Analytic construction of multi-brane solutions in cubic string field theory for any brane number Open Access

Abstract

1. Introduction

The string field |$\Psi$| (as well as the elements |$(K,B,c)$| of the |${K\!Bc}$| algebra) is subject to the self-conjugateness condition |$\Psi^\ddagger=\Psi$| with |$\ddagger$| denoting the composition of the BPZ and the Hermitian conjugations. Therefore, |$U$| in Eq. (1.1) is chosen to be unitary in the sense that |$U^\ddagger=U^{-1}$|⁠. In fact, |$U$| in Eq. (1.1) is the most generic form of unitary |$U$|⁠, which is the sum of two terms, one containing |$Bc$| and the other without it.

In fact, |${\mathcal{N}}$| (1.8) is equal to the minus of the action of |${\Psi_{\varepsilon}}$|⁠, |$-S=-\int\left(\frac12{\Psi_{\varepsilon}}{Q_\textrm{B}}{\Psi_{\varepsilon}}+\frac13{\Psi_{\varepsilon}}^3\right)$|⁠, divided by the D25-brane tension |$1/(2\pi^2)$| only when the EOM test (1.9) vanishes, |${\mathcal{T}}=0$|⁠.²

The EOM test is also passed; namely, |${\mathcal{T}}=0$| in these two cases. It was shown that the origin of non-trivial |${\mathcal{N}}$| in these solutions is the singularity coming from the zero or pole of |$G(K)$| at |$K=0$| [5–8].

Namely, |${\mathcal{N}}$| is not an integer and the EOM test is not passed (⁠|${\mathcal{T}}\ne 0$|⁠) for the present type of solutions with |$N\ge 2$|⁠.

In Ref. [8], we proposed that the |$3$|-brane solution with |${\mathcal{N}}=2$| and |${\mathcal{T}}=0$| can be constructed in the form (1.2) by making use of the singularities both at |$K=0$| and |$K=\infty$|⁠, and taking, e.g., |$G(K)=(1+K)^2/K$|⁠. However, multi-brane solutions with larger |${\mathcal{N}}$||$(=3,4,5,\ldots)$| and |${\mathcal{T}}=0$| seem not to exist in the form of Eq. (1.2).

In this paper, we present an analytic expression of multi-brane solutions carrying any integer |${\mathcal{N}}$| and satisfying the EOM test |${\mathcal{T}}=0$|⁠. We start with the most generic form of unitary string field |$U$| consisting only of |$(K,B,c)$| and examine the pure-gauge configuration |$\Psi=U{Q_\textrm{B}} U^{-1}$|⁠, which manifestly satisfies the self-conjugateness condition. For considering the most generic unitary |$U$|⁠, we adopt a convenient notation for expressing a string field, which is given as the sum of products of |$(K,B,c)$|⁠. Then, by referring to the successful examples of the tachyon vacuum and the |$2$|-brane solutions given by Eqs. (1.1), (1.2), and (1.10), we make a natural ansatz on the functions of |$K$| defining |$U$|⁠. As a result, |$U$|⁠, which is expected to represent |$(N+1)$|-branes, is specified by |$(N+1)$| real parameters |$\left(\alpha_0,\alpha_1,\ldots,\alpha_N\right)$|⁠, among which only |$[N/2]$| are independent.⁵ We carry out the calculation of |${\mathcal{N}}$| (1.8) and |${\mathcal{T}}$| (1.9) for this type of solution, and find that these two quantities are again given in the form (1.11): the anomalous terms |$A_N$| and |$B_N$| are polynomials in |$(2\pi i)^2$| of order |$[N/2]$| and |$[N/2]-1$|⁠, respectively (⁠|$A_N$| starts with the |$(2\pi i)^2$| term). This is also the case for |$A_N$| and |$B_N$| of Eq. (1.12) for the solution (1.2). A different point in the present |$U$| is that the coefficients of the polynomials are not constants but are linear functions of |$\alpha_k$|⁠. Moreover, the coefficient |$f_n(\alpha_k)$| multiplying |$(2\pi i)^{2n}$| is common between |$A_N$| and |$(2\pi i)^2 B_N$| up to a constant factor. Therefore, both |$A_N=0$| and |$B_N=0$|⁠, namely, |${\mathcal{N}}=N$| and |${\mathcal{T}}=0$|⁠, are realized by choosing as |$\{\alpha_k\}$| the solution to |$f_n(\alpha_k)=0$| (⁠|$n=1,2,\ldots,[N/2]$|⁠). In fact, we find that |$\alpha_k$| and hence the solution |$\Psi=U{Q_\textrm{B}} U^{-1}$| are uniquely determined in this way for any integer values of |$N$| (⁠|$=2,3,4,5,\ldots$|⁠) that we have tested. For example, the |$3$|-brane solution is given by Eq. (5.5) with |$G=(1+K)/K$|⁠.

However, we have not succeeded in determining |$\alpha_k$| for a generic |$N$|⁠. The reason is that the expressions of |${\mathcal{N}}$| and |${\mathcal{T}}$| that we will obtain in this paper are too complicated to get |$f_n(\alpha_k)$| in a closed form for a generic |$N$|⁠. Even more, the fact that |$f_n(\alpha_k)$| are common between |$A_N$| and |$(2\pi i)^2B_N$| is merely an “experimental fact” obtained by the evaluation of |$A_N$| and |$B_N$| for various values of |$N$|⁠. However, there is no doubt that we can determine |$\alpha_k$| so that our solution can realize both |${\mathcal{N}}=N$| and |${\mathcal{T}}=0$| for any integer |$N$|⁠. The technical problem of giving |$f_n(\alpha_k)$| for a generic |$N$| will be resolved by mathematical sophistication.

Even if the solution |$\{\alpha_k\}$| to |$f_n(\alpha_k)=0$| is found for a generic integer |$N$|⁠, there still is an important problem in our construction of solutions. In this paper, as the EOM test, we consider only |${\mathcal{T}}$| (1.9), namely, the EOM test against the candidate solution |$\Psi$| itself. However, it is known that the 2-brane solution given by |$U$| of Eq. (1.1) with |$G=G_\textrm{2-brane}$| (1.10) does not pass the EOM test against the Fock states [7], and this property is inherited by the multi-brane solutions in this paper consisting solely of |$(K,B,c)$| (see Sect. 6). This problem of the failure of the EOM test against the Fock states might be resolved by some improvements of the solution, or by some consistent truncation of the space of fluctuations around multi-branes, which excludes the Fock states.

The rest of this paper is organized as follows. In Sect. 2, first introducing our convenient notation for expressing the sum of products of |$(K,B,c)$|⁠, we determine the form of the most generic unitary string field |$U$|⁠, and present our assumption on the form of |$U$| that is specified by |$\alpha_k$|⁠. Then, in Sects. 3 and 4, we obtain |${\mathcal{N}}$| and |${\mathcal{T}}$|⁠, respectively, as functions of |$\alpha_k$|⁠. In particular, we calculate |${\mathcal{N}}$| not directly but in a way where the role of the singularity at |$K=0$| as the origin of non-trivial |${\mathcal{N}}$| is manifest. In Sect. 5, we examine the conditions |${\mathcal{N}}=N$| and |${\mathcal{T}}=0$| on our solution to determine |$\alpha_k$| for various values of |$N$|⁠. We summarize the paper and discuss future problems in Sect. 6. In the appendices, we present technical details used in the text.

2. Assumptions on the solution

In this section, we first introduce our convenient notation for expressing string fields in the framework of the |${K\!Bc}$| algebra. Then, we obtain the form of the most generic unitary string field |$U$| for our candidate solution |$\Psi=U{Q_\textrm{B}} U^{-1}$| of the pure-gauge type. After these preparations, we restrict |$U$| to a particular form that is specified by real parameters |$(\alpha_0,\alpha_1,\ldots,\alpha_N)$|⁠.

2.1. Convenient notation

In Eq. (2.2) and in the following, we use notations such as |$(Bc)_{ab}\left(=B_ac_{ab}\right)$|⁠, |$(cB)_{ab}\left(=c_{ab}B_b\right)$|⁠, and |$(cK)_{ab}\,(=c_{ab}K_b)$|⁠. The advantage of the present notation is that we can put the |$K$| dependences at any place without any ambiguity.

2.2. The most generic unitary |$U$|

The derivation of these two conditions as well as those of some of the equations in this subsection are given in Appendix A.

2.3. Assumptions on |$\Gamma_a$| and |${\mathcal{F}}_{ab}$|

It is impossible to evaluate |${\mathcal{N}}$| and |${\mathcal{T}}$| for |$\Psi=U{Q_\textrm{B}} U^{-1}$| given by Eq. (2.11) without any assumptions on |$\Gamma_a$| and |${\mathcal{F}}_{ab}$|⁠. Here, on the basis of known facts, we would like to make plausible assumptions on the form of |$\Gamma_a$| and |${\mathcal{F}}_{ab}$|⁠, which is expected to realize |${\mathcal{N}}=N$| and |${\mathcal{T}}=0$| for each positive integer |$N$|⁠.

Secondly, by replacing |$G_a$| in Eq. (2.17) with |$G_a^N=\left((1+K_a)/K_a\right)^N$|⁠, we get |${\mathcal{N}}$| and |${\mathcal{T}}$| given by Eqs. (1.11) and (1.12). As we saw there, |${\mathcal{N}}$| for |$N\ge 2$| is a polynomial in |$(2\pi i)^2$| starting with the zeroth term |$N$|⁠. This seems to suggest that the replacement |$G_a\mapsto (G_a)^N$| is, though not perfect, fairly close to the final answer realizing |${\mathcal{N}}=N$|⁠.

Though |${\mathcal{F}}_{ab}$| (2.19) itself is not of a factorized form with respect to the |$K_a$| and |$K_b$| dependences, it should be suitably expressed as a sum of factorized terms by, e.g., Taylor expansion, for calculating correlators containing |${\mathcal{F}}_{ab}$|⁠.

In the rest of this paper, we shall first obtain |${\mathcal{N}}$| and |${\mathcal{T}}$| for the present solution as functions of |$\{\alpha_k\}$|⁠, and then examine whether there exists |$\{\alpha_k\}$| satisfying both |${\mathcal{N}}[\alpha_k]=N$| and |${\mathcal{T}}[\alpha_k]=0$| for each positive integer |$N$|⁠.

3. Expression of |${\mathcal{N}}[\alpha_k]$|

As preparation for examining |${\mathcal{N}}$| (1.8) for our candidate solution proposed above, we in this section obtain a calculable concrete expression of |${\mathcal{N}}[\alpha_k]$| for a given |$\{\alpha_k\}$|⁠.

3.1. |${\mathcal{N}}$| in terms of |$\Gamma_a$| and |${\mathcal{F}}_{ab}$|

The derivation of Eq. (3.1) is given in Appendix B.

3.2. |${\mathcal{N}}$| for a given |$\{\alpha_k\}$|

The formula (3.10) is valid for any |$U$| (2.7) given in terms of |$\left(\Gamma_a, {\mathcal{F}}_{ab}\right)$|⁠. In this subsection, we use Eqs. (3.10) and (3.5) to calculate |${\mathcal{N}}$| for our particular choice of |$\Gamma_a$| and |${\mathcal{F}}_{ab}$|⁠, Eqs. (2.18) and (2.19), specified by |$\{\alpha_k\}$|⁠. An important point in this calculation is that Eq. (3.10) is multiplied by |$\varepsilon$|⁠, which should be taken to |$+0$| in the end. This implies that we are allowed to keep only the most singular part of |$W_{123}$| (3.11) with respect to |$\varepsilon$|⁠.

Note that |$F_{P,Q}(z)$| and hence |$f_{P,Q}$| are anti-symmetric with respect to |$(P,Q)$|⁠.

The terms of non-trivial power of |$z^2$| are the “anomalous” part. We present the analysis of the anomalous part as well as that of |${\mathcal{T}}[\alpha_k]$| in Sect. 5 after obtaining a calculable expression of |${\mathcal{T}}[\alpha_k]$| in the next section.

4. Expression of |${\mathcal{T}}[\alpha_k]$|

As in the previous section, all the quantities in the rest of this section should be regarded as |${K_{\varepsilon}}$|-regularized ones, and we omit the corresponding subscript |$\varepsilon$|⁠. For example, |$T_{1234}$| means |$(T_{1234})_\varepsilon$|⁠.

Note that |$H_Q(z)$| is a polynomial in |$z^2$|⁠.

5. Solutions with |${\mathcal{N}}=N$| and |${\mathcal{T}}=0$|

As independent elements among |$\alpha_k$| (⁠|$k=0,1,\ldots, N$|⁠) subject to the constraints (2.20) and (2.21), we take the first |$[N/2]$| ones, |$(\alpha_0,\alpha_1,\ldots,\alpha_{[N/2]-1})$|⁠. Since our solution for |$N=1$| is unique, |$(\alpha_0,\alpha_1)=(1/2,1/2)$|⁠, and agrees with that of Ref. [9] satisfying |${\mathcal{N}}=1$| and |${\mathcal{T}}=0$|⁠, let us start with the |$N=2$| case. In the following, |$z^2$| implies |$(2\pi i)^2$|⁠.

5.1. |$\alpha_k$| for |$N=2,3,4,5$|

• |$N=2$| For |$N=2$|⁠, |${\mathcal{N}}$| and |${\mathcal{T}}$| are given by

  \begin{equation} {\mathcal{N}}=2+\alpha_0 z^2, \qquad {\mathcal{T}}=-12\alpha_0 . \end{equation}

  (5.2)
  Therefore, |${\mathcal{N}}=2$| and |${\mathcal{T}}=0$| are simultaneously realized by taking |$\alpha_0=0$|⁠:

  \begin{equation} \left(\alpha_0,\alpha_1,\alpha_2\right) =(0,1,0) . \label{eq:alpha_N=2} \end{equation}

  (5.3)
  In this case, |${\mathcal{F}}_{ab}$| (2.19) and |$E_{abc}$| (2.12) for the |$3$|-brane solution are

  \begin{equation} {\mathcal{F}}_{ab}=1-G_aG_b, \qquad E_{abc}=1-\frac{G_a+G_c}{G_b}+\frac{1}{G_b^2} . \label{eq:pF_E_for_3-brane} \end{equation}

  (5.4)
  Explicitly, the solution is given by

  \begin{equation} \Psi_\textrm{3-brane}=cK\!\left(1+\frac{1}{G^2}\right)\!Bc -GcK\frac{1}{G}Bc-cK\frac{1}{G}Bc\,G , \label{eq:Psi_3-brane} \end{equation}

  (5.5)

  with |$G(K)$| of Eq. (2.16).
• |$N=3$| For |$N=3$|⁠, we obtain

  \begin{equation} {\mathcal{N}}=3+3\left(\alpha_0+\frac16\right)z^2, \qquad {\mathcal{T}}=-36\left(\alpha_0+\frac16\right)\!. \end{equation}

  (5.6)
  |${\mathcal{N}}=3$| and |${\mathcal{T}}=0$| are simultaneously realized by taking |$\alpha_0=-1/6$|⁠:

  \begin{equation} \left(\alpha_0,\alpha_1,\alpha_2,\alpha_3\right) =\left(-\frac16,\frac23,\frac23,-\frac16\right)\!. \end{equation}

  (5.7)
• |$N=4$| For |$N=4$|⁠, we obtain

  \begin{align} {\mathcal{N}}&=4+\left(1+8\alpha_0+2\alpha_1\right)z^2+\frac12\alpha_0\,z^4 , \nonumber\\ {\mathcal{T}}&=-12\left(1+8\alpha_0+2\alpha_1\right) -\frac{10}{3}\alpha_0\,z^2 . \end{align}

  (5.8)
  Demanding |${\mathcal{N}}=4$| and |${\mathcal{T}}=0$|⁠, |$\alpha_k$| are uniquely determined by two equations, |$1+8\alpha_0+2\alpha_1=0$| and |$\alpha_0=0$|⁠, as

  \begin{equation} \left(\alpha_0,\alpha_1,\alpha_2,\alpha_3,\alpha_4\right) =\left(0,-\frac12,2,-\frac12,0\right)\!. \end{equation}

  (5.9)
• |$N=5$| For |$N=5$|⁠, we have

  \begin{align} {\mathcal{N}}&=5+\frac52\left(1+6\alpha_0+2\alpha_1\right)z^2 +\frac12\left(6\alpha_0+\alpha_1\right)z^4 , \nonumber\\ {\mathcal{T}}&=-30\left(1+6\alpha_0+2\alpha_1\right) -\frac{10}{3}\left(6\alpha_0+\alpha_1\right)z^2 , \end{align}

  (5.10)

  and the conditions |${\mathcal{N}}=5$| and |${\mathcal{T}}=0$| uniquely determine |$\alpha_k$| as

  \begin{equation} \left(\alpha_0,\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5\right) =\left(\frac16,-1,\frac43,\frac43,-1,\frac16\right)\!. \end{equation}

  (5.11)

5.2. |$\alpha_k$| for |$N\ge 6$|

In fact, |$\{\alpha_k\}$| given above for |$N=2,3,4,5$| have been determined by Eq. (5.13). For larger |$N$|⁠, the conditions (5.13) provide us with sufficient conditions to uniquely determine |$\{\alpha_k\}$|⁠, and the resultant |$\alpha_k$| is a rational number, as far as we have checked. Here, we present |${\mathcal{N}}$| and |${\mathcal{T}}$| and the solution to Eq. (5.13) in the cases |$N=6$|⁠, |$7$|⁠, and |$11$|⁠, as examples.

• |$N=6$| In this case, |${\mathcal{N}}$| and |${\mathcal{T}}$| are certainly of the form of Eq. (5.12):

  \begin{align} {\mathcal{N}}&=6+\left(4+27\alpha_0+12\alpha_1+3\alpha_2\right)z^2 +\frac12\left(21\alpha_0+6\alpha_1+\alpha_2\right)z^4 +\frac{1}{24}\alpha_0 z^6 , \nonumber\\ {\mathcal{T}}&=-12\left(4+27\alpha_0+12\alpha_1+3\alpha_2\right) -\frac{10}{3}\left(21\alpha_0+6\alpha_1+\alpha_2\right)z^2 -\frac{7}{30}\alpha_0 z^4 . \end{align}

  (5.14)
  The solution to Eq. (5.13) is given by:

  \begin{equation} \left(\alpha_0,\alpha_1,\alpha_2,\alpha_3, \alpha_4,\alpha_5,\alpha_6\right) =\left(0,\frac23,-4,\frac{23}{3},-4,\frac23,0\right)\!. \end{equation}

  (5.15)
• |$N=7$| In this case, |${\mathcal{N}}$| and |${\mathcal{T}}$| are again of the form of Eq. (5.12):

  \begin{align} {\mathcal{N}}&=7+7\left(1+6\alpha_0+3\alpha_1+\alpha_2\right)z^2 +\frac14\left(1+110\alpha_0+40\alpha_1+10\alpha_2\right)z^4 +\frac{1}{24}\left(8\alpha_0+\alpha_1\right)z^6 , \nonumber\\ {\mathcal{T}}&=-84\left(1+6\alpha_0+3\alpha_1+\alpha_2\right) -\frac53\left(1+110\alpha_0+40\alpha_1+10\alpha_2\right)z^2 -\frac{7}{30}\left(8\alpha_0+\alpha_1\right)z^4 . \end{align}

  (5.16)
  The solution to Eq. (5.13) is

  \begin{equation} \left(\alpha_0,\alpha_1,\alpha_2,\alpha_3, \alpha_4,\alpha_5,\alpha_6,\alpha_7\right) =\left(-\frac{3}{10},\frac{12}{5},-\frac{32}{5},\frac{24}{5}, \frac{24}{5},-\frac{32}{5},\frac{12}{5},-\frac{3}{10}\right)\!. \end{equation}

  (5.17)
• |$N=11$| In the case of |$N=11$|⁠, |${\mathcal{N}}$| and |${\mathcal{T}}$| are of the form (5.12) with |$f_n(\alpha_k)$| and |$t_n$| given by

  \begin{align} f_1&=11\left(\frac52+15\alpha_0+10\alpha_1+6\alpha_2 +3\alpha_3+\alpha_4\right)\!, \nonumber\\ f_2&=\frac34\left(9+510\alpha_0+290\alpha_1+150\alpha_2+66\alpha_3 +20\alpha_4\right)\!, \nonumber\\ f_3&=\frac{7}{24}\left(\frac{1}{14}+113\alpha_0 +47\alpha_1+17\alpha_2+5\alpha_3+\alpha_4\right)\!, \nonumber\\ f_4&=\frac{1}{720}\left(220\alpha_0+55\alpha_1+10\alpha_2 +\alpha_3\right)\!, \nonumber\\ f_5&=\frac{1}{40\,320}\left(12\alpha_0+\alpha_1\right)\!, \end{align}

  (5.18)

  and

  \begin{equation} \left(t_1,t_2,t_3,t_4,t_5\right) =\left(12,\frac{20}{3},\frac{28}{5},\frac{36}{7}\right)\!. \end{equation}

  (5.19)
  The solution to Eq. (5.13) is

  \begin{equation} \left(\alpha_0,\alpha_1,\alpha_2,\alpha_3,\alpha_4\right) =\left(-\frac{691}{210},\frac{1382}{35},-\frac{20\,528}{105}, \frac{10\,652}{21},-\frac{24\,384}{35}\right)\!. \end{equation}

  (5.20)

Summarizing this section, as far as we have checked for various positive integer |$N$|⁠, |${\mathcal{N}}$| and |${\mathcal{T}}$| take the form of Eq. (5.12) in terms of common linear functions |$f_n(\alpha_k)$|⁠, and the condition (5.13) uniquely determines |$\{\alpha_k\}$|⁠.¹⁰ Of course, there are many questions to be answered and subjects to be studied, which we shall discuss in the next section.

6. Summary and discussions

In this paper, we have presented an analytic expression of the multi-brane solutions of CSFT for arbitrary (positive integer) brane numbers. We started with the most generic unitary and real string field |$U$| (2.7) with |$\Gamma_a$| and |${\mathcal{F}}_{ab}$| satisfying Eqs. (2.8) and (2.9), and considered as a candidate solution the pure-gauge string field |$U{Q_\textrm{B}} U^{-1}$|⁠. As |$\Gamma_a$| and |${\mathcal{F}}_{ab}$| for multi-brane solutions, we adopted the ansatz of Eqs. (2.18) and (2.19) using |$G(K)$| with a simple pole at |$K=0$|⁠. For the |$(N+1)$|-brane solution, we in this paper demanded the following: First, the energy density of the solution calculated from the action should be that of the |$(N+1)$|-brane. Concretely, |${\mathcal{N}}$| (1.8) should be equal to the integer |$N$|⁠. Second, the EOM test against the solution itself given by |${\mathcal{T}}$| (1.9) should vanish. In the previous constructions of multi-brane solutions based on the singularity at |$K=0$|⁠, these two conditions were hard to be realized in the cases of |$N\ge 2$|⁠. In the present construction, our solution is specified by real parameters |$\alpha_k$| subject to Eqs. (2.20) and (2.21), and the problem is whether there exists |$\{\alpha_k\}$| that realizes |${\mathcal{N}}=N$| and |${\mathcal{T}}=0$|⁠. We calculated |${\mathcal{N}}[\alpha_k]$| and |${\mathcal{T}}[\alpha_k]$| in the |${K_{\varepsilon}}$|-regularization to find that there indeed exists |$\{\alpha_k\}$| satisfying the two conditions for any |$N=2,3,4,5,\ldots$| as far as we have tested. For |$N\ge 6$|⁠, the two conditions, |${\mathcal{N}}=N$| and |${\mathcal{T}}=0$|⁠, cannot uniquely fix |$\alpha_k$|⁠, and we proposed to demand stronger conditions (5.13) on |$\alpha_k$|⁠, which give a sufficient number of equations to uniquely determine |$\alpha_k$| as rational numbers.

Here, we add a remark for preventing a possible misunderstanding of the reader about our construction of solutions. One might think that our construction is almost trivial and meaningless since we are imposing only the two conditions (5.1) on the solutions, and this is always possible if the candidate solution has enough parameters (⁠|$\alpha_k$| in our case). However, we should recall that our candidate solution is “almost a solution” since it is of the pure-gauge form |$\Psi=U{Q_\textrm{B}} U^{-1}$|⁠, which automatically satisfies the EOM if there is no subtlety at |$K=0$|⁠. The non-integer nature of |${\mathcal{N}}$| and, possibly, the failure of the EOM test against itself, for a generic |$\{\alpha_k\}$| would be manifestations of the non-regularity of |$U$| at |$K=0$|⁠, as we explained in the introduction. The two conditions that we impose should be regarded as conditions necessary for making the pure-gauge configuration a more regular one.

We have certainly succeeded in constructing |$(N+1)$|-brane solutions satisfying the two conditions for |$N=2, 3, 4, 5,\ldots$|⁠. However, our analysis in this paper is still at an “experimental” level. Namely, we have confirmed the existence of the “natural” choice of |$\{\alpha_k\}$| determined by Eq. (5.13) only for sample values of |$N$|⁠. Although there is no doubt that such |$\{\alpha_k\}$| giving a desired multi-brane solution exists for any integer |$N$|⁠, we should present a general proof for our expectation. For this, we have to show that the expressions of |${\mathcal{N}}$| and |${\mathcal{T}}$| given in Eq. (5.12) in terms of common functions |$f_n(\alpha_k)$| are valid for any |$N$|⁠. It is of course desirable that the solution |$\alpha_k$| to Eq. (5.13) is explicitly given for a generic |$N$|⁠.

Even if these technical problems are resolved, there still remain important questions on our construction of multi-brane solutions:

• What is the meaning of the stronger conditions (5.13) on |$\alpha_k$|? Possibly, these conditions could be derived by considering other natural requirements on the solution, e.g., the requirement that the energy density of the solution evaluated from the gravitational coupling [10–13] be equal to that of the |$(N+1)$|-brane. Besides, since the number of conditions (5.13) depends on |$N$| (and is equal to |$[N/2]$|⁠), requirements related to the fluctuation modes on the solution might be the origins of the conditions.

• Is there any profound mathematical meaning in |${\mathcal{F}}_{ab}$| given by Eq. (2.19) in terms of |$\alpha_k$| satisfying the condition (5.13)? Recalling that |${\mathcal{N}}$| (1.8) for the present pure-gauge type solution |$\Psi=U{Q_\textrm{B}} U^{-1}$| has an analogy to the winding number |${\mathcal{W}}[g]$| (1.14) of the mapping |$g(x)$| from a three-manifold |$M$| to a Lie group, it would be interesting if the present construction realizing arbitrary integer |${\mathcal{N}}$| gives some hint for uncovering the meaning of |${\mathcal{N}}$| as a “winding number”, as we explained in the introduction.

• In this paper, as the EOM tests, we considered only that against the solution itself given by |${\mathcal{T}}$| (1.9). Let us define the EOM test against a generic string field |${\mathcal{O}}$| with |$N_\textrm{gh}=1$| by

  \begin{equation} {\mathcal{T}}[{\mathcal{O}}]=\int\!{\mathcal{O}}*\left({Q_\textrm{B}}{\Psi_{\varepsilon}}+{\Psi_{\varepsilon}}^2\right)\!. \label{eq:calT[calO]} \end{equation}

  (6.1)

  It is known that the |$N=1$| (2-brane) solution does not pass the EOM test against the Fock vacuum; |${\mathcal{T}}[(e^{-\frac{\pi}{4}K}c\,e^{-\frac{\pi}{4}K})_\varepsilon] =O(1/\varepsilon)\ne 0$| [7]. This property also persists in our |$N\ge 2$| solutions irrespective of the choice of |$\alpha_k$|⁠, as we have already mentioned in the introduction. Instead, our solutions pass the EOM test against the unitary transformed Fock vacuum: |${\mathcal{T}}[(Ue^{-\frac{\pi}{4}K}c\,e^{-\frac{\pi}{4}K}U^{-1})_\varepsilon]=0$|⁠. On the other hand, the tachyon vacuum solution (⁠|$N=-1$|⁠) passes all the EOM tests. For full understanding of the problem of the EOM test, it would be necessary to solve the problem of the fluctuation modes around the solution (see Ref. [14]).

Among the above three questions/problems, the last one is the most serious one from the viewpoint of constructing complete solutions. However, we expect that, even if the third problem remains unresolved, our finding in this paper gives a useful hint in considering the topological aspects of CSFT, as we stated in the introduction and in the above second question. We finish this paper by giving some comments concerning our solution:

• In the particular case of |$N=2$|⁠, our |$U$| with |$\alpha_k$| of Eq. (5.3) has the following manifestly unitary expression:

  \begin{equation} U=\exp\left(\frac12\bigl\{\left[{B},{c}\right],g(K)\bigr\}\right)\!, \end{equation}

  (6.2)

  where |$g(K)$| is defined by

  \begin{equation} e^{g(K)}=G(K)=\frac{1+K}{K} . \end{equation}

  (6.3)
  In relation to this, the following |$U$| is also unitary for any self-conjugate |$f(K)$|⁠:

  \begin{equation} U=\exp\bigl(f(K)\left[{B},{c}\right]f(K)\bigr) . \end{equation}

  (6.4)
  This |$U$| is rewritten into the standard form (2.7) and the corresponding |$\Gamma_a$| and |${\mathcal{F}}_{ab}$| are

  \begin{equation} \Gamma_a=e^{f(K_a)^2}, \qquad {\mathcal{F}}_{ab}=\frac{2\left(\ln\Gamma_a\ln\Gamma_b\right)^{1/2}}{ \ln\Gamma_a+\ln\Gamma_b}\left(1-\Gamma_a\,\Gamma_b\right)\!. \end{equation}

  (6.5)
  Note that this |${\mathcal{F}}_{ab}$| is equal to |${\mathcal{F}}_{ab}=1-\Gamma_a\Gamma_b$| in Eq. (5.4) for |$N=2$| (recall that |$\Gamma=G$| when |$N=2$|⁠) multiplied by the front term consisting of |$\ln\Gamma$|⁠. However, we find that, due to the presence of the |$\ln\Gamma$| term in |${\mathcal{F}}_{ab}$|⁠, both |${\mathcal{N}}$| and |${\mathcal{T}}$| are divergent in the limit |$\varepsilon\to +0$|⁠. In fact, if we take |$\Gamma=G(K)=(1+K)/K$|⁠, |${\mathcal{N}}$| diverges as

  \begin{equation} {\mathcal{N}}=O\!\left(\frac{1}{\varepsilon^2\ln^2(1/\varepsilon)}\right)\!. \end{equation}

  (6.6)

  In this respect also, our |${\mathcal{F}}_{ab}$| given by Eq. (2.19) is a good choice.

• The product |$U^{(3)}=U^{(1)}U^{(2)}$| of two unitary |$U^{(1)}$| and |$U^{(2)}$| is of course unitary and is written in the form (2.7) with |$\Gamma_a$| and |${\mathcal{F}}_{ab}$| satisfying Eqs. (2.8) and (2.9). In fact, |$(\Gamma_a,{\mathcal{F}}_{ab})$| of |$U^{(3)}$| is given in terms of those of |$U^{(1)}$| and |$U^{(2)}$| by

  \begin{equation} \Gamma_a^{(3)}=\Gamma_a^{(1)}\Gamma_a^{(2)}, \qquad {\mathcal{F}}^{(3)}_{ab}={\mathcal{F}}^{(1)}_{ab}+\Gamma^{(1)}_a{\mathcal{F}}^{(2)}_{ab}\Gamma^{(1)}_b . \end{equation}

  (6.7)

  This relation implies that, even if |$(\Gamma_a^{(1,2)},{\mathcal{F}}_{ab}^{(1,2)})$| are of the form of Eqs. (2.18) and (2.19), |$(\Gamma_a^{(3)},{\mathcal{F}}_{ab}^{(3)})$| is no longer so and cannot realize integer |${\mathcal{N}}=N^{(3)}=N^{(1)}+N^{(2)}$| and |${\mathcal{T}}=0$| in general. We have already seen this phenomenon of the violation of the additivity of |${\mathcal{N}}$| in the case of |$N^{(1)}=N^{(2)}=1$| in Ref. [6].

• In this paper, we considered explicitly only |$(N+1)$|-brane solutions with positive integer |$N$|⁠. However, “ghost brane” solutions with |$N\le -2$| can also be constructed in the same manner.

Funding

Open Access funding: SCOAP|$^3$|⁠.

Appendix A. Calculations for Sect. 2.2

In this appendix, we present the derivations of some of the equations in Sect. 2.2, in particular, the conditions (2.8) and (2.9) for the unitarity of |$U$| (2.7). Though the calculations are straightforward, they may be helpful as examples of the convenient notation of this paper.

Appendix B. The formula (3.1)

Appendix C. Derivation of Eq. (3.28)

Appendix D. Proof of Eq. (3.30)

As we saw in Sect. 3.2, |${\mathcal{N}}$| for our solution is given as a polynomial in |$z^2=(2\pi i)^2$|⁠. In this appendix, we show Eq. (3.30); namely, that the |$z^0$| term of |${\mathcal{N}}$| is equal to |$N$|⁠.

Appendix E. Derivation of Eqs. (4.11)–(4.13)

Footnotes

References