BV Analysis of Tachyon Fluctuation around Multi-brane Solutions in Cubic String Field Theory

We study whether the tachyon mode exists as a physical fluctuation on the 2-brane solution and on the tachyon vacuum solution in cubic open string field theory. Our analysis is based on the Batalin-Vilkovisky formalism. We first construct a set of six string states which corresponds to the set of fields and anti-fields containing the tachyon field. Whether the tachyon field can exist as a physical fluctuation is determined by the 6x6 matrix defining the anti-bracket in the present sector. If the matrix is degenerate/non-degenerate, the tachyon field is physical/unphysical. Calculations for the pure-gauge type solutions in the framework of the KBc algebra and using the Ke-regularization lead to the expected results. Namely, the matrix for the anti-bracket is degenerate/non-degenerate in the case of the 2-brane/tachyon-vacuum solution. Our analysis is not complete, in particular, in that we have not identified the four-fold degeneracy of tachyon fluctuation on the 2-brane solution, and moreover that the present six states do not satisfy the hermiticity condition.


Introduction
After the discovery of exact tachyon vacuum solution [1] in cubic string field theory (CSFT) followed by its concise understanding [2] in terms of the KBc algebra [3], there have been considerable developments in the construction of multi-brane solutions [4,5,6,7]. The identification of a solution as the n-brane one representing n pieces of D25-branes has been done from its energy density consideration. However, for the complete identification, we have to show that the the physical excitations on the solution are those of the open string and, in particular, that each excitation has n 2 degeneracy. For the tachyon vacuum solution (n = 0), a general proof has been given for the absence of physical excitations [8]. On the other hand, for n-brane solution with n ≥ 2, no formal existence proof nor an explicit construction of the excitations has been given. 1 In this paper, we present an explicit analysis of fluctuations around multi-brane solutions in the framework of the Batalin-Vilkovisky (BV) formalism [10,11]. Our analysis is not a complete one, but is rather a first step toward the final understanding. First, our analysis is restricted only to the tachyon vacuum solution and the 2-brane one. Second, we do not solve the general excitation modes on the solution. Our analysis is restricted to the tachyon mode among all the excitations.
Let us explain our analysis in more detail. We are interested in the kinetic term of the action of CSFT expanded around a multi-brane solution: where Q is the BRST operator in the background of the solution, and Φ is the fluctuation around the solution. Previous arguments have been mainly on the presence of the homotopy operator A on the tachyon vacuum solution satisfying QA = I with I being the identity string field. If there exists a well-defined A, it implies that there are no physical excitations at all. In this paper, we carry out a different kind of analysis. We consider a candidate tachyon field χ(x) as a fluctuation around a class of multi-brane solutions, and examine whether χ represents a genuine physical excitation or it is unphysical. In the former case, the lagrangian of χ contained in (1.1) should be the ordinary one: 2 1 See [9], for a construction of multi-brane solutions and the fluctuation modes around them by introducing the boundary condition changing operators. 2 The space-time metric used in this paper is the mostly plus one; g µν = diag(−1, 1, 1, · · · , 1).
On the other hand, if χ is unphysical, it should be a member of unphysical BRST quartet fields χ, C, C, B with the lagrangian given by a BRST-exact form [12]: where the BRST transformation δ B (satisfying the nilpotency δ 2 B = 0) is defined by In CSFT which has been constructed in the BV formalism, the lagrangian for unphysical χ is not of the type (1.3) containing the auxiliary field B, but is rather the one obtained by integrating out B: This is invariant under the redefined BRST transformation δ ′ B : Note that δ ′ B is nilpotent only on-shell; in particular, we have (δ ′ B ) 2 C = i ( − m 2 ) C. Our analysis is carried out within the framework of the BV formalism. We first construct six first-quantized string states u i (k) carrying center-of-mass momentum k µ . These six states correspond to the three fields (χ, C, C) in (1.5) as well as their anti-fields (χ ⋆ , C ⋆ , C ⋆ ). We call the six states u i the tachyon BV states. Then, we examine the 6 × 6 matrix ω ij = u i u j given by the CSFT integration. In fact, this ω ij is the matrix defining the anti-bracket in the BV formalism, and it determines whether χ is physical or unphysical. If ω ij is non-degenerate, namely, det ω ij = 0, χ is unphysical. More precisely, after the gauge-fixing by removing the anti-fields, we obtain the lagrangian (1.5) of an unphysical system. On the other hand, if ω ij is degenerate, it implies that the six states u i are not independent, and therefore, some of the fields/anti-fields necessary for making χ unphysical are missing. Concrete analysis shows that the lagrangian for χ in the case of degenerate ω ij is the physical one (1.2).
We consider multi-brane solutions in CSFT given formally as the pure-gauge UQ B U −1 with U specified by a function G(K) of K in the KBc-algebra (see (3.2)). The point is that the eigenvalues of K are in the range K ≥ 0, and various physical quantities associated with the solution such as the energy density are not well-defined due to the singularity at K = 0. Therefore, we introduce the K ε -regularization of replacing K in UQ B U −1 by K ε = K + ε with ε being a positive infinitesimal [5]. Then, the regularized solution (UQ B U −1 ) K→Kε is no longer exactly pure-gauge, and the zero or the pole of G(K) at K = 0 is interpreted as the origin of the non-trivial energy density of the apparently pure-gauge configuration [4,5,6,7]. 3 Namely, There is another important technical point in our BV analysis. The matrix ω ij = u i u j and the EOM test against the commutator Ψ T = [u i , u j ] are functions of k 2 of the momentum k µ carried by u i . Then, a problem arises: Some of these quantities contain terms depending on ε of the K ε -regularization in a manner such as ε min(2k 2 −1,1) , which diverges in the limit ε → 0 for a smaller k 2 and tends to zero for a larger k 2 . Therefore, we define them as the "analytic continuation" from the region of sufficiently large k 2 (namely, sufficiently space-like k µ ) to drop this type of ε-dependent terms.
Next, we comment on the "cohomology approach" to the problem of physical fluctuation around a multi-brane solution. In this approach, we consider the BRST cohomology KerQ/ImQ, namely, we solve Qu 1 (k) = 0 for u 1 (k) which carries ghost number one and is not Q-exact. However, the meaning of (non-)equality in Qu 1 = 0 and u 1 = Q( * ) is subtle for multi-brane solutions of the pure-gauge type discussed in this paper due to the singularity at K = 0. To make these equations precise, we should introduce the K ε -regularization and consider their inner-products (CSFT integrations) with states in the space of fluctuations. We would like to stress that our BV analysis indeed gives information for solving the BRST cohomology problem within the K ε -regularization. (The present BV analysis can identify some of the non-trivial elements of KerQ/ImQ. However, it cannot give the complete answer to the cohomology problem since we consider only a set of trial BV states.) We will explain the interpretation of our results of the BV analysis in the context of the cohomology approach in Secs. 4.4 and 5.2. We also comment that the analysis of the BRST cohomology around the tachyon vacuum by evaluating the kinetic term of the action of the fluctuation in the level truncation approximation [13,14,15] has some relevance to the present BV approach.
Then, finally in the Introduction, we state our results obtained in this paper. For the tachyon vacuum solution, we find that the matrix ω ij is non-degenerate. This implies that our candidate tachyon field is an unphysical one belonging to a BRST quartet. On the other hand, for the 2-brane solution, ω ij turns out to be degenerate, implying that the tachyon field is a physical one. These results are both what we expect for each solution. However, we have not succeeded in identifying the whole of the 2 2 tachyon fields which should exist on the 2-brane solution. In addition, the six tachyon BV states in this paper have a problem that they do not satisfy the hermiticity requirement (see Sec. 3.5).
The organization of the rest of this paper is as follows. In Sec. 2, we recapitulate the BV formalism used in this paper, and give examples of the BV states on the unstable vacuum. In Sec. 3, we present the construction of the six tachyon BV states on a generic pure-gauge type solution, and prepare various formulas necessary for the BV analysis. In Sec. 4, we carry out the calculation of the EOM against u i and [u i , u j ] and of each component of ω ij on the 2-brane solution to confirm the existence of a physical tachyon field. Next, in Sec. 5, we repeat the same analysis for the tachyon vacuum solution. There we find that the candidate tachyon field is unphysical. We summarize the paper and discuss future problems in Sec. 6. In the Appendices, we present various technical details used in the text.

BV formalism for CSFT
The action of CSFT on the unstable vacuum [16], 4 and the BV equation holds due to (i) the nilpotency Q 2 B = 0 of the BRST operator Q B , (ii) the derivation property of Q B on the * -product, (iii) the property Q B (· · · ) = 0, (iv) the associativity of the * -product, and (v) the cyclicity A 1 * A 2 = (−1) A 1 A 2 A 2 * A 1 valid for any two string fields A 1 and A 2 . 5 The BV equation is a basic requirement in the construction of gauge theories including SFT. The BV equation implies the gauge invariance of the action. Moreover, it gives a consistent way of gauge-fixing and quantization of the theory.
In this paper, we are interested in CSFT expanded around a non-trivial solution Ψ S satisfying the EOM: Expressing the original string field Ψ in (2.1) as Ψ = Ψ S + Φ, (2.5) with Φ being the fluctuation, we obtain The second term on the RHS of (2.6) should vanish due to the EOM (2.4). However, for multi-brane solutions in CSFT, this EOM term cannot vanish for all kinds of fluctuations Φ as stated in the Introduction. This is the case even for the tachyon vacuum solution. In this paper, we restrict the fluctuation Φ around Ψ S to those for which the EOM term of (2.6) vanishes. We will see later that the EOM term must also vanish against the commutator among the fluctuations. 4 We have put the open string coupling constant equal to one. 5 (−1) A = +1 (−1) when A is Grassmann-even (-odd).
The last term of (2.6) is the action of the fluctuation: The only difference between the two actions (2.1) and (2.7) is that the BRST operator Q B in the former is replaced with Q Ψ S , the BRST operator around the solution Ψ S . The operation of Q Ψ S on any string field A with a generic ghost number is defined by which is formally equivalent to (2.2) for the original S, also holds since Q Ψ S satisfies the same three basic properties as Q B does; (i), (ii) and (iii) mentioned below (2.3). Among them, the nilpotency Q 2 Ψ S = 0 is a consequence of the EOM; namely, we have from (2.8) On the other hand, the other two properties (ii) and (iii) hold for any Ψ S irrespectively of whether it satisfies the EOM or not. In the following, we omit the subscript Ψ S in S Ψ S unless necessary.

BV equation in terms of component fields
Here, we consider the BV equation (2.9) for the action (2.7) in terms of the component fields. 6 Let {u i (k)} be a "complete set" of states of fluctuation around Ψ S (here, we take as Ψ S a translationally invariant solution, and k µ is the center-of-mass momentum of the fluctuation).
Note that each u i (k) is a string field. Then, we expand the fluctuation field Φ as where ϕ i (k) is the component field corresponding to the state u i (k), and k is short for d 26 k/(2π) 26 . In (2.11), u i (k) may carry any ghost number N gh (u i ), and the ghost number of the corresponding ϕ i must satisfy N gh (u i ) + N gh (ϕ i ) = N gh (Φ) = 1. (2.12) Then, we define the matrix ω ij (k) and its inverse ω ij (k) by (2.14) Here, we are assuming that ω ij is non-degenerate, namely, that the inverse matrix ω ij exists. 7 In particular, the number of the basis u i (k) must be even. Note that ω ij and ω ij are nonvanishing only for (i, j) satisfying N gh (u i ) + N gh (u j ) = 3, and therefore, Note also that these matrices are symmetric in the following sense: The completeness relation of the set {u i } reads for any string field A, and hence we have 8 . (2.18) Using (2.18) in (2.9), we obtain the BV equation in terms of the component fields: It is convenient to take the Darboux basis where the matrix ω ij (k) takes the following form: (2.20) Denoting the corresponding component fields, namely, the pair of fields and anti-fields, as (2.21) 7 Precisely speaking, our assumption here is that det ω ij (k) is not identically equal to zero as a function of k µ . ω ij being degenerate at some points in the k µ space is allowed. 8 The sign factor (−1) ϕ j in (2.18) is due to the fact that the CSFT integration is Grassmann-odd. In this paper, δ/δϕ j for a Grassmann-odd ϕ j is defined to be the left-derivative − → δ /δϕ j .
Then, the gauge-fixed action S and the BRST transformation δ B under which S is invariant are given by where | L denotes the restriction to the Lagrangian submanifold defined by the gauge-fermion Υ[φ]: The simplest choice for Υ is of course Υ = 0.

Examples of BV basis on the unstable vacuum
For CSFT on the unstable vacuum, the BV basis {u i (k)} consists of an infinite number of first quantized string states of all ghost numbers. Though the whole BV basis is infinite dimensional, we can consider a subbasis with non-degenerate ω ij and consisting of a finite number of states which are connected by the operation of Q B and are orthogonal (in the sense of ω ij = 0) to any states outside the subbasis.
Here, we present two examples of BV subbasis with non-degenerate ω ij . For our later purpose, we present them using the KBc algebra in the sliver frame. The KBc algebra and the correlators in the sliver frame are summarized in Appendix A. In the rest of this paper, we omit "sub" for the BV subbasis and simply write "BV basis" since we will not consider the full BV basis.

Unphysical BV basis of photon longitudinal mode
Our first example is the unphysical BV basis associated with the longitudinal mode of the photon on the unstable vacuum Ψ S = 0. Namely, we consider the unphysical model obtained by restricting the photon field to the pure-gauge, A µ (x) = ∂ µ χ(x). The corresponding BV basis consists of the following six states: where α is a constant, and V k is the vertex operator of momentum k µ at the origin: These six states u i (k) are all chosen to be hermitian in the sense that Among the six u i , u 1A is the photon state with longitudinal polarization k µ . The operation of Q B on the six states (2.24) is given as follows: The non-trivial components of the 6 × 6 matrix ω ij (k) are given by and therefore ω ij is non-degenerate. 9 Moreover, the present basis {u i } is already Darboux as seen from (2.29). Then, expanding the string field Ψ as (2.30) and using (2.28) and (2.29), we find that the kinetic term of the CSFT action (2.1) is given by Though ω ij (k) is degenerate at k 2 = 0, this is not a problem as we mentioned in footnote 7.
Finally, the gauge-fixed action S 0 and the BRST transformation δ B in the gauge C ⋆ = χ ⋆ = C ⋆ = 0 are given using (2.22) by and This is the m 2 = 0 version of the unphysical system given in (1.5) and (1.6).

BV basis of the tachyon mode
Our second example is the BV basis for the tachyon mode on the unstable vacuum. It consists only of two states: the tachyon state u 1 and its BRST-transform u 2 : The 2 × 2 matrix ω ij is non-degenerate since we have Expressing the string field as the kinetic term reads which does not contain the anti-field φ ⋆ . The gauge-fixed action S 0 is the same as S 0 , the ordinary kinetic term of the tachyon field φ. The BRST transformation of φ is of course equal to zero;

Tachyon BV states around a multi-brane solution
We consider the fluctuation around a multi-brane solution Ψ ε given as the K ε -regularization of the pure-gauge UQ B U −1 [5]: where U and its inverse U −1 are specified by a function G(K) of K: Here and in the following, O ε for a quantity O containing K denotes the K ε -regularized one; Although the EOM is satisfied automatically by the pure-gauge UQ B U −1 , the K ε -regularization breaks the EOM by the O(ε) term: As we saw in [5], this O(ε) breaking of the EOM can be enhanced by the singularity at K = 0 to lead to non-trivial results for the EOM against Ψ ε itself: We found that (3.4) vanishes for G(K) having a simple zero, a simple pole or none at all at K = 0, which we expect to represent the tachyon vacuum, the 2-brane and the 1-brane, respectively, from their energy density values. For G(K) with higher order zero or pole at K = 0, (3.4) becomes non-vanishing. Therefore, in this paper, we consider the following two G(K) as concrete examples: which correspond to the tachyon vacuum and the 2-brane, respectively.
For our purpose of studying the fluctuation, it is more convenient to gauge-transform Ψ ε by Note that P ε is apparently of O(ε) since, without the K ε -regularization, the present gauge transformation transforms the pure-gauge UQ B U −1 back to zero. The fluctuation around P ε and that around Ψ ε are related by for S Ψ S of (2.7). Note the following property of Q Ψ S (2.8): The EOM of P ε is given by Though the EOM against the solution itself, (3.4), is not a gauge-invariant quantity, we have confirmed that P ε * (Q B P ε + P 2 ε ) vanishes in the limit ε → 0 for the two G(K) in (3.5).

Six tachyon BV states around P ε
We are interested in whether physical fluctuations exist or not around the classical solutions P ε specified by two G(K) in (3.5). Our expectation is of course that there are no physical fluctuations at all for G tv , while there are quadruplicate of physical fluctuations for G 2b . In this paper, we consider this problem in the framework of the BV formalism by focusing on the tachyon mode. In the following, Q denotes Q Pε , the BRST operator around P ε : , the action of the fluctuation Φ around P ε : Our analysis proceeds as follows: 1. We first present a set of six BV states {u i (k)} containing the tachyon state. This set of BV states is similar to (2.24) for the photon longitudinal mode.
2. We evaluate the matrix ω ij (k) (2.13) for the six BV states, and obtain the kinetic term of the action (3.11), by expanding Φ in terms of the six states.
3. If the matrix ω ij is non-degenerate, det ω ij = 0, we conclude that the tachyon field is an unphysical one. On the other hand, if ω ij is degenerate, det ω ij = 0, and, furthermore, the kinetic term (3.12) is reduced to (1.2), the tachyon field is a physical one.
As a concrete choice of the six tachyon BV states u i (k), we take (3.14) By Q B , they are related by Note that there appear in (3.14) e −αKε instead of e −αK . Namely, each state in (3.14) is multiplied by an extra factor e −2αε . Though this is merely a c-number factor which is reduced to one in the limit ε → 0, it makes the expressions of various O(1/ε) quantities simpler as we will see in Sec. 4.
Second, L = L(K ε ) and R = R(K ε ) in (3.13) are functions of only K ε . Though they are quite arbitrary at this stage, we will determine later, for each classical solution P ε , their small K ε behavior from the requirement that the EOM against u 1A/B and that against the commutators u 0 , u 1A/B hold. Finally, ξ in (3.13) is a parameter related to the arbitrariness in the definitions of u 1A and u 1B .
The action of the BRST operator Q (3.10) on the six states of (3.13) is given by where EOM ε is defined by and given explicitly by (3.9).
The set of six BV states (3.13) has been constructed by comparing its BRST transformation property (3.16) with that of the six BV states (2.24) for the longitudinal photon and by taking into account that EOM ε and P ε are both apparently of O(ε). First, u 1A is the tachyon state on the unstable vacuum, and u 0 is u 1A multiplied by the "homotopy operator" B/K ε of Q B . We start with u 0 , which is u 0 dressed by L and R −1 , and divided Qu 0 into the difference of u 1A and u 1B as given by the first equation of (3.16). If we ignore the apparently of O(ε) terms, u 1A is the dressed tachyon state, and u 1B , which is multiplied by (1 − k 2 ) vanishing at the tachyon on-shell k 2 = 1, corresponds to k 2 u 1B in the first equation of (2.28) for the massless longitudinal photon. We have distributed [P ε , L u 0 R −1 ] in iQu 0 to u 1A and u 1B with coefficients specified by the parameter ξ. Then, we consider Qu 1A and Qu 1B , which are equal to each other if Q 2 = 0 and hence EOM ε = 0 holds (see the second equation of (3.16)). We have chosen u 2A and u 2B as the part of Qu 1A which is multiplied by (1 − k 2 ) and the rest, respectively. For u 2A and u 2B , we have no clear correspondence with the BV states of the longitudinal photon. Finally, u 3 is naturally defined from Q(u 2A , u 2B ) as given in the last equation of (3.16).
Our choice (3.13) of the six states u i is of course not a unique one. For instance, in (3.14), the part (ε/K ε ) e −αKε cV k e −αKε in u 1B may be moved to u 1A to replace u 1A , u 1B and u 2A in (3.13) with the following ones: For u 2A and u 2B , we may take more generic linear combinations of the three terms However, here in this paper, we carry out the BV analysis by adopting the states of (3.13) with u i given by (3.14). In this sense, our analysis is rather an "experiment" and is not a comprehensive one. We do not know whether the conclusion of tachyon being physical or unphysical can be changed by taking another set of tachyon BV states. 10 As we will see later, L(K ε ) and R −1 (K ε ) appearing in the definition of u i (3.13) play a crucial role in making the EOM terms to vanish on the 2-brane. However, the pair (L, R) is not uniquely determined by this requirement alone. Therefore, we put a superscript (a) on (L, R) and the corresponding states u i in (3.13) to distinguish different choices of (L, R). For example, we write u Then, the matrix ω ij (2.13) now has another index (a, b): ij (k) should not be regarded as a matrix with its left index (i, a) and right one (j, b); it is still a 6 × 6 matrix with a fixed pair of (a, b). When we consider the action (3.11) in the final step of our analysis, we put (a) = (b) by taking a particular (L, R).
We see that all the components of ω 1A , and using and (3.16), we obtain the following relations: In deriving the last two relations, we have assumed the vanishing of the EOM terms: Its determinant is given by 1A,2B . (3.28) Using (3.16) and assuming (3.25), we also obtain the following useful formulas: where we have omitted (2π) 26 δ 26 (k ′ + k) on the RHS.

Formulas for the EOM tests and ω (a,b) ij
For the BV analysis for a given G ε , we need to evaluate (i) the EOM test of P ε against u 1A/B and u 1A,2B . For u 1A and u 1B containing the parameter ξ (see (3.13)), it is convenient to introduce w where we have allowed the parameter ξ to depend on the index a of L (a) and R (a) . For L (a) (K ε ) and R (a) (K ε ), we assume that their leading behaviors for K ε ∼ 0 are and give their remaining K ε -dependences as Laplace transforms:  1/R (s a ) should be determined by the requirement of the EOM as stated before. Concerning the choice of (m a , n a ), it would be natural to consider the case m a = n a since the overall order of the BV states u i (3.13) with respect to K ε for K ε ∼ 0 is not changed from the case without L (a) and R (a) . We will restrict ourselves to the case m a = n a in the concrete calculations given in Secs. 4 and 5.
Then, the three kinds of quantities necessary for the BV analysis are expressed as the following integrations over the Laplace transform variables: The explicit expressions of E ℓ , E are lengthy and hence are summarized in Appendix B. They are given as sliver frame integrations containing a single or no B. Though some of their defining expressions contain two or more B, we have used the KBc algebra to reduce them to sliver frame integrations with a single B.
The three E ℓ are not independent, but they satisfy the following relation: for distinguishing v L , v 1/R , ξ defining the state u i and that defining u j . However, when we express the fluctuation in terms of the basis {u i } and the corresponding component fields, we choose one particular v L , v 1/R , ξ . Namely, when we consider the action (3.12), there appear only ω (a,a) ij with (a) = (b). Therefore, we here omit the superscript (a, a) and simply write ω ij .
When ω ij is non-degenerate and the determinant (3.28) is not identically equal to zero, |Ω| = 0, it is convenient to move to the Darboux basis by switching from where the inverse matrix Ω −1 is given by The new set {u 0 , u 1A , u 1B , u 2P , u 2Q , u 3 } is in fact a Darboux basis since we have Instead of (3.30), (u 2P , u 2Q ) satisfies For expressing Φ in terms of the Darboux basis, it is more convenient to use still another one { u i } with tilde, which is defined by multiplying the states corresponding to the fields and anti-fields by ̟(k) and its inverse, respectively:  with ̟(k) given by Then, expressing Φ as

48) the kinetic term (3.12) is given in terms of the component fields and anti-fields as
The action (2.31) for the photon longitudinal mode on the unstable vacuum is essentially the special case of (3.49) with ̟(k) = k 2 , and the gauge-fixing process for (3.49) goes in the same manner as for (2.31). Adopting the gauge L with χ ⋆ = C ⋆ = C ⋆ = 0, the gauge-fixed action and the BRST transformation are given by and If ̟(k) has a zero at k 2 = −m 2 , the action (3.50) describes a totally unphysical system with mass m explained in the Introduction.
The above argument leading to (3.49) does not apply if we ω ij is degenerate. In such a case, the system can describe a physical one in general.

(Non-)hermiticity of the BV states
Our tachyon BV basis {u i } given by (3.13) has in fact a problem that it does not satisfy the hermiticity condition. We will explain it in this subsection.
In the original CSFT action (2.1), the the string field Ψ is assumed to be hermitian; Ψ † = Ψ, or more generally, Ψ † = W (K)ΨW (K) −1 with W (K) depending only on K. This constraint ensures the reality of the action (2.1) and, at the same time, prevents the duplication of each fluctuation modes. Then, let us consider the hermiticity for the action (3.11) of the fluctuation Φ around P ε . First, P ε (3.6) satisfies the hermiticity in the following sense: Therefore, the fluctuation Φ in (3.11) must satisfy the same hermiticity: If (3.54) holds, it follows that QΦ with Q defined by (3.10) also satisfies the same hermiticity 11 and hence that the action (3.11) is real. In the expansion (2.11) of Φ in terms of the basis {u i (k)} and the component fields ϕ i (k), the hermiticity of Φ, (3.54), is realized by imposing and ϕ i (k) † = ϕ i (−k). However, our BV states (3.13) do not satisfy this hermiticity condition. 11 In deriving (3.55), we use the property Q B K = 0 and Q B A † = −(−1) A Q B A † valid for any string field A.
One way to realize the hermiticity (3.56) is to take, instead of the states u i (3.13), the following ones U i : on the 2-brane no longer holds for any v L , v 1/R , ξ with (m, n) = (1, 1). Therefore, we will continue our analysis by using the original non-hermitian basis {u i }, though this is certainly a problem to be solved in the future.

BV analysis around the 2-brane solution
In this section, we carry out the BV analysis of the six states of (3.13) for the 2-brane solution given by G 2b in (3.5). Our analysis consists of the following three steps: 1. Evaluation of the EOM of P ε against u 1A/B and u 1A/B (recall (3.25) for the necessity of the latter). From the vanishing of these EOMs, we determine the allowed set of (L(K ε ), R(K ε )).

(m, n) = (1, 1)
In this case, E ℓ are given up to O(ε) terms by where C 1,2 (s) are defined by C 1 (s) =s + α + 1, C 2 (s) = (s + α + 1) 2 + 1. (4.7) We defer further arguments on the EOM against u 1A/B for (m, n) = (1, 1) till we discuss the EOM against u This implies that we have to choose ξ = −1 to make the 1/ε 2 part of the EOM test against u 1A and u 1B to vanish. Namely, we have to take Then, the combinations of E ℓ relevant to u 1A/B are given as follows:  5)-(B.7) for a generic G ε . As in the previous subsection, we consider only the two cases; (m a , n a ) = (m b , n b ) = (1, 1) and (0, 0). We will explain the calculations in the case of (1, 1) in great detail. The same method will be used also in the calculation of ω Let us explain how we evaluate (4.12) and other E (a,b) ℓ,0 for a generic momentum k µ . Let us consider, as an example, the contribution of the c e −(α+sa)K term of the commutator c, e −(α+sa)K to the first integral of (4.12): with e −ε(π+ s) defined by e −ε(π+ s) = e −ε(π+sa+s b +sa+s b ) . (4.14) In (4.13), t 1 , t 2 and t 3 are the Schwinger parameters for 1/(1 + K ε ) and the two 1/K ε , respectively, and the function F (t 1 , t 2 , t 3 ) is given by where G is the product of the ghost correlator and the matter one on the infinite cylinder of circumference ℓ = w 1 + w 2 + w 3 + w 4 : The explicit expressions of the correlators are given in Appendix A.
One way to evaluate (4.13) in the limit ε → 0 is to (i) make a change of integration variables from (t 2 , t 3 ) for 1/K ε to (u, x) by (t 2 , t 3 ) = (u/ε) (x, 1 − x), (ii) carry out the x-integration first, (iii) Laurent-expand the integrand in powers of ε to a necessary order, and finally (iv) carry out the integrations over u and t 1 . In fact, we obtained the results (4.6), (4.8) and (4.10) by this method. However, it is hard to carry out explicitly the x-integration in (4.13) before Laurentexpanding with respect to ε due to the presence of the k 2 -dependent matter correlator in (4.16). On the other hand, Laurent-expanding the (t 1 , u, x)-integrand with respect to ε before carrying out the x-integration sometimes leads to a wrong result. Namely, the integration regions where x or 1 − x are of O(ε) can make non-trivial contributions.
Our manipulation for obtaining the correct result for (4.13) is as follows. Eq. (4.13), which is multiplied by ε, can be non-vanishing due to negative powers of ε arising from the two 1/K ε at the zero eigenvalue K = 0. In the RHS of (4.13), this contribution comes from any of the following three regions of the (t 2 , t 3 )-integration: Concretely, (4.13) is given as the sum of the contributions from the three regions: with each term given by where Ser ε denotes the operation of Laurent-expanding the function with respect to ε to a necessary order. 14 In each region, we have put t 2 + t 3 = u/ε and limited the integration region of u to (ε, ∞) since the other region (0, ε) cannot develop a negative power of ε. As given in (4.19)-(4.21), the three regions of (4.17) are specified by two parameters, ζ and η, which we assume to be of O(ε 0 ). Explicitly, the evaluation of the terms (I)-(III) goes as follows: Term (I) For (4.19), the Laurent expansion gives The leading term of the y-integration is of order (u/ε) max(5−2k 2 ,0) , and we obtain where we have used that 15 ∞ ε du e −u ε u g = O ε min(g, 1) .

Term (II)
The Laurent expansion in (4.20) gives Since the x-integration in the range ζ ≤ x ≤ 1 − η is finite, we obtain The Laurent expansion in (4.21) gives Carrying out the y-integration, we get where the last term is the contribution of the y = ηu/ε term.
Summing the three terms, (4.23), (4.26) and (4.28), and carrying out the t 1 -integration of (4.18), we finally find that (4.13) is given by (4.29) This result can also be checked by numerically carrying out the integrations of (4.13) for given values of ε, k 2 and other parameters in (4.13).
The evaluation of the other term of the first integral of (4.12), namely, the term containing the e −(α+sa)K c part of the commutator c, e −(α+sa) , is quite similar. In fact, the two terms of the commutator almost cancel one another, and the whole of the first integral of (4.12) turns out to be simply of O ε min(2k 2 −1,1) . 16 Next, the second integral of (4.12) is given by (4.30) The evaluation of this term is much easier than that of the first integral explained above since there is only one Schwinger parameter t 2 for 1/K ε . We have only to Laurent-expand the integrand with respect to ε after making the change of integration variables from t 2 to u = εt 2 , and carry out the (t 1 , u)-integrations. After all, the whole of E (a,b) A,0 (4.12) is found to be given by where the first (second) term on the RHS corresponds to the first (second) integral of (4.12).

(a,b)
A,0 for a generic k 2 as the "analytic continuation" from the region of sufficiently large k 2 (k 2 > 1/2 in the present case). Thus, E

(a,b)
A,0 is simply given by the last term of (4.31). Eq. (4.31) has been obtained by keeping only the first term of the Laurent expansion. The subleading term which has an extra positive power (ε/u) p contributes O ε min(2k 2 −1+p,1) . This vanishes for k 2 > 1/2 and does not affect our definition of E C,0 is similar except two points. First, they contain terms with three 1/K ε . For such terms, we have to carry out the integration over the three Schwinger parameters by considering 2 3 − 1 = 7 regions with at least one large parameters (see Appendix C). Second, the obtained E (a,b) ℓ,0 (ℓ = B, C) both contain 1/ε terms, and, therefore, e −ε(π+ s) (4.14) multiplying them makes non-trivial contribution to their O(ε 0 ) terms. Then, we get the following results: 17 (4.33) In particular, E (a,b) 1B , is given by where S a+b and S b+ã are defined by S a+b = s a +s b + 2α, S b+ã = s b +s a + 2α. 1B hold for any ξ. Furthermore, the EOM against u 1A/B also holds for any ξ as seen from (4.6).
The condition (4.36) restricts the first few terms of the series expansion of 1/R(K ε ) with respect to K ε . In fact, expanding the expression (3.34) for 1/R(K ε ) in powers of K ε and using the condition (4.36), we obtain ℓ,0 for (m a , n a ) = (m b , n b ) = (0, 0) is much harder than that for (1, 1). Here, however, we need only their 1/ε 2 part: This result implies that the 1/ε 2 part of the EOM against u Assuming that v Therefore, we express the fluctuation Φ around the solution P ε in terms of only u 1A and u 2A which are non-trivial and independent:  Since we have ω 1A,2A (k 2 = 1) = 1, the expansion (4.48) and the action (4.49) are essentially the same as (2.37) and (2.38), respectively, for the tachyon field on the unstable vacuum. The present χ represents a physical tachyon field.
Finally, let us interpret the above result in the context of the BRST cohomology problem. Using the truncation (4.47) and discarding the EOM terms in the BRST transformation formula (3.16), we obtain the following equations for the remaining u 1A and u 2A : The first equation and the fact that u 0 ∼ 0, namely, that there is no candidate BRST parent of u 1A , imply that u 1A at k 2 = 1 is a physical state belonging to KerQ/ImQ.

BV analysis around the tachyon vacuum solution
In this section, we repeat the BV analysis of the previous section by taking G tv (3.5) which represents the tachyon vacuum. We expect of course that the matrix ω ij of the six BV states u i is non-degenerate and therefore the excitations they describe are unphysical ones. As (m, n) for (L, R), we consider here again only the two cases, (1, 1) and (0, 0). The result (5.1) implies that the EOMs against u 1A and u 1B cannot be satisfied for any ξ in the case (m, n) = (1, 1). On the other hand, EOMs against u 1A/B both hold for an arbitrary ξ in the case (0, 0). Therefore, in the rest of this section, we consider only the latter case (m, n) = (0, 0).

EOM against u
Next, E in the case (m, n) = (0, 0) are found to be given by Namely, E where the last term should be discarded by analytic continuation.
Next, for ω 1A,2B , we have to specify (ξ a , ξ b ). Here, we consider the simplest case of (ξ a , ξ b ) = (0, 0), for which we need to calculate only W Finally, our result is interpreted in the BRST cohomology problem as follows. On the massshell k 2 = 1, ω ij is reduced to 20 This ̟(k) is not necessarily non-negative, and this may be a problem for the hermiticity of u i related to u i by (3.46) containing ̟(k). For example, in the simplest case of L = R = 1, we have ω 1A,2A (k) ≡ 1 and hence ̟(k) is negative for k 2 < 1. Though the hermiticity of the original BV states u i itself is a problem as we mentioned in Sec. 3.5, one way to resolve the negative ̟ problem would be to Wick-rotate to the Euclidean space-time where we have k 2 ≥ 0 (the negative ̟ region, 0 ≤ k 2 < 1, should be regarded as an artifact of the tachyon).
From this we find that u 1B ∼ 0 and u 2B ∼ 0 at k 2 = 1. Then, from (3.16), we obtain the following BRST transformation rule for the remaining (u 0 , u 1A , u 2A , u 3 ): This implies, in particular, that the candidate physical state u 1A is a trivial element of KerQ/ImQ. Of course, this cannot be a proof of the total absence of physical excitations with m 2 = −1 on the tachyon vacuum.

Summary and discussions
In this paper, we carried out the analysis of the six tachyon BV states for the 2-brane solution and for the tachyon vacuum solution in CSFT. This set of six states was chosen from the requirement that the EOM of the solution holds against the states and their commutators. We found that the matrix ω ij defining the BV equation is degenerate and therefore the tachyon mode is physical for the 2-brane solution. On the other hand, ω ij is non-degenerate on the tachyon vacuum solution, implying that the candidate tachyon field is in fact unphysical there. These results are in agreement with our expectation and the general proof of the non-existence of physical excitations on the tachyon vacuum [8].
Our analysis in this paper is incomplete in several respects. First, we have not identified all of the four tachyon fields of the same m 2 = −1 which should exist on the 2-brane solution. Secondly and more importantly, we must resolve the problem that our six tachyon BV states (3.13) do not satisfy the hermiticity condition (3.56). Even if we put aside this problem, there are a number of questions to be understood concerning our tachyon BV states: • The construction of our tachyon BV states (3.13) is not a unique one. In particular, the division of iQu 0 into u 1A and u 1B and that of Qu 1A/B into u 2A and u 2B (see (3.16)) have much arbitrariness which is not reduced to a linear recombination among the two states. We have to confirm that the (non-)existence of physical tachyon fluctuation does not depend on the choice of the tachyon BV states so long as they satisfy the EOM conditions. (Or we have to establish a criterion for selecting a particular set of the BV states besides the EOM conditions.) In this paper, we introduced one parameter ξ representing an arbitrariness of the tachyon BV states (recall (3.13)). For the 2-brane solution and for (m, n) = (1, 1) and v 1/R satisfying (4.36) from the EOM conditions, we found in Sec. 4.3 that the matrix ω ij (k) is totally independent of the parameter ξ, implying that a physical tachyon fluctuation exists for any ξ. For the tachyon vacuum solution and for (m, n) = (0, 0), the results for ω 0,3 given in (5.4) and (5.7) are independent of ξ. Though we have to evaluate other ω ij for confirming the non-degeneracy of the 2 × 2 part Ω (3.27) with det Ω = ω 0,3 ̟, the fact that ω 0,3 (k) = 0 supports that the present set of the tachyon BV states is an unphysical one for any ξ.
Besides the analysis presented in Secs. 4 and 5, we carried out the analysis also for the BV states (3.13) using another choice of u i given by (3.18). The results for this BV states are mostly the same as those for the BV states using u i of (3.14). First, for the 2-brane solution, the EOM conditions are all satisfied for (m, n) = (1, 1) and v 1/R satisfying (4.36), and we obtain, in the particular case of ξ = 1, where ω 1A,2A is given by the following W with f defined by (4.42). This result should be compared with (4.45) for the choice (3.14) of u i adopted in Sec. 4. Eq. (6.1) implies that ω ij is degenerate and the tachyon is physical. However, the fact that ω 1A,2A = −1 at k 2 = 1 implies that the the tachyon field kinetic term given by (4.49) has the wrong sign, namely, that the physical tachyon is a negative norm one. Of course, we have to resolve the hermiticity problem before taking this problem seriously. Secondly, for the tachyon vacuum solution and for (m, n) = (0, 0) and ξ = 1, we found that ω ij is non-degenerate and hence the fluctuation is an unphysical one. The main difference from the case of u i given by (3.14) is that ω ij are of O(1/ε); for example, ω 0,3 = −(π/2) 2(1−k 2 ) [1 + 6/ (π 3 (2k 2 − 1) ε)] for L = R = 1.
These two results, one concerning the parameter ξ and the other for another choice (3.18) of u i , may support the expectation that the (un)physicalness of the tachyon fluctuation is insensitive to the details of the choice of the BV states. In any case, we need a deeper understanding and general proof of this expectation.
• We have restricted our analysis of the kinetic term S 0 (3.12) only to the six tachyon BV states and ignored the presence of all other states. For this analysis to be truly justified, we have to show that the complete set of the BV states of fluctuation can be constructed by adding to our set of tachyon BV states its complementary set of BV states which are orthogonal (in the sense of ω ij = 0) to the former set.
• As (m, n) specifying the leading small K ε behavior of L(K ε ) and R(K ε ), we have considered only the two cases, (1, 1) and (0, 0). We should examine whether there are other allowed (m, n) passing the EOM tests, and if so, we must clarify the relationship among the BV bases with different (m, n).
Finally, we have to extend our analysis to more generic n-brane solutions (including the exotic one with n = −1), and also to fluctuations other than the tachyon mode.
c(z 1 )c(z 2 )c(z 3 ) ℓ = ℓ π 3 sin π ℓ (z 1 − z 2 ) sin π ℓ (z 1 − z 3 ) sin π ℓ (z 2 − z 3 ) . (A.7) Finally, the matter correlator is given by e ik·X(z,z) e ik ′ ·X(z ′ ,z ′ ) matt  In each region, we Laurent-expand the integrand with respect to ε by regarding the specified integration variables kept fixed. The integration ranges, [0, 1] for x and p, and [0, u/ε] for y and z, should be appropriately modified to avoid overlaps among the seven regions as given in (4.19)-(4.21) for x and y. Finally, the u-integration should be carried out in the range u > ε as given there.