained in [10]) or if it is degenerate. For example, whenU = 0 the projector in the phantom term is the identity string field, and the resultingdecomposition (2.3) into a pure gauge and phantom piece,2 = 11()+ (2 1) 12(), (2.9)2We use Tr[] to denote the 1-string vertex (Witten integral) and TrV [] to denote the 1-string vertexwith a midpoint insertion of V [11].3For example, U = Q(BK) = K2 is a singular gauge transformation from the perturbative vacuum toitself, but the inverse of K2 + would be defined as an integral over wedge states which at best convergesin a distributional sense. In this case the identity (2.3) should be treated with care, or it may be moreconvenient to either make a different choice of U or a different type of splitting between pure gauge andphantom terms, analogous to, but different from, (2.3). An example of this is discussed in the next section. 3 JHEP06(2012)084is too trivial to be useful. The examples we have analyzed concern a choice of U derivedfrom b of the formb = Bf(K) (2.10)where f(K) is a state in the wedge algebra with f(K) > 0. (See [2, 18] and appendix Aof [8] for definitions of K,B, c and the algebraic notation we use here.) In this case, wehave always found that the phantom term is proportional to the sliver state, and the 0limit simplifies upon application of the L level expansion4 to fields inside the phantomterm while ignoring subleading levels which do not contribute. We will see how this worksin later examples. For more exotic choices of U we do not know what form the boundarycondition changing projector or the phantom term might take, or whether they will beuseful for analytic calculations of observables.5Note that the phantom term is a property of a pair of solutions and a singular gaugetransformation relating them. In this sense, a solution by itself does not have a phantomterm. That being said, some solutions like Schnabls solution seem to be naturallydefined as a limit of a pure gauge configuration subtracted against a phantom term. Othersolutions, like the simple tachyon vacuum [8] and marginal solutions, can be defineddirectly without reference to a singular gauge transformation or its phantom term. Itwould be interesting to understand what distinguishes these two situations, and why.3 Relation to Schnabls phantom termThe phantom term defined by equations (2.3) and (2.4) is different from the phantom termas it conventionally appears in Schnabls solution [1] or some of its extensions [4, 5, 9]. Thestandard phantom term can be derived from the identity2 =1XNU(Q+1)U +XN2. (3.1)where we define X:U 1X. (3.2)In the N limit, the second term in (3.1) is the phantom term:limNXN2 = X2. (3.3)This is different from (2.4), though both phantom terms are proportional to the boundarycondition changing projector. The major difference between the identities (2.3) and (3.1)is that 1() is exactly gauge equivalent to 1 for all > 0, whereas the correspondingterm in (3.1),1XNU(Q+1)U, (3.4)4L is the BPZ odd component of Schnabls L0, the zero mode of the energy momentum tensor in thesliver coordinate frame [1, 19]. It is a derivation and a reparameterization generator, and computes (twice)the scaling dimension of operator insertions in correlation functions on the cylinder. See [9] for discussionof the L level expansion.5We thank the referee for suggesting a clarification of this point. 4 JHEP06(2012)084is not gauge equivalent to 1, or even a solution, for any finite N . In this sense (2.3) ismore natural, and this is the definition of the phantom term we will use in subsequentcomputations. However, the phantom term can be defined in many ways using manydifferent identities similar to (2.3) and (3.1), and for certain purposes some definitions mayprove to be more convenient than others.To make the connection to earlier work, let us explain how the identity (3.1) leadsto the standard definition of Schnabls solution as a regularized sum subtracted against aphantom term. We can use Okawas left gauge transformationU = 1cB, (3.5)to map from the perturbative vacuum 1 = 0 to Schnabls solution2 = =cKB1 c. (3.6)Substituting these choices into (3.1) gives the expression = N1n=0n +cNKB1 c, (3.7)where [1]n ddnn, n cBnc. (3.8)To simplify further, expandK1 =n=0Bnn!(K)n, (3.9)inside the second term of (3.7), where Bn are the Bernoulli numbers. The expansion inpowers of K is equivalent to the L level expansion alluded to earlier. Usually the higherpowers of K in (3.9) can be ignored in the N limit [5], so we can effectively replacethe sum by its first termK1 1. (3.10)Then the N limit of (3.1) reproduces the usual expression for Schnabls solution = limN[Nn=0n + N], (3.11)where N is the phantom term.4 Example 1: Identity-like marginalsWe start with a simple example: computing the shift in the closed string tadpole amplitudebetween the identity-like solution for the tachyon vacuum [8, 20, 21],1 = c(1K), (4.1) 5 JHEP06(2012)084and the identity-like solution for a regular marginal deformation [12],2 = cV, (4.2)where V is a weight 1 matter primary with regular OPE with itself. Both these solutionsare singular. For example, we cannot evaluate the tadpole directly becauseTrV [cV ] (4.3)requires computing a correlator on a surface with vanishing area. However, with the phan-tom term, we can circumvent this problem with a few formal (but natural) assumptions.We can relate the above solutions with a left gauge transformation6U = Q12B= 1 +Bc(K + V 1). (4.4)The shift in the open string background is described by the phantom term:712() =+ U(2 1) ( 1 )=(c+ + (K + V )Bcc)(1K V )=(c+ 0dt et t/V Bcc)(1K V ). (4.5)where in the third line we defined the states tV et(K+V ). (4.6)These are wedge states whose open string boundary conditions have been deformed by themarginal current V [? ]. In the 0 limit the phantom term islim012() = V Bcc(1K V ), (4.7)Note that the phantom term corresponds to a nondegenerate surface with the boundaryconditions of the marginally deformed BCFT, and so will naturally reproduce the expectedcoupling to closed strings. This is in spite of the fact that both solutions we started withwere identity-like. Also note that the phantom term vanishes in the Fock space (since Bkills the sliver state), but still it is nontrivial.Now we can use the phantom term to compute the shift in the closed stringtadpole amplitude:TrV [12()]. (4.8)6We construct the left gauge transformation U using b = B in (2.2), though in principle other choicesare possible. We have checked that the final result is the same choosing b = Bf(K) for f(K) > 0, thoughthe derivation is slightly more complicated.7In the following examples the dependence on simplifies if we make a reparameterization /relative to (2.3), where by definition 1 . 6 JHEP06(2012)084Since the amplitude vanishes around the tachyon vacuum, only the marginally deformedD-brane should contribute. Plugging (4.5) in, we findTrV [12()] = TrV[cB(K + V 1)c + + (K + V )Bcc]. (4.9)The first term in the trace formally vanishes.8 ThenTrV [12()] = TrV[+ (K + V )Bcc]= 0dt et TrV [t/V Bcc]. (4.10)With the reparameterization we can scale the deformed wedge state inside the trace tounit width:TrV [t/V Bcc] = TrV[(t) 12L (VBcc)]= TrV [VBcc]. (4.11)Integrating over t givesTrV [12()] = TrV [VBcc] . (4.12)Note that this is manifestly independent of . In the general situation, explicitly proving-independence requires much more work than is needed to compute the result, and itis easier to assume gauge invariance and take the 0 limit. At any rate, furthersimplifying (4.12), we can replace the ghost factor Bcc in the trace with c,9. Mappingfrom the cylinder to the unit disk givesTrV [12()] = 12iexp[ 20d V (ei)]V(0)c(1)disk. (4.14)This is exactly the closed string tadpole amplitude for the marginally deformed D-brane,as defined in the conventions of [11].5 Example 2: Energy for Schnabls solutionIn this section we compute the energy for Schnabls solution, =cKB1 c. (5.1)8This first term in (4.9) formally vanishes because it is the trace of a state which has negative scalingdimension plus a state which is BRST exact with respect to the BRST operator of the marginally deformedBCFT. However, rigorously speaking the trace is undefined, since computing it requires evaluating a corre-lator on a surface with vanishing area. This is a remnant of the fact that our marginal solution and tachyonvacuum are too identity-like. However, since the offending term is proportional to , and (4.8) is formallyindependent of , we will set = 0 and ignore this term.9This follows from the fact that the derivation B annihilates the 1-string vertex [8], and12B(V cc)= VBcc+V c. (4.13) 7 JHEP06(2012)084The original computation of the energy, based on the expression (3.11), was given in [1](see also [2, 3]). Our computation will be quite different since we define the phantom termin a different way.We take the reference solution to be the perturbative vacuum, and map to Schnablssolution using Okawas left gauge transformationU = 1cB= Q0(B1 K). (5.2)The regularized phantom term is() =+ U =cB1 K1 c. (5.3)In the 0 limit the ratio 1 approaches the sliver state (see later), so we can replace thefactor K1 with its leading term in the L level expansion. Then (2.3) gives a regularizeddefinition of Schnabls solution: = lim0+[ +cB1 c]. (5.4)Note that here is precisely the pure gauge solution discovered by Schnabl [1]: = cKB1 c. (5.5)Using (3.8) we can express this regularization in the form = lim0+n=0n[ n + n]. (5.6)Clearly this is different from the standard definition of Schnabls solution, (3.11).To calculate the action we use (2.7):S =16Tr[()Q()] 13Tr[()Q]. (5.7)A quick calculation shows that the second term can be ignored in the 0 limit, essentiallybecause the phantom term vanishes when contracted with well-behaved states. ThereforeS =16lim0+Tr[()Q()]. (5.8)Substituting the phantom term (5.3) gives an expression of the formTr[()Q()]= Tr[C11 C21 ]+Tr[C31 C41 ], (5.9) 8 JHEP06(2012)084whereC1 = [c,]KB1 , (5.10)C2 = cccK1 , (5.11)C3 = cK[c,]KB1 , (5.12)C4 = ccK1 . (5.13)To understand what happens in the 0 limit, note that the factor 1 inside thetrace (5.9) approaches the sliver state:lim0+1 = . (5.14)To prove this, expand the geometric series1 = n=0nn. (5.15)and expand the wedge state in the summand around n =:n = +1n+ 1(1) +1(n+ 1)2(2) + . . . , (5.16)where (1),(2), . . . are the coefficients of the corrections in inverse powers of n+1 (actually(2) is the first nonzero correction in the Fock space). Plugging (5.16) into the geometricseries and performing the sums gives1 = + ln (1) +Li2(2) + . . . . (5.17)Only the sliver state survives the 0 limit. This means that for small equation (5.9) isdominated by correlation functions on the cylinders with very large circumference. In thislimit, it is useful to expand the fields C1, . . . C4 into a sum of states with definite scalingdimension (the L0 level expansion). To leading order this expansion givesC1 =(cB)+ . . . , (L0 = 1), (5.18)C2 = 12(cc2c)+ . . . , (L0 = 0), (5.19)C3 =(KccB)+ . . . , (L0 = 1), (5.20)C4 = (cc)+ . . . , (L0 = 1). (5.21)Now consider the following: If a cylinder of circumference L has insertions of total scalingdimension h separated parametrically with L, rescaling the cylinder down to unit circum-ference produces an overall factor of Lh, which vanishes in the large circumference limit 9 JHEP06(2012)084if h is positive. Since the sum of the lowest scaling dimensions of C1 and C2 is positive,the corresponding term in (5.9) must vanish. The sum of the lowest scaling dimensions ofC3 and C4 is zero, so the corresponding term in (5.9) is nonzero and receives contributiononly from the leading L0 level of C3 and C4. Therefore the action simplifies toS = 16lim0Tr[KccB1 cc1 ]. (5.22)Expanding the geometric series givesS = 16lim02L=2LL1k=1Tr[KccBLkcck]. (5.23)Scaling the total wedge angle inside the trace to unity,S = 16lim02L=2LL(1LL1k=1Tr[KccB1kL cckL]). (5.24)Expanding the factor in parentheses around L =, the sum turns into an integral:1LL1k=1Tr[KccB1kL cckL]= 10dx Tr[KccB1xccx]+O(1L)+ . . . . (5.25)The order 1/L terms and higher do not contribute in the 0 limit, as explained in (5.17).ThereforeS = 16(lim02L=2LL) 10dx Tr[KccB1xccx]= 16 10dx Tr[1xBccQ(xBcc)]. (5.26)A moments inspection reveals that this integral is precisely the action evaluated on thesimple solution for the tachyon vacuum [8], expressed in the form [23]simp = c B1 +Kcc. (5.27)ThereforeS = 16Tr[simpQsimp] =122, (5.28)in agreement with Sens conjecture.6 Example 3: Tadpole shift between two marginalsIn this section we use the phantom term to compute the shift in the closed string tadpoleamplitude between two Schnabl-gauge marginal solutions [13, 14]:1 =cV1B1 + 1K V1c,2 =cV2B1 + 1K V2c, (6.1) 10 JHEP06(2012)084where V1 and V2 are weight 1 matter primaries with regular OPEs with themselves (butnot necessarily with each other).10 Our main interest in this example is to understandhow the boundary condition changing projector works when connecting two distinct andnontrivial BCFTs; In this case the projector has a rather nontrivial structure and possiblesingularities from the collision of matter operators at the midpoint [10]. This is the firstexample of a phantom term which does not vanish in the Fock space (at least in the casewhere the V1-V2 OPE is regular). This example also gives an independent derivation ofthe tadpole amplitude for Schnabl-gauge marginals, which previously proved difficult tocompute [15]. Another computation of the tadpole for the closely related solutions ofKiermaier, Okawa, and Soler [? ] appears in [24].We will map between the marginal solutions 1 and 2 using the left gauge transfor-mationU = Q12(B1 K)= 1cB11 + V11K11 + 1K V2Bc. (6.2)This choice of U is natural to the structure of the marginal solutions, since it factorizes intoa product of left gauge transformations through the Schnabl-gauge tachyon vacuum [10].The regularized boundary condition changing projector is+ U= + cB1 + V11K P1+ 1 + 1K V2 Bc P2+ 211 + 1K V2 B[c,]11 + V11K P12. (6.3)The first two terms, P1 and P2, are regularized boundary condition changing projectorsfrom 1 to the tachyon vacuum (Schnabls solution), and from the tachyon vacuum to 2,respectively. These terms vanish in the Fock space in the 0 limit. The third term P12is the nontrivial one: In the 0 limit it approaches the sliver state in the Fock space,with the boundary conditions of 2 on its left half and the boundary conditions of 1 onits right half; it represents the open string connecting the BCFTs of 2 and 1 [10]. If V1and V2 have regular OPE, P12 is a nonvanishing projector in the Fock space. If the OPEis singular, P12 may be vanishing or divergent because of an implicit singular conformaltransformation of the boundary condition changing operator between the BCFTs of 2and 1 at the midpoint. Part of our goal is to see how this singularity is resolved when wecompute the overlap. The phantom term is12() = (+ P1 + P2 + P12)(2 1). (6.4)10For example, we could choose V1 = e1X0to be the rolling tachyon deformation, and V2 = e1X0to be the reverse rolling tachyon; or we could choose V1 and V2 to be Wilson line deformations along twoindependent light-like directions. In both these examples the V1-V2 OPE is singular. 11 JHEP06(2012)084First let us consider the contribution to the tadpole from P12:TrV [P12(2 1)]. (6.5)Plugging everything in givesTrV [P12(2 1)] (6.6)= TrV[211+ 1K V2 B[c,]11+V11K (V211+ 1K V2 V1 11+ 1K V1)c]= TrV[Bcc11 + V11K (V211 + 1K V2 V1 11 + 1K V1) next step11 + 1K V2 ].In the second step we inserted a trivial factor of cB next to the commutator [c,], whichallows us to remove the c ghost from the difference between the solutions. Now lets lookat the factor above the braces. Re-express it with a few manipulations(V211 + 1K V2 V1 11 + 1K V1)= (11 + V11KV11 K 1)K1 + K1 (1 KV211 1K V2 1)= 11 + V11KK1 K1 11 1K V2= (1 11 + V11K)K1 +K1 (1 11 1K V2). (6.7)Express the factors on either side of the underbrace in equation (6.6) in the form:11 + V11K =11 + V11K 11 11+V11K11 + 1K V2 = 11 11+ 1KV2 11 + 1K V2. (6.8)Plugging everything into (6.6), the factors in parentheses above cancel against the factorsin parentheses in (6.7). Thus the contribution to the tadpole from P12 simplifies toTrV [P12(2 1)] = TrV[1 + V11K K1V1K1 11 + 1K V2Bcc]TrV[Bcc11 + V11KK1 1 + 1K V2 V2 K1]. (6.9) 12 JHEP06(2012)084In the 0 limit, we claim that the factors above the braces approach the sliver state withboundary conditions deformed by the corresponding marginal current, multiplied by thefactor K1 . If V1 and V2 have regular OPE, we can expand the factors outside the bracesin the L level expansion and pick off the leading term in the 0 limit. This givesprecisely the difference in the closed string tadpole amplitude between the two solutions.Unfortunately this argument does not work when V1 and V2 have singular OPE, sincecontractions between V1 and V2 produce operators of lower conformal dimension whichmake additional contributions. It is not an easy task to see what happens in this case inthe 0 limit, but there is no reason to believe that (6.9) should calculate the shift in thetadpole amplitude. This is a remnant of the midpoint singularity of the boundary conditionchanging projector when the boundary condition changing operator between the BCFTs of2 and 1 has nonzero conformal weight. To fix this problem we need to account for thetachyon vacuum contributions to the phantom term. Let us focus on the contributionfrom P1:TrV [P1(2 1)] = TrV[1 + V11K V211 + 1K V2Bcc] TrV [P11]= TrV[1 + V11K K1 ( 11 + 1K V2+ 1)Bcc]TrV [P11]. (6.10)Note that this precisely cancels the problematic contractions between V1 and V2 in thefirst term in (6.9). A similar thing happens for the second term when we consider thecontribution of P2. Therefore, in total the overlap isTrV [12()] = TrV [2 1] TrV [P11] + TrV [P22]+TrV[1 + V11K K1 Bcc]TrV[BccK1 1 + 1K V2 ]. (6.11)Note that we have not yet taken the 0 limit, so this formula is valid for all . Now wesimplify further by taking 0, and a short calculation shows that the first three termsvanish. Therefore consider the contribution from the fourth term:TrV[1 + V11K K1 Bcc]= L=0LTrV[( 1V11 K)L K1 Bcc].(6.12) 13 JHEP06(2012)084Expanding the summand perturbatively in V1,TrV[( 1V11 K)L K1 Bcc]= TrV[LK1 Bcc]TrV[(L1k=0L1k[1V11 K]k)K1 Bcc]+TrV[(L2k=0L2kl=0L2kl[1V11 K]k[1V11 K]l)K1 Bcc] . . . . (6.13)To derive the 0 limit we expand this expression around L = . Then each term inthe perturbative expansion is a correlator on a very large cylinder, and we can pick out theleading L level of every field in the trace whose total width is fixed in the L limit.This gives:TrV[( 1V11 K)L K1 Bcc]= TrV [LBcc] + 1TrV[L1k=0L1k V1k Bcc] 12TrV[L2k=0L2kl=0L2kl V1k V1lBcc]+ . . .+O(1L). (6.14)Scaling the circumference of the cylinders with 1L , the sums above turn into integrals whichprecisely reproduce the boundary interaction of the marginal current [? ]:TrV[( 1V11 K)L K1 Bcc]= TrV [Bcc] + 1 10dxTrV [1x V1xBcc] 12 10dx 1y0dy TrV [1xy V1x V1y Bcc] + . . .+O(1L)= TrV [e(K+1V1)Bcc] +O(1L). (6.15)Plugging this back into (6.12) and taking the 0 limit giveslim0TrV[1 + V11K K1 Bcc]= TrV [e(K+V1)Bcc]. (6.16) 14 JHEP06(2012)084A similar argument for the final term in (6.11), and removing the B ghost follow-ing (4.13) giveslim0TrV [12()] = TrV [e(K+V2)c] + TrV [e(K+V1)c]= A2(V)A1(V). (6.17)which is the expected shift in the closed string tadpole amplitude between the twomarginal solutions.AcknowledgmentsT.E. gratefully acknowledges travel support from a joint JSPS-MSMT grant LH11106.This research was funded by the EURYI grant GACR EYI/07/E010 from EUROHORCand ESF.References[1] M. Schnabl, Analytic solution for tachyon condensation in open string field theory, Adv.Theor. Math. Phys. 10 (2006) 433 [hep-th/0511286] [INSPIRE].[2] Y. Okawa, Comments on Schnabls analytic solution for tachyon condensation in Wittensopen string field theory, JHEP 04 (2006) 055 [hep-th/0603159] [INSPIRE].[3] E. Fuchs and M. 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