Solitons in Open N=2 String Theory

The open N=2 string theory is defined on the four-dimensional space-time with the split signature (+,+,-,-). The string field theory action of the open N=2 string theory is described by the four-dimensional Wess-Zumino-Witten (WZW_4) model. Equation of motion of the WZW_4 model is the Yang equation which is equivalent to the anti-self-dual Yang-Mills equation. In this paper, we study soliton-type classical solutions of the WZW_4 model in the split signature by calculating the action density of the WZW_4 model. We find that the action density of the one-soliton solutions is localized on a three-dimensional hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently, some kind of three-branes in the open N=2 string theory. We also prove that in the asymptotic region of the space-time, the action density of the n-soliton solutions is a ``nonlinear superposition'' of n one-solitons. This suggests the existence of intersecting n three-branes in the N=2 strings. Finally we make a reduction to a (1+2)-dimensional real space-time to calculate energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed.


Introduction
The Anti-Self-Dual Yang-Mills equation in four dimensions has been of great interest in elementary particle physics and mathematics. In the Euclidean signature, it has quite important soliton solutions, instantons which are crucial to reveal non-perturbative aspects of quantum field theory. In the split signature (+, +, −, −), it has a close relationship to integrable systems. It is well known that by imposing appropriate reduction conditions for the gauge fields, the anti-self-dual Yang-Mills equations in the split signature can be reduced to various lower-dimensional integrable equations, such as the KdV equation, the Non-Linear Schrödinger equation, the Toda equations and so on [43,30]. The integrability of these equations is well understood in the geometrical framework of twistor theory [30]. Soliton solutions are mostly of codimension-one in the sense that the energy density of the one-soliton solutions is localized on a codimension-one hyperplane in the space-time (See e.g. [24,31]).
The anti-self-dual Yang-Mills equation is realized in string theories which are classified according to the number N of the world-sheet supersymmetry. Under the condition that the critical dimension of the target space is positive and that the string world-sheet theory has an appropriate conformal symmetry, the maximal number is found to be N=2 ( [13] §4.5 and references therein). In the case of the N=2 string theories, the condition of conformal anomaly cancellation determines the critical dimension to be four, and the Virasoro constraints forbid any excited physical states except for massless scalars [13]. Hence, the string field theory can be reduced to the conventional field theory. The worldsheet N=2 supersymmetry induces a complex structure on the four-dimensional target space and hence the Minkowski target space is forbidden [39]. This is the reason why non-trivial string field theories are realized only when the metric has the split signature. (In the Euclidean case, momentum of the massless scalar fields becomes identically zero.) Therefore the N=2 string theory is closely related to the Ward conjecture and integrable systems.
The space-time action of the open N=2 string theory is described by the four dimensional Wess-Zumino-Witten (WZW 4 ) model [39,28,21,22] (see also [29,32,33,40]). Equation of motion of this model is the Yang equation which is equivalent to the antiself-dual Yang-Mills equation [19]. Hence solutions of the Yang equation are classical solutions of the open string field theory action of the N=2 strings. It is surprising that the action of the string field theory is explicitly written down in terms of massless scalar fields only. Exact analysis of the classical solutions leads to exact analysis of classical aspects of the string field theory.
Recently a new type of soliton solutions of the Yang equation has been constructed by using the Darboux transformation [36,10] in four-dimensional flat spaces with all kinds of signatures, that is, the Euclidean, the Minkowski and the split signatures [15]. These soliton solutions have localized Yang-Mills action densities on three-dimensional hyperplanes, and hence can be interpreted as codimension-one solitons. Furthermore, asymptotic analysis has been also made in [16,19,20], which suggests the existence of intersecting three-dimensional branes. In the case of the split signature, these solutions are supposed to be relevant to the open N=2 string theory. Therefore, to analyze the solitonic behavior (including the interacting region) for the WZW 4 action is much more appropriate than the Yang-Mills action.
In this paper, we study the classical soliton solutions in the WZW 4 model. The WZW 4 action (2.15) consists of the non-linear sigma model (NLσM) term and the Wess-Zumino (WZ) term [7]. We calculate the action densities of the NLσM model term and the Wess-Zumino term for the soliton solutions. 4 For the one-soliton solutions, we find that the WZW 4 action density is localized on a three-dimensional hyperplane. This suggests that there would be a codimension-one solitonic object, or equivalently, some kind of three-brane in the open N=2 string theory. For the multi-soliton solutions, we clarify the asymptotic behavior and conclude that the n-soliton solution possesses n isolated and localized lumps of the action density, and can be interpreted as intersecting n soliton walls. More precisely, each lump of the action density is essentially the same as a onesoliton because it preserves its shape and "velocity" with a possible position shift (called the phase shift) of the peak in the scattering process. We evaluate the distribution of the NLσM term for the two-soliton solutions successfully and find that there is no singularity in the interacting region. This is consistent with the existence of smooth intersecting three-branes in the whole region. Finally, we make a reduction to a (1 + 2)-dimensional real space-time to calculate energy densities of the soliton solutions. In (2+2)-dimensions, the concept of energy is ambiguous because of the existence of two time directions. This is the reason why in this paper we discuss action density instead of the energy density. We compute the energy densities of the one-soliton and two-soliton solutions in the (1 + 2) dimensions to confirm that they are localized on the same hyperplanes as those the action densities are localized. This suggests that the locus where the action density is localized could be considered as existence of a physical object. This paper is organized as follows. In section 2, the WZW 4 model is introduced and our conventions are set up. In section 3, soliton solutions of the Yang equation are reviewed and some properties of the solutions, such as the flip symmetry, singularities and an asymptotic behavior are discussed. In section 4, the action density for the one-and two-soliton solutions is calculated. In section 5, asymptotic analysis of the n-soliton solution is given. In section 6, we reduce the WZW 4 model from (2 + 2)-dimensions to (1 + 2)-dimensions and calculate the Hamiltonian density for the one and two-soliton solutions. Section 7 is devoted to conclusion and discussion. Appendix A is a brief review of the quasideterminant. In Appendix B, a statement in footnote 8 is proved (See section 5). Appendix C is a proof of unitarity of the n-soliton solutions on the Euclidean space. Appendix D includes miscellaneous formulas and detailed calculations.

Four-Dimensional Wess-Zumino-Witten Model
In this section, we review the four-dimensional Wess-Zumino-Witten (WZW 4 ) model. In order to treat it in a unified way, we introduce a four-dimensional space with complex coordinates (z, z, w, w) and the flat metric: ds 2 = g mn dz m dz n = 2(dzd z − dwd w), m, n = 1, 2, 3, 4. The space C 4 can be reduced to the three kinds of real spaces by imposing suitable reality conditions on (z, z, w, w). For example, the Euclidean real space E is given by z = z, w = −w, and the Ultrahyperbolic real space U by (1) z, z, w, w ∈ R or (2) z = z, w = w, which are denoted respectively by U 1 and U 2 . Our choices are shown in terms of real coordinates x µ (µ = 1, 2, 3, 4) as follows:

4)
(2.5) In this paper, we mainly consider the case of the Ultrahyperbolic space U 1 . 5 Let M 4 be a four-dimensional flat space and σ be a map from M 4 to G = GL(N, C) or its subgroup. The action of the WZW 4 model consists of two parts as follows: S WZW 4 := S σ + S WZ , (2.6) where M 5 := M 4 ×[0, 1] and σ(z, z, w, w, t), t ∈ [0, 1] is a homotopy such that σ(z, z, w, w, 0) = Id and σ(z, z, w, w, 1) = σ(z, z, w, w), and ω is the Kähler two-form on M 4 given by The exterior derivatives are defined as follows: (2.10) The first part S σ is called the non-linear sigma model (NLσM) term and the second part is called the Wess-Zumino (WZ) term. In the Wess-Zumino term, we use an abbreviated notation: This is derived as follows. Let us consider an infinitesimal variation of the dynamical variable σ such that δσ| ∂M 4 = 0 and dδ = δd. Then, Note that (δσ)σ −1 is a g-valued zero-form while (dσ)σ −1 is a g-valued one-form, where g is the Lie algebra of G. The cyclic symmetry of trace implies Since dω = 0, we have The variation of the sigma model term is where we use ∂ ∂ + ∂∂ = 0 due to ∂ 2 = 0, ∂ 2 = 0 and d = ∂ + ∂. The first and second terms become a surface integration due to dω = 0 and the fact that: Therefore we get the final form of the total action variation and the equation of motion is obtained: Finally we rewrite the WZW 4 action without integration over M 5 . By the cyclic property of the trace we have (2.12) The Kähler two-form ω is closed and H 2 (M 4 , R) = 0 and hence there exists a one-form A on the flat space-time such that Note that A is not uniquely determined and has ambiguity with respect to the following degree of freedom: A → A + dκ, where κ is an arbitrary zero-form. The Wess-Zumino term is written as If there exists a homotopy such that σ(t = 0) = Id and σ(t = 1) = σ, the second term vanishes and we obtain From now on, we use this form of action. Since our soliton solutions allow such a homotopy as above, we can use (2.15) for computing the action density.

Component Representation of WZW 4 Action Density
Let us write down explicit representations of the WZW 4 action density (2.15) in the flat four-dimensional real spaces.
In terms of the local complex coordinates (2.1) ∼ (2.4), the NLσM action is described as follows: where ∂ m := g mn ∂ n and the metric is given by (2.1). This can be represented explicitly in terms of real coordinates on U, E: where the real space metrics are given in (2.3) and (2.5). The NLσM action density is read from the integrand as L σ : Similarly the Wess-Zumino action is described as follows: where θ m := (∂ m σ) σ −1 . Here we choose the potential one-form A as A = (i/4) (zd z − zdz − wd w + wdw). This can be reduced to the three kinds of real spaces. For example, in the Ultrahyperbolic space U 1 , it is where θ µ := (∂ µ σ) σ −1 . The Wess-Zumino action density L WZ can be read from the integrand.

Useful Formulas for G = GL(2, C)
In this subsection, we focus on the case of G = GL(2, C). Additionally we impose the condition ∂ m |σ| = 0 on σ because our soliton solutions σ obtained in section 3.2 satisfy this condition. Then the WZW action density becomes a concise determinant form as follows.
By Jacobi's Formula: Tr [(∂ m σ)σ −1 ] = ∂ m |σ|/|σ| = ∂ m log |σ|, we find that the condition: ∂ m |σ| = 0 is equivalent to the condition: Tr [(∂ m σ)σ −1 ] = 0 which can be expressed in terms of the matrix elements: (2.20) In this paper, we always take the following parametrization for the soliton solution σ: under the condition ∂ m |σ| = 0. Note that this reparametrization is not unique and there is a relation between the five variables: In this setting, the quadratic term (2.20) can be rewritten as Similarly, the cubic term is: By the permutation property of determinants (Cf: (2.25)), we have Therefore under the condition ∂ m |σ| = 0, the Wess-Zumino term can be further simplified as

Darboux Transformation and Soliton Solutions
In this section, we review the soliton solutions of the Yang equation which are constructed by applying the Darboux transformation [36,10].

Darboux Transformation for the Yang Equation
Let us assume that G = GL(N, C) in this subsection. The Yang equation (2.11) can be rewritten as the following differential equation: There exists a Lax representation of (3.1) given by the following linear system [36]: The spectral parameter ζ here must be generalized to an N ×N constant matrix otherwise the Darboux transformation would be a trivial transformation. This is a key point to define a nontrivial Darboux transformation as we will see later.
It is not hard to verify that the compatibility condition L(M(f )) − M(L(f )) = 0 implies the Yang equation (3.1). The existence of N-independent solutions of the linear system (3.2) is an assumption here, however we will show later that it actually exists for the soliton solution cases. Then, f can be rewritten as an N × N matrix which consists of the N-independent solutions as column vectors of length N.
The Darboux transformation is defined as an auto-Bäcklund transformation of the linear system (3.2). Firstly, we start with a solution σ of the Yang equation, and a solution f = f (ζ) of the linear system (3.2). Secondly, we prepare a special solution ψ(Λ) := f (Λ) for a fixed spectral parameter matrix ζ = Λ. Then the following Darboux transformation keeps the linear system (3.2) invariant in form, that is, As mentioned before, the transformation (3.3) becomes trivial if the spectral parameter is a scalar matrix where Λ commutes with ψ. The Darboux transformation maps the input data (σ, f (ζ), ψ(Λ)) to the output data (σ ′ , f ′ (ζ)) and therefore we get a new solution σ ′ of the Yang equation successfully. In the same way, these output data can be reused as the next input data (σ ′ , f ′ (ζ), ψ ′ (Λ ′ )) for the Darboux transformation. Here we define a special solution ψ ′ (Λ ′ ) := f ′ (Λ ′ ) by choosing a suitable spectral parameter matrix ζ = Λ ′ . Continuing this process, we get a series of input-output data : (σ, f, ψ) → (σ ′ , f ′ , ψ ′ ) → · · · . Therefore by applying n iterations of the Darboux transformation, we can get n exact solutions σ [j] of the Yang equation and express them in terms of the quasideterminants in a compact form [36,10]. For our purpose in this paper, it is sufficient to choose a trivial seed solution σ [1] = 1: where each ψ j = ψ j (Λ j ) (j = 1, 2, · · · , n) is a solution (N × N matrix) of the initial linear system (σ = 1): Hence the problem of solving the Yang equation reduces to solving the equation (3.6).
(The label [n+1] in the n-soliton solution is omitted in most part of this paper except for Appendix C.) In the main part of this paper, we do not explain details of the quasideterminant, but provide the definition and properties of the quasideterminant in Appendix A. The detailed computations can be found in Appendix B and C.

Soliton Solutions for G = GL(2, C)
From now on, we focus only on the soliton solutions for G = GL(2, C). An example of the multi-soliton solution is given by [16]: where the two kinds of spectral parameters λ j , µ j (j = 1, 2, · · · , n) are complex constants with the following mutual relationship on each real space: The powers L j of the exponential function are linear in the complex coordinates: The representations of L j in real coordinates are We use the notation ℓ (j) µ to simplify the coefficients of L j in the following sections, that is, L j := ℓ (j) µ x µ . We remark that the determinant of the n-soliton solution σ is constant [16,19]: which satisfies the requirement ∂ µ |σ| = 0. Therefore, we can apply the formulas (2.23) and (2.25) to the n-soliton solutions. Especially on the Ultrahyperbolic space U 1 , the n-soliton solution σ satisfies σσ † = σ † σ = |σ| [16,19] and hence after the scale transformation: σ → |σ| 1/2 σ, σ belongs to SU(2). On the Euclidean space E, σ can take values in U(2), which is proved in Appendix C. By the definition (A.5) of the quasideterminant, the n-soliton solution σ (Cf: (3.5) and (3.7)) can be represented in the form of (2.21): and e 1 := (1, 0) t , e 2 := (0, 1) t , 0 := (0, 0) t , and (A) k is the k-th row of a square matrix A. The data ∆ and ∆ jk are determinants of 2n × 2n matrices. We remark that ψ j can be decomposed into, for instance where X j := L j +L j , iΘ j := L j −L j . The second factor diag(e −L j , e −L j ) can be eliminated in the n-soliton solutions (3.5) due to the property of the quasideterminant (A.7). Hence the n-soliton solutions (3.5) depend only on X j and Θ j . The expansion coefficients for the real coordinates are denoted by: where Θ jk := Θ j − Θ k . Note that the flip of space-time coordinates ,−x 4 ) corresponds to the following flips of the new variables: Under this flip, we find the following symmetry (Cf: D.1): Here let us discuss the singularities of the solution σ. Under the decomposition (2.21), possible singularities correspond to zeros of ∆. For the one-soliton solution (Cf: (D.6)), ∆ = 2 cosh X 1 and hence there is no singularity. 6 For the two-soliton solution (Cf: (D.7)), we can evaluate the value of ∆ on the Ultrahyperbolic space U 1 as follows: where a, b, c are real constants defined in Table 1 of section 4.2. Therefore, the denominator is anywhere positive and σ is proved to be non-singular on U 1 . On the other hand, on the Euclidean space E, σ has singularities because it has zero locus due to the fact that cosh(X 1 ± X 2 ) ≥ 1, |cosΘ 12 | ≤ 1 and a, b have opposite signs (See Table 1 in section 4.2). However this problem can be solved successfully by choosing suitable initial data ψ, which will be discussed in section 5.
Finally we comment on an asymptotic behavior in the region that r 2 := (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 + (x 4 ) 2 is large enough in order to prove that the Wess-Zumino action density decays exponentially in the asymptotic region. We note that the n-soliton solution (3.5) is a meromorphic function of (ξ 1 , · · · , ξ n , η 1 , · · · , η n ) where ξ K := e X K , η K := e iΘ K . Let us discuss the absolute value of the Wess-Zumino action density. In fact, we will see in section 5 that the action density tends to zero in the asymptotic region. This implies that for any variable ξ K , the polynomial degree of the denominator is greater than that of the numerator. (η K is not essential because of |η K | = 1.) Let us consider a specific asymptotic direction where the most dominant terms are ξ i 1 1 · · · ξ in n in the numerator and ξ j 1 1 · · · ξ jn n in the denominator where i k ≤ j k . Then the action density behaves O(ξ i 1 −j 1 1 · · · ξ in−jn n ). At least for one K, i K < j K and hence this implies that it decays exponentially.
Let us take the two-soliton case for example. If we consider the asymptotic limit such that X 1 is finite and |X 2 | ≫ 1 (Cf: (4.15)), the most dominant factor is e ±X 2 and the denominator and the numerator have the same order of ξ 1 ≡ e ±X 2 . However, due to the identity (4.17), the most dominant term in the numerator vanishes and hence the Wess-Zumino action density is O(ξ −k 1 ), where k is some positive integer. Therefore we can conclude the Wess-Zumino action density decays exponentially.
Therefore on U 1 , the Wess-Zumino action converges for the one-and two-soliton solutions because it has no singularity and decays exponentially. For the n-soliton solution, this is an open problem which is discussed in section 7.

Evaluation of Action Density
In this section, we compute the action density of the SU(2) WZW 4 model for the oneand two-soliton solutions and find that the corresponding action densities are real-valued on each space. We also find that for the one-soliton solution, the NLσM action density is localized on a three-dimensional hyperplane and the Wess-Zumino action density identically vanishes. For the two-soliton solution, we complete the calculation of the NLσM term, and reach to a compact form. In particular, the two peaks of the action density are localized on nonparallel two three-dimensional hyperplanes. As for the Wess-Zumino term, we show that the action density is asymptotic to zero. Especially on the Ultrahyperbolic space U 1 , no singularity appears in the action densities for the two-soliton case as indicated in section 3.

One-Soliton Solutions
In this subsection, we compute the action densities of the SU(2) WZW 4 model for the one-soliton solutions.
To calculate the NLσM action density explicitly, we substitute the data of one-soliton (D.6) and (3.11) into (2.23) for m = n = µ and then obtain the following result: where d 11 is determined by (3.8) and (3.9) ∼ (3.10), for instance, Table 1). Hence d 11 and L σ are clearly real-valued. Note that the action density vanishes identically in the case of α 1 , β 1 , λ 1 ∈ R on U 1 and hence the result is trivial. For the nontrivial cases, the peak of the action density lies on the three-dimensional hyperplane described by the linear equation X 1 = 0 on each space.
The Wess-Zumino action density can be calculated by substituting the data of onesoliton (D.6) and (3.11) into (2.25) directly. Then we have These facts imply that and therefore the Wess-Zumino action density is identical to zero for the one-soliton case.
In fact, the identity (4.6) holds even when σ is not a classical solution of the WZW 4 model because the condition (3.6) is not used in our discussion. More explicitly, as long as the power L j of exponential function in ψ j is an arbitrary linear function of x µ , the Wess-Zumino action density vanishes identically. All the above results also hold in the case of E.

Two-Soliton Solutions
In this subsection, we calculate the action densities of SU(2) WZW 4 model explicitly for the two-soliton solutions. By the result of Appendix D.3 together with (3.8) and (3.9) ∼ (3.10), we get the following compact form of the NLσM action density for two-soliton solution : where a, b, c, d jk , e jk are defined in the following Table 1 for each space. (The difference between E and U 1 appears only in the coefficients like the one-soliton case.) Note that the coefficients in Table 1 also guarantee the NLσM action density to be real-valued on U 1 and E.
Next, let us explain why (4.7) can be interpreted as two intersecting one-solitons in the asymptotic region. Due to the solitonic property, individual one-solitons will regain all their features (wave shape, velocity, amplitude, etc.) outside the scattering region except for respective differences of an additional position shift. Theoretically, each one-soliton can be separated completely from the other one-soliton in the asymptotic region in which it dominates the asymptotic behavior mainly. Therefore, we can consider the following type of asymptotic limit: in which the first one-soliton, localized on the hyperplane X 1 = 0, dominates the asymptotic behavior mainly. Such asymptotics will be discussed more systematically in section In the asymptotic limit (4.8), the action density (4.7) is dominated by (4.9) Since sinhX 1 and coshX 1 are finite and sechX 2 → 0 and tanhX 2 → ±1 as X 2 → ±∞, we have (4.10) Now we conclude that where the position shift factors (or the phase shift factors) are The cases (3) and (4) are obtained by the same argument and we just skip the detail here. By the above analysis, we find that the NLσM action density (4.7) has two peaks which are localized on nonparallel two three-dimensional hyperplanes described by the linear equation X 1 ± δ 1 = 0 and X 2 ± δ 2 = 0. More general discussion for n-soliton case is mentioned in Appendix D.5.
As for other asymptotic regions which differ from the cases (1) ∼ (4), no solitonic effect contributes to the action density, that is, the action density is asymptotic to zero in these regions. This will be proved in section 5.
Let us proceed to calculate the Wess-Zumino action density for the two-soliton solution. By substituting (3.11), (D.7) and (3.14) into (2.25) for (m, n, p) = (µ, ν, ρ), we have Here each ingredient of B µνρ can be calculated in the same way as the previous section. For example, the result of the first determinant factor in (4.13) is The definition of the coefficients and the result of the remaining determinant factors can be found in Appendix D.4. Furthermore, we can also show that the Wess-Zumino action density is real-valued on U 1 and E. (Cf: Appendix D.4). By the same technique used in the previous section, we consider the asymptotic limit such that |X 2 | ≫ |X 1 | for finite X 1 , and find that where C µνρ := r ν . This is asymptotic to where the phase shift factor is δ 1 := (1/2)log(a/b). In fact, the coefficient C µνρ in (4.16) satisfies the following relation: Therefore the cubic term (4.13) identically vanishes in the asymptotic region, and the Wess-Zumino action density is asymptotic to zero for the two-soliton case: More general discussion for n-soliton case can be found in Appendix D.5. The Wess-Zumino action density for the two-soliton is a smooth function and nonsingular and hence bounded. Moreover, it decays to zero exponentially as mentioned in section 3.2. Therefore, we conjecture that the Wess-Zumino action S WZ would be zero exactly.

Asymptotic Analysis of n-Soliton Solutions
Due to the problem of singularity of two-soliton solution on the Euclidean space E, in this section, we consider a modified n-soliton solution and discuss the corresponding asymptotic behaviors in a systematic way. The modified n-soliton solution is where the spectral parameters (λ j , µ j ) are rewritten by (λ The slight difference between (5.1) and (3.7) is an additional constant factor ǫ taking values in {±1}. The case of ǫ = +1 coincides with (3.7). We will show that the nonsingular n-solitons can be constructed completely for all n ∈ N by suitable choices of the constant ǫ with respect to the Ultrahyperbolic space U 1 and the Euclidean space E.
First of all, we define two types of the asymptotic region for the n-soliton solutions. Let us consider the asymptotic region of the four-dimensional space where The asymptotic region is divided into 2 n regions by the n hyperplanes X j = 0 (j = 1, 2 · · · , n) depending on X j > 0 or X j < 0. In order to label these regions, it is convenient to introduce a new notation ε j ∈ {±1, 0}. Then the 2 n asymptotic regions can be denoted by R(ε 1 , · · · , ε n ) in which ε j = +1, and ε j = −1 correspond to the following cases (+) and (−) respectively: 7 Then we can unify the asymptotic regions as We will see that the Wess-Zumino action density vanishes in R.
On the other hand, there is the other type of the asymptotic region along the hyperplane X j which corresponds to the case of ε j = 0. To make the asymptotic region be four-dimensional, let us define the asymptotic region along X K as a tubular neighborhood of R(ε 1 , · · · , ε K = 0, · · · , ε n ) which is denoted by R K (ε 1 , · · · , ε n ). In this region, the value of X K is considered to be finite. We will see that the NLσM and the Wess-Zumino action densities coincide with a one-soliton configuration in R K .
The two type of asymptotic regions can be expressed in terms of the following sets: is large enough. X j are all positive or negative. (j = 1, · · · , n) e 1 , e 2 , e 3 are linearly independent vectors tangent to the hyperplane: X K = 0 a is a finite vector. For the asymptotic regions of type R, the behavior of (5.1) is dominated by all X j for |X j | is large enough. By (5.2), we find that σ is asymptotic to a constant matrix for each asymptotic region of the type R: The suffix (±) in C j corresponds to the signature of ε j . Therefore, the action densities L σ and L WZ identically vanish in the type R asymptotic regions.
On the other hand, since X K is kept to be finite for the type R K asymptotic regions, we have [16] This actually leads to the following one-soliton type solution [16]: (5.4) and the coefficients a K , b K , c K , d K can be expressed in terms of the spectral parameters as: In fact, we will see later that the coefficients a K , b K , c K , d K determine the position shift (known as the phase shift) of the one-soliton solution (5.4) in each asymptotic region of type R K . Furthermore, these coefficients also determine whether the singularities of nsolitons exist in the action density of the WZW 4 model. First of all, we can calculate the asymptotic form of the following µ-th component of the quadratic term by using (5.4).
(The summation is not taken over µ.) The result is where cschx := 1/ sinh x. Apparently, for a K d K /b K c k < 0, the singularities exist on the entire three-dimensional hyperplane X K + δ K = 0. Now let us find out the condition such that the NLσM action density is non-singular. For the Ultrahyperbolic space U 1 , the reality condition is λ j . By (5.5), we have Comparing it with (5.6), we can conclude that in the case of ǫ = +1, the NLσM action density of the n-soliton is definitely asymptotic to a non-singular one-soliton for the Ultrahyperbolic signature. This fact implies that for all n ∈ N, the n-soliton solution (3.7) gives a class of non-singular NLσM action densities for the Ultrahyperbolic signature. Similarly, the reality condition of the Euclidean space E : λ The ratio (5.8) is positive in the following two cases: (1) n is odd and ǫ = +1 or (2) n is even and ǫ = −1. Then the NLσM action densities are non-singular. On the other hand, the ratio (5.8) is negative in the following two cases: (3) n is even and ǫ = +1 9 or (4) n is odd and ǫ = −1. 10 Then the NLσM action densities are singular. It is quite interesting that in the Euclidean signature, singular and non-singular solutions are generated alternately by the Darboux transformations with respect to initial solutions ψ j for ǫ = ±1.
In summary, for all n ∈ N, non-singular NLσM action densities of the n-soliton can be constructed by taking ǫ = +1 for all n ∈ N (Cf: (5.7)) on the Ultrahyperbolic space U 1 , and by taking ǫ = +1 for all odd n and ǫ = −1 for all even n (Cf: (5.8) and the cases (1) and (2)) on the Euclidean space E. They would share the same asymptotic form in R K on each real space: where d KK is defined in Table 1 (Cf : U 1 and E) and the phase shift factor is Since the result of (5.9) is valid for arbitrary K in {1, 2, . . . , n}, we can regard the behavior of non-singular n-soliton as a "non-linear superposition" of n non-singular and mutually nonparallel one-solitons on each real space in which each one-soliton in the asymptotic region R K maintains its form invariant but is shifted by δ K , called the phase shift factor which results from a non-linear effect.
In conclusion, in the asymptotic region, the n-soliton solution possesses n isolated and localized lumps of the NLσM action density, and we can interpret it as n intersecting soliton walls. The phase shift factors are also obtained explicitly. The scattering process of the n-soliton solution is quite similar to that of the KP solitons [37,24,25]. On the other hand, the Wess-Zumino action density identically vanishes in the asymptotic region because in the asymptotic region R, the action density identically vanishes, and in the asymptotic region R K , the n-soliton solution is reduced to the one-soliton (5.4) whose Wess-Zumino action density is identically zero as proved in section 4.1.

Reduction to (1 + 2)-Dimensions
So far, we discuss the action density of the WZW 4 model for n-soliton solutions and find that it is localized on nonparallel n codimension-one hyperplanes in four dimensions. However, to understand better the physical meaning of our soliton solutions, it would be a good idea to calculate the energy density of the soliton solutions and compare with the action density. For this purpose, we assume the translation invariance in the x 2 direction. The WZW 4 model Lagrangian is reduced to the following one (Cf: (2.17) and (2.27)): Tr where we reset (t, x, y) := (x 1 , x 3 , x 4 ) and θ µ := (∂ µ σ)σ −1 (µ = t, x, y). The equation of motion is the Ward chiral model [44] or the space-time monopole equation [4,12] in (1 + 2) dimensions in the Yang form. The n-soliton solution of (6.1) is obtained by imposing the condition α j = λ j β j (Cf: (3.9)) on the powers L j of the n-soliton solution (3.7). Then, the powers of n-soliton solution is actually reduced to L j = (β j / √ 2) (λ 2 j + 1)t + (λ 2 j − 1)x + 2λ j y in the (1+2)-dimensional space-time. Let us consider three angular coordinates φ i (x) (i = 1, 2, 3) which parametrize SU(2) ≈ S 3 where σ(x) belongs to. Then the Hamiltonian density can be obtained by the Legendre transformation of the Lagrangian: Tr(θ t ) 2 + Tr(θ x ) 2 + Tr(θ y ) 2 , This Hamiltonian physically makes sense because it is positive definite due to the fact that θ µ is an anti-hermitian matrix. This is a conserved energy density by definition. Note that the contribution of the Wess-Zumino term to H tot vanishes identically: H WZ = 0, or equivalently, H tot = H σ . Let us calculate the energy density of the reduced soliton solution from the Hamiltonian density H tot . For the one-soliton solution, the Hamiltonian H tot is: This is in the same form as the reduced NLσM action density L σ up to an overall coefficients (Cf: (4.1)). Therefore, the peaks of H tot and L σ are localized on the same two-dimensional hyperplane X = 0 in the (1 + 2)-dimensional space-time. In this sense, L σ also can be interpreted as an analogue of the energy density in physical reality.
For the two-soliton solution, the Hamiltonian density for the two-soliton solution is calculated by using the result of Appendix (D.3), we have where a, b, c are the same coefficients defined in Table 1 and As for the NLσM term, the Hamiltonian density H tot for the two-soliton is also in the same form as the reduced NLσM action density L σ up to the differences of the coefficients d ij and e ij (Cf: (4.7)). Therefore, the two peaks of H tot are localized on the same hyperplanes X 1 ±δ 1 = 0 and X 2 ±δ 2 = 0 as those of L σ (Cf: (4.11)). The phase shift factors are also perfectly the same. There is no singularity as well. This result implies that there is no essential difference between H tot and L σ for describing the solitonic properties.
The peaks of the energy density of the two-soliton solutions is shown in Figure below.
On the other hand, as for the Wess-Zumino term, there is a mismatch that the Hamiltonian density H WZ is identical to zero, while we cannot confirm the action density L WZ is. The physical meaning of this mismatch should be clarified in the future work.

Conclusion and Discussion
In this paper, we calculated the action density of the WZW 4 model for the classical soliton solutions. We found that for the one-soliton solutions, the NLσM action density is localized on a three-dimensional hyper-plane and the Wess-Zumino action density identically vanishes. This suggests the existence of a three-brane in the open N=2 string theory. For the two-soliton solutions, the NLσM action density has a beautiful compact form which represents an intersecting two one-solitons. The Wess-Zumino action density does not vanish in the interaction region but does vanish in the asymptotic region. For the n-soliton solutions, we clarified asymptotic behavior and found that the NLσM action density describes "nonlinear superposition" of intersecting n one-solitons and the Wess-Zumino action density asymptotically vanishes. The nonlinear interaction gives rise to phase shifts which were evaluated explicitly. We also calculated the Hamiltonian (energy) density of the one and two-soliton solutions of the reduced WZW model in (1+2)-dimensions. We found that the energy density of the Wess-Zumino term identically vanishes, and the energy density of the NLσM term has the same profile as the action density for our soliton solutions. The peaks of the energy densities perfectly coincide with those of the action density including the phase shift factor.
We also discussed whether singularities exist for the n-soliton solutions. For the one and two-soliton solutions, we proved that there is no singularity. For the n-soliton solutions (n ≥ 3), it is unsolved, however, we can argue as follows. The existence of singularities in the solution is equivalent to the existence of zeros in the data ∆ which is a polynomial of e X j and e iΘ j . Because X j and Θ j are linear functions of the real coordinates, possible singularities would lie on the intersection of X j = C j and/or Θ j = D j where C j and D j are constants. These possibilities are mostly forbidden because there is no singularity of the action density of the n-soliton solutions in the asymptotic region where the intersection still exists. The intersection of just four hyperplanes defined by X j = C j or Θ j = D j (j = 1, 2, 3, 4) gives rise to an isolated singularity. This possibility might arise when the parameters in the solutions are appropriately tuned, which should be clarified in the future. The next step is to clarify roles and properties of the soliton solutions in the open N=2 string theory. At least we can see that they are not D-branes because the number of the solitons is not related to the rank of gauge group. It is worth studying the topological charge and mass of the solitons, and explicit calculation of infinite conserved densities [2,6,23,21] for the n-soliton solutions. It is also interesting to construct resonance solutions of the solitons which represent the three-brane reconnections, or in other words, annihilation and creation of the three-branes. Then a classification of the soliton solutions could be possible like the positive Grassmannian description of the KP solitons by Kodama and Williams [26]. The moduli space of the n-soliton solutions could be described in a geometrical framework. Extension of the model to noncommutative spaces would allow the presence of background B-fields in the open N=2 string theory [42,27,17,14]. The isolated singularities mentioned above might be resolved and new physical objects appear on the noncommutative spaces such as noncommutative U(1) instantons [35]. Sen's conjecture on the tachyon condensation (for a review see [38]) could be confirmed by the solution generating technique [18] in the context of the open N=2 string theory.
Furthermore, the WZW 4 model can be realized in the context of the twistor string theory [45]. Recently Bittleston and Skinner show that a meromorphic Chern-Simons theory on the twistor space in six dimensions has a double fibration structure which gives rise to the WZW 4 model by solving along fibers in one direction and the four-dimensional Chern-Simons theory by symmetry reduction in another direction [1]. These models are connected to each other and have a close relationship to integrable systems [3]. The KP equation has not yet obtained as a symmetry reduction of the anti-self-dual Yang-Mills equation so far, however, this six-dimensional Chern-Simons theory might give a "unified theory" of integrable systems including both the Sato theory [41] of the KP equation and the twistor descriptions of classical integrable systems. This might give a stringy viewpoint to various aspects of integrability and duality. The relation to mirror symmetry is also exciting [34].

Acknowledgments
MH thanks string group members at Nagoya university for useful comments at the String Journal Club on July 21, 2022. MH is also grateful to the YITP at Kyoto University, where he had fruitful discussions at the conference on Strings and Fields 2022 (YITP-W-22-09) on August 19, 2022. The work of HK and SCH is supported in part by Grant-in-Aid for Scientific Research (#18K03274). The work of SCH is supported by the Iwanami Fujukai Foundation.

A Brief Review of Quasideterminants
In this subsection, we excerpt some necessary pre-knowledge of quasideterminant mentioned in section 2 of the previous paper [16]. It is a brief review of the work of Gelfand and Retakh [9] (See also e.g. [8,19]).
The quasideterminant is defined for an n × n matrix X where matrix elements belong to a noncommutative ring. The quasideterminant is a noncommutative generalization of the matrix determinant in this sense, however, rather has a direct relation to the inverse matrix of X.
Let X = (x ij ) be an n × n invertible matrix over a noncommutative ring and Y = (y ij ) be the inverse matrix of X: XY = Y X = 1. The existence of Y is assumed. Then the (i, j)-th quasideterminant of X is defined as the inverse of an element of Y = X −1 : This has a convenient expression as follows: When the matrix elements belong to a commutative ring, e.g. C, the quasideterminant can be represented as a ratio of ordinary determinants by virtue of the Laplace formula on inverse matrices: where X ij is a matrix obtained from X by deleting i-th row and j-th column. In order to find another representation of the quasideterminant, let us consider the inverse matrix formula for the 2 × 2 block matrix divided as follows: where A is a square matrix, d is a single element and s := d − CA −1 B is called the Schur complement. The quantity s −1 is just the (n, n)-element of X −1 and hence the quasideterminant |X| nn is s. If we decompose X into a 2 × 2 block matrix where x ij corresponds to the single element d, the (i, j)-th quasideterminant can be expressed in the form of the Schur complement: By using this, explicit representations of the quasideterminants can be obtained iteratively. We note that the quasideterminant is well-defined in the case that each matrix element x ij in (A.2) take values in GL(N, C). (Then, X is an nN × nN matrix.) The following example of the N = 2 case can be expressed finally by the ratios of determinants due to (A.4) and (A.3): The final form corresponds to the parametrization (2.21) of σ for the soliton solution (3.5): which leads to the soliton data (3.12).
Here we summarize some properties and identities of the quasideterminant, which are relevant to discussions in this paper.
Proposition A.1 [8,9,19] Let A = (a ij ) be a square matrix of order n in (1), while in (2) and (3), appropriate partitions are made so that all matrices in quasideterminants are square.

(Proof)
Without loss of generality, we consider K = 1 case due to the fact that the quasideterminant |σ| ij does not depend on permutations of rows and columns in the matrix σ [9]. For j = 1, ψ j Λ m j can be decomposed into which is asymptotic to By the fact that and (A.7), the right common factors C (±) j of each column of σ can be omitted completely, and hence σ = which is called the asymptotic form of the n-soliton solution σ. By the derivative formula [11] of the quasideterminant (A j : j-th column of A, E j : j-th row of identity matrix I.) we have Now we can conclude that On the other hand, by the derivative formula of the quasideterminant on σ we have By a similar argument as (B.3), (B.4), we have , we can omit the right common factor E (±) j from the j-th column (j = 2 ∼ n), and take the right common factor E (±) j of the last column out of the quasideterminant. Then we obtain By the Jacobi identity (A.9), we have which are constant matrices. Therefore, we can conclude that which is asymptotic (Cf: (B.4), (B.14)) to = 0 (The j-th column is identical to the last column) By (B.7) and (A.7), the right common factors C (±) j of each column can be omitted completely, and hence (Proof) For n = 1, we have that is, the one-soliton solution σ [2] ∈ U (2). Assume that the n-soliton solution σ [n+1] ∈ U(2) for 1 ≤ n ≤ k − 1. For n = k and by the Darboux transformation [10], we have By the Jacobi identity (A.9), By the Jacobi identity (A.9) ∈ U(2). On the other hand, || By the homological relation (A.10) Note that the second equality from the bottom is obtained by using the homological relation (A.10) and (A.7). By (C.3), (C.5) and (C.6), we can conclude that We remark that in the case of E the condition |λ j | = 1 comes from the condition that Λ j Λ † j is a scalar matrix together with the reality condition µ j = −1/λ j . In the case of U 2 , the condition that Λ j Λ † j is a scalar matrix and the reality condition µ j = +1/λ j leads to trivial solutions because Λ j becomes a scalar matrix and hence the Darboux transformation becomes trivial. The data of the n-soliton solutions has the following symmetry: Therefore, (∆, ∆ 11 , ∆ 12 , ∆ 21 , ∆ 22 ) By the fact that Corollary D.2 Proposition D.1 implies: is an even function with respect to L j . ( is an odd function with respect to L j . (Proof)

D.3 Exact Calculation of NLσM Term (Two-Soliton)
For preparation, we introduce some symmetries between the coefficients of the soliton data before our calculation. Our observation is as the following Remark 1 ∼ 3 which can be checked simply from (D.7) and (D.8).

Remark 1
(D.10) which implies the relation directly by taking some simple addition and subtraction over (D.11). On the other hand, by taking sum of squares over L.H.S. and R.H.S. of (D.11) and using the relation (D.12), we get the following nontrivial identity.
Now let us start our main calculation of the NL sigma model term for two-soliton. By using the soliton data (D.7) and after a little bit tedious calculation, we can conclude that By (D.9) and (D.10), we find that the coefficients of the leading terms exp(±2(X 1 ± X 2 )) are identical to zero and the remaining terms can be rewritten as (D.14) =: Ξ Next, we want to show that the constant term Ξ above can be absorbed completely into the non-constant terms. By the definition of r By (D.13) and (D.11), Comparing with (D.14), we have µ A(−1, i)A(i, 1) e X 1 +X 2 +iΘ 12 2 + e − (X 1 +X 2 +iΘ 12 )