Open String Fields as Matrices

We show that the action expanded around Erler-Maccaferri's N D-brane solution describes the N+1 D-brane system where one D-brane disappears due to tachyon condensation. String fields on multi-branes can be regarded as block matrices of a string field on a single D-brane in the same way as matrix theories.


Introduction
Open string field theory has the possibility of revealing non-perturbative aspects of string theory.Recently, Erler and Maccaferri have proposed a method to construct classical solutions, which are expected to describe any open string background [1].This indeed implies that open string field theory is able to give a unified description of various D-branes regarded as nonperturbative objects of string theory.
Multi-brane solutions in Ref. [1] provide a correct vacuum energy and gauge invariant observables.Accordingly, in order to prove whether the theory describes a multi-brane system, it is necessary to clarify open and closed string spectra in the background of the solution.However, it is difficult to give a definite answer to this problem, because there are some subtleties concerning BRST cohomology in the background [1].
There is another question related to the degree of freedom of string fields in the background.We have one string field in the theory on a single D-brane.However, in the case that the multibrane solution provides the background of the N D-branes, the number of string fields increases to N 2 around the solution.Intuitively, N 2 fluctuation fields in the multi-brane background seem to be introduced as redundant degrees of freedom.Here, it is natural to ask how to generate N 2 string fields or Chan-Paton factors from one string field.
On the other hand, it is well-known that matrix theories are able to describe various Dbranes [2,3].In matrix theories, D-branes are created by classical solutions as block diagonal matrices.After expanding a matrix around the solution, block matrices can be understood as representing open strings connecting each D-brane.Here, it should be noted that there are similarities between the matrix and the open string field: the matrix is deeply tied to the open string degree of freedom and an open string field is interpreted as a matrix in which the left and right indices correspond to the left and right half-strings [4].Then, it seems plausible that N 2 string fields on N D-branes are embedded like block matrices in a string field on a D-brane.
The purpose of this paper is to clarify the origin of the N 2 string fields in the background of an N D-brane solution.We will show that the theory expanded around the solution is regarded as an open string field theory on N + 1 D-branes, but in which a D-brane vanishes as a result of tachyon condensation.Then, the N 2 string fields will be given as block matrices in a string field as an infinite-dimensional matrix.Consequently, we can expect that the N D-brane solution correctly reproduces the open and closed string spectra in the N D-brane background.
The paper is organized as follows.In Sect.2, after a brief explanation of multi-brane solutions by Erler-Maccaferri [1], we will introduce projection operators acting on a space of string fields.Then, we will analyze a string field theory expanded around the N D-brane solution in terms of the projectors.In Sect.3, we will give concluding remarks.
2 Open string field theory around multi-brane solutions From the action, the equation of motion is given by To construct multi-brane solutions for N D-branes, Erler and Maccaferri introduced N pairs of regularized boundary-conditions-changing operators, Σ a and Σa (a = 1, and where Q T is a modified BRST operator on the tachyon vacuum.From (2.5), we find that where ψ T denotes the tachyon vacuum solution of (2.2).Here, we only assume that Σ a and Σa satisfy Eqs.(2.4) and (2.5) (or equivalently (2.6)) for a tachyon vacuum solution ψ T , regardless of wedge-based [5,6] or identity-based [7,8] solutions.Using ψ T , Σ a and Σa , Erler-Maccaferri provided a multi-brane solution as [1] Ψ We can calculate the action for Ψ 0 with the help of (2.4) and (2.5): Then, the solution Ψ 0 provides a correct vacuum energy for N D-branes.Expanding the string field around the solution as Ψ = Ψ 0 + ψ, we can obtain the action for the fluctuation ψ: where the operator Q Ψ 0 denotes the shifted BRST operator by the solution Ψ 0 .
1 Σ a and Σa are constructed by boundary-condition-changing (bcc) operators, σ a and σa , satisfying the operator product expantion: σa (z ′ )σ b (z) → δ ab (z ′ → z).In the Minkowski background, a zero momentum condition for the bcc operators is not necessarily required.So, the simplest bcc operators are given as where k µ a satisfy k 2 a = 0 and k a • k b < 0 (a = b).For example, we can take

Projectors
To clarify the physical interpretation of S[ψ; Q Ψ 0 ], we introduce N projection states as follows: where the same indices a are not summed.Here we have to notice that, as pointed out in Ref. [1], Σa should be multiplied to Σ a from the left and so these projectors should be dealt with carefully.More precisely, we define the projections for arbitrary string fields A and B as follows: From (2.4) and (2.11), we can easily find that This is a sufficient definition of the projectors for later calculation.But it suggests that we need to insert some infinitesimal worldsheet to separate Σ a and Σa .We will discuss this point further in the last section.
In addition to P a , we define the 0th projection as a complementary projector: where 1 denotes the identity string field.By definition, these N + 1 projections satisfy where the Greek indices are used for values 0, 1, • • • , N. From (2.5), it follows that Q T P α = 0 and then we have Moreover, we can find some relations among P α , Σ α and Σα : With the help of these projectors, the string field Ψ can be partitioned into (N +1)×(N +1) blocks: where Ψ αβ is defined as the (α, β) sector of Ψ, i.e., Ψ αβ ≡ P α ΨP β .According to Ref. [1], the second term in (2.7) is a solution to the equation of motion at the tachyon vacuum.From (2.16), the second term is represented as Accordingly, it turns out that the N D-brane solution at the tachyon vacuum is given as a block diagonal matrix.This is a similar result to the case of matrix theories [2,3].

Background described by the solution
Now, we consider the fluctuation ψ around the N D-brane solution.According to the previous subsection, ψ can be written by matrix representation: where φαβ = P α ψP β .φαβ represents a block matrix of ψ with infinite dimension.
Here, we consider change of variables of φαβ .φab can be rewritten as φab = P a φab P b = Σ a ( Σa φab Σ b ) Σb .
Similar to the equation Σb given in Ref. [1], by using (2.6) and (2.15), we have where the operator Q ψ 1 ψ 2 is defined as |A| Aψ 2 for two classical solutions ψ 1 and ψ 2 [1], and then Q T0 ≡ Q ψ T 0 and Q 0T ≡ Q 0 ψ T .Using these relations, we can obtain a matrix representation of Q Ψ 0 ψ: Consequently, the action expanded around Ψ 0 can be rewritten as where each action is given by and This system should be physically equivalent to the N D-brane system because Q T and Q T0 have trivial cohomology2 and therefore this result is consistent with the expectation that the solution (2.7) is regarded as an N D-brane solution.
Let us consider an on-shell closed string coupling to an open string field.In the complex plane, a closed string vertex operator is given by V(z, z) = c(z)c(z)V matt (z, z), where V matt is a vertex operator with the conformal dimension (1, 1) in the matter sector.We can give a BRST invariant state using V as where the point z = i corresponds to the midpoint of an open string.Since the vertex is inserted at the midpoint, the state V commutes with any string field A: V A = AV .For the open string field Ψ, an interaction term with the closed string vertex is given as a gauge invariant overlap [11]: In the background of the N D-brane solution, using (2.4) and (2.16), we can easily find couplings of the fluctuation fields to the closed string as This correctly provides a closed string interaction to open strings on the N + 1 D-branes.Next, we consider the correspondence between gauge symmetries in the original action (2.1) and the expanded action (2.26).The original gauge transformation is given by ( Since Ψ = Ψ 0 + ψ, the gauge transformation for ψ is given by where we note that Λ is the same parameter as in (2.32).Here, we decompose Λ into Λαβ = P α ΛP β by the projectors.Then, changing variables as Λab = Σ a Λ ab Σb , Λ0a = λ a Σa , Λa0 = Σ a λa , (2.34) and writing Λ00 = λ, we find that Σ a (δ Λ χa )P 0 + P 0 (δ Λ χ)P 0 , (2.35) where the gauge transformations for the components are given as (2.36)

Concluding remarks
We have shown that the theory expanded around the N D-brane solution given by Erler-Maccaferri describes an N + 1 D-brane system with a vanishing D-brane due to the tachyon condensation.By projectors made of regularized bcc operators, an open string field in the original theory is divided into multi-string fields with matrix indices.Then, these indices can be regarded as Chan-Paton factors in the N D-brane background.We have found that N 2 string fields on N D-branes are embedded in a string field as block matrices.Similarly, gauge transformation parameters in the expanded theory are represented as block elements of a gauge parameter string field in the original theory.
From the matrix representation (2.19), the string fields φαβ are mutually independent variables and then the degrees of freedom of φαβ are equivalent to those of the string field ψ.Then, it is natural to expect that the path integral measure of the fluctuation ψ is given by the product of measures of φαβ .As seen in the previous section, we can rewrite φαβ as φ ab , χ, χ a , and χa by linear transformations.Therefore, the measure of ψ is expressed by the measures of the string fields on the N + 1 D-branes: Hence, the matrix interpretation of open string fields ensures that the quantum measure for the N D-brane system is correctly derived from the classical solution in the string field theory.
Finally, we should comment on the multiplicative ordering of Σ a and Σa in the projectors.As in (2.11), we have defined the projectors such that Σ a does not operate on Σa , because bcc operators break associativity, as discussed in Ref. [1].To get a more definite result, we should separate these states by some worldsheet.This is a similar approach to that adopted in Ref. [12] to remedy the problem due to another nonassociativity.Accordingly, we need to regularize P a by inserting some worldsheet between Σ a and Σa .In the case that ψ T is given by the Erler-Schnabl solution, one possible choice for regularization is where ǫ is a positive infinitesimal parameter.It is noted that B/(1+K) is a homotopy operator for Q T and this construction is parallel to that of the regularized bcc operators from σ and σ [1].It can easily be seen that P a P b = δ ab P a and Q T P a = 0.In this regularization, the limit ǫ → 0 should be taken after calculating the correlation functions related to trace (or integration) of string fields.It should never be done in string fields; e.g., the state P a A keeps the parameter ǫ until correlation functions are calculated.Evidently, the state with the regularization parameter is regarded as a kind of distribution as in Ref. [13] and indeed it is outside the usual Fock space like the phantom term in Schnabl's tachyon vacuum solution [5].We hope that, in terms of the projectors, it will be possible to obtain a deeper understanding of a space of string fields, in particular, the topology in the space beyond the single Fock space in string field theories [14].

2. 1
Erler-Maccaferri's solution for N D-branes The action of bosonic cubic open string field theory is 27), φ represents a matrix (φ ab ) and the trace denotes the sum of the diagonal elements with indices a, b.Obviously, (2.27) represents the action for N D-branes; namely, φ ab is a string field of an open string attached on the ath and bth D-branes.Moreover, in the action (2.28), χ is a string field on a D-brane with tachyon condensation, and χ a and χa represent string fields of an open string attaching on a D-brane with tachyon condensation and on one of the N D-branes, on which φ ab also attach.Accordingly, the actions (2.27) and (2.28) describe the theory for N + 1 D-branes in which a D-brane vanishes due to tachyon condensation.