Two different candidates for the nonperturbative formulation of string theory have been known—string field theory (SFT) and matrix model. It is commonly believed that they are just different descriptions of the underlying theory. Therefore, it is important to investigate the relationship between these formulations. However, a few examples have been known to reduce open SFT (OSFT) [1] to certain matrix models. First is OSFT for topological A or B model [2] which reduces to Chern-Simons matrix model [3] or ordinary Hermitian matrix model [4] respectively. Second is OSFT for (2,1) minimal string [5] which reduces to Kontsevich matrix model [6]. In addition, less direct examples for c = 1 [7], c = 0 [8], and critical [9] strings have been known. In those examples, each matrix model is obtained from different OSFT associated with particular boundary conformal field theory (BCFT). A systematic way to derive different matrix models from OSFT in fixed background has not yet known. Finding such method is important to study the background independence of OSFT.

In this paper, we present an example for such method. Our idea is simple: instead of varying BCFT, we start with a general setup in critical string, and approximate the string field to its first few components. An example we will study is the level truncation in the universal sector [10] of critical (D = 26) OSFT in which the string field is approximated at level n as (1)Ψ=ψ0+ψ1+ψ2+⋯+ψn. ¹

Given this approximation, the OSFT action [1] immediately reduces to a matrix action²

where M_n is a Hermitian matrix. The approximation is known to work well in the first few level and improves quickly as the level increases [11–15].

A possible reason that such method has not yet been examined is the lack of understanding of Chan-Paton factors in OSFT. Although Chan-Paton factors can be introduced to OSFT consistently, their origin has not yet been explained. Recently, Erler and Maccaferri proposed a new construction of classical solutions of OSFT in terms of the regularized boundary conditions changing (BCC) projectors [16]. Their construction covers a wide range of backgrounds including multiple D-branes, which is the main interest of present paper. Following their work, Kishimoto, Masuda, Takahashi, and Takemoto (KMTT) demonstrated that N Chan-Paton factors naturally arise from the decomposition of string field in terms of the regularized BCC projectors [17]. SFT expanded around the multiple D-branes solution was interpreted as a system of N + 1 D-branes. We will employ their formulation as our foundation.

This paper is organized as follows. Section 2 introduces KMTT formulation. In Section 3, we derive a one-matrix model in terms of the level truncation. In Section 4, we derive the partition function of the model both in finite and large N. Section 5 summarizes our results and further discussions are given.

Let us briefly review the KMTT formulation [17] of OSFT. They begin with Erler-Maccaferri solution for multiple D-branes [16]

where Ψ_T is the tachyon vacuum solution and Σ_a and Σ¯a are the regularized BCC projectors which obey

Here Q_T is the kinetic operator at the tachyon vacuum defined by

The index a in Σ_a or Σ¯a is a Chan-Paton factor which labels D-branes. The solution (Equation 3) carries degrees of freedom required for multiple D-branes. We further expect that non-Abelian gauge gauge symmetry is realized in OSFT expanded around (Equation 3). KMTT [17] introduced the decomposition of the string field which realizes this:

where χ, χ_a, χ¯a and χ_ab are component string fields. By expanding OSFT action around Ψ₀ according to Equation (6), they derived the matrix action

where Q_T0 is the kinetic operator defined by Q_T0Ψ = Q_BΨ + Ψ_TΨ. The trace in (Equation 8) runs over indices of ϕ_ab, and identical indices in Equation (10) are to be summed over³. KMTT [17] claimed that (Equation 7) describes N + 1 D-branes rather than N D-branes. In addition to the N unstable D-branes, there is a “D-brane” at the tachyon vacuum described by χ. According to their interpretation, ϕ_ab connects two unstable D-branes a and b while χ_a connects D-brane a with the D-brane at the tachyon vacuum. χ represents fluctuation on the D-brane at the tachyon vacuum.

A remarkable feature of the action (Equation 7) is the presence of the “vector” sector described by χ_a and χ¯a, which has not been found in literature. The action S₃ is quadratic for the vectors therefore can be integrated out in the path integral. With assuming suitable gauge fixing, we perform path integral for χ_a and χ¯a and obtain a determinant factor

in the partition function. In the following sections, we will evaluate this determinant by truncating the string field rather than imposing conventional gauge condition such as linear gauge [18].

Although the string fields presented in previous section carry Chan-Paton factors, they also depend on infinitely many labels which distinguish each state in a BCFT. In order to make our analysis tractable, we introduce an approximation in which dynamical variables reduce to matrices. Let us consider the string field in the universal sector⁴ truncated at level n, where the level is defined by eigenvalue of the kinetic operator of OSFT. Denoting the level k base element of the string field ψ_k, we truncate each component in Equation (6) as

where Mab(k) is a Hermite matrix, ξa(k) is a complex vector, ξ¯a(k) is its complex conjugate, and t^(k) is a real number. Note that these component fields do not depend on any other variables. As a concrete example of such truncation, we choose the expansion examined in Zeze [19] for dressed B₀ gauge [20]. In this case, the level k base is given by

where K, B, c are the elements of the KBc algebra [21]. We would like to study level 1 truncation in which string fields are given by

A matrix action obtained by this truncation is rather complex since it includes both level 0 and 1 fields. However, it is possible to reduce degrees of freedom further. It is know that level 1 fields cannot have cubic terms since [19]

Thus, level 1 fields are quadratic in the action and can be integrated out using equations of motion. With the help of the explicit values of products of the basis ψ₀ and ψ₁ evaluated in Zeze [19], a nontrivial solution of equations of motion for Mab(1),ξa(1),ξ¯a(1), and t⁽¹⁾ turns out to be

By this assignment, four string fields in Equation (12) become proportional to Erler-Schnabl solution [20]:

Thus, the four fields⁵ can be written as

Although only few fields are included in the approximation, we expect that it well captures the essence of the dynamics of D-branes, as is the case in the conventional level truncation analysis. An evidence that supports our expectation is that the truncation interpolates between two analytic solutions: one is the Erler-Maccaferri's multiple D-branes denoted as (Mab,ξa,ξ¯a,t)=(0,0,0,0) while the other is the perturbative vacuum, also denoted as (δ_ab, 0, 0, −1).

Let us derive the truncated action. The requirement that Ψ_T reproduces the correct value of the D-brane tension is represented by

Applying these to Equation (7), we find

where we have rescaled the open string coupling as

For later convenience, we shift M to M + 1 and omit the prime in g′. Then, ignoring constant shift, we obtain an action

where W(x)=12x2+13x3. In this way, the truncation (Equation 12) reduces OSFT action to a cubic matrix action coupled with a scalar and complex vectors⁶. The partition function for Equation (25) can be obtained by employing the standard technique of matrix model⁷. As readily found in Equation (25), the action is invariant under U(N) transformation M→UMU†, ξ→Uξ, ξ¯→ξ¯U†, where U denotes an U(N) matrix. This is a residual gauge symmetry of KMTT action [17]. This U(N) symmetry can be fixed by diagonalizing M_ab to its eigenvalues λ_a with inserting Vandermonde determinant (λa−λb)2 in the partition function. After performing Gaussian integral for ξ, we obtain a partition function

where

This can describes a system of N + 1 particles moving in the potential W. An eigenvalue λ_a feels repulsive forces from other eigenvalues through the last term of Equation (27). Also, it is attracted toward −t − 1 due to the second term in Equation (27).

It is interesting to compare our result with the matrix formulation of c < 1 noncritical string theories. It has been recognized that Hermitian one-matrix model serves nonperturbative definition of (2, 2k + 1) minimal string theory. As an example, let us consider the action studied in Kutasov et al. [23]:

where ψ and ψ¯ are fermionic vectors. They can be integrated out so that insert a factor

in the matrix integral. In double scaling limit, this determinant is identified with a FZZT brane [24, 25], and the fermionic vectors are identified with fermionic strings between a FZZT brane and a stuck of N ZZ branes [26]. In contrast, our action (Equation 25) contains bosonic vectors ξ and ξ¯ rather than fermionic ones. Integration with respect to them yields a factor

in the partition function. Comparison between Equation (29) and Equation (30) naturally identifies the determinant Equation (30) as a ghost FZZT brane [27] which cancels the effect of a FZZT brane⁸. Unfortunately, corresponding observable in minimal string theory is not yet identified.

In this section, we will evaluate the effective potential for t in terms of saddle point equations for eigenvalues. In large N limit of t'Hooft expansion, saddle point configurations are leading contributions to the matrix integral. Even for finite N, saddle point configurations also offer a good approximation to the matrix integral when g is small. In either case, saddle point equations are obtained from the variation of Equation (27):

In this paper, we proposed a systematic method to derive matrix models from level truncated OSFT. Obtained matrix model contains U(N) vectors and a scalar in addition to Hermite matrix. We have evaluated the effective potential of the scalar both for finite and large N. Increase of the potential height was observed at small coupling. In Section 3, we have interpreted our model as a system of ZZ branes and a ghost FZZT brane.

We would like to discuss further issues to be explored. First, we would like to present an alternative but rather heuristic interpretation of our result. Let us go back to the inverse of the determinant :

The basic idea is that this quantity can be regarded as a propagator with ϕ and χ insertions. Recall that this factor is obtained by an integrating out χ_a and χ¯a which connect a D-brane with the tachyon vacuum where the world-sheet boundary disappears. Therefore, it is natural to think that Equation (70) amounts to a disk amplitude with single world-sheet boundary of a D-brane. Schematically, such disk amplitude can be written

As is well known, small t limit of such amplitude corresponds to closed string propagation [30]. Therefore, the inverse determinant (Equation 70) encodes gravitational force between N D-branes and the tachyon vacuum⁹.

Second issue is abut the level truncation. We have seen that the approximated OSFT action at first few levels yields a one-matrix model which can be interpreted as c < 1 noncritical string theory [31]. It is interesting to improve the approximation by including higher level fields to obtain multi-matrix models. We speculate that that the improved matrix action continues to be dual to some closed string theory on more nontrivial background. Finally, we will recover original OSFT with infinitely many matrices in infinite level limit. We also expect that this matrix model describes a critical string theory in nontrivial background through AdS/CFT like duality [32]. Thus, the level truncated OSFT offers a way to describe closed strings in approximated geometry.

Last issue is the matrix description of OSFT based on the left-right splitting of open strings which have been examined in past [33–36]. Our model based on the KMTT decomposition looks quite differently from these models. However, as mentioned in the previous paragraph, our model will recover full OSFT in infinite level. Thus, the left-right splitting models and our model both describe same OSFT. We expect that all ingredients of KMTT, including Chan-Paton factors and regularized BCC projectors, are embedded in the left-right type matrix model in quite nontrivial manner.

Together with recent development which deals with different backgrounds as classical solutions [16], our result presents further evidence for SFT as a formulation of nonperturbative string theory. We hope that further developments in this direction will shed light on the landscape of string theory.

The author confirms being the sole contributor of this work and approved it for publication.