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Cauchy matrix approach to the SU(2) self-dual Yang--Mills equation

The Cauchy matrix approach is developed to solve the SU(2) self-dual Yang--Mills equation. Starting from a Sylvester matrix equation coupled with certain dispersion relation for infinite coordinates, the self-dual Yang--Mills equation under Yang's formulation is constructed. By imposing further constraints on complex independent variables, a broad class of explicit solutions are obtained.


Introduction
The Yang-Mills theory is the most important development in physics in the second half of last century (see [41]). The pioneer work is that of Yang and Mills [45], which laid a foundation of nonabelian gauge theories that explain the electromagnetic, the strong and weak nuclear interactions. The theory has also geometric interpretations in its nature, where the concept of gauge fields is found to be identical to fiber bundles, e.g. [2,9,18,42] (also see [8,39] and the references therein).
The Yang-Mills equation of motion is [28] (please refer Section 2 for notations) which corresponds to the case where action functional S and energy functional E take local minima in the sense of semi-classical approximation. It is hard to solve this equation exactly. Even one could find solutions to (1.1), it is still difficult to verify S or E reaches local minima. However, it can be shown that (see [6,28]) when the gauge field strength F µν is self dual, for given integer topological charge q, the gauge fields are absolute minima of the action S. The corresponding solutions are called instantons and monopoles (for static gauge fields). The paper [28] well collected early approaches to the solutions of the self-dual Yang-Mills (SDYM) equation before 1980, such as the approach based on the so called Corrigan-Fairlie-'t Hooft-Wilczek ansatz [12,40], the Atiyah-Hitchin-Drinfeld-Manin construction [3] on instantons with integer topological charge q and depending on (for SU(N ) case) at least 8q − 3 parameters [4], and the Bäcklund transformation approach [13] based on the Atiyah-Ward ansatz [5]. Both [3] and [5] follows Ward's observation in 1977 [35] on the connection between self-dual gauge fields and twistor theory. One can also refer [28] and the references therein for more details. The SDYM equation is an integrable system. It has a Lax pair and has Painlevé property for any gauge group [19,36], therefore some methods based on integrability have been employed to solve the SDYM equation, such as a direct transform approach by solving Lax pair [7], Bäcklund transformation based on Riemann-Hilbert problem [33], an approach inspired from Sato's theory [31], bilinear method [26,29], and Darboux transformation [25].
In this paper we aim to solve the SU(2) SDYM equation via a direct method, the Cauchy matrix approach. The Cauchy matrix approach is a method to construct and study integrable equations by means of the Sylvester-type equations. In this approach integrable equations are presented as closed forms of some recurrence relations involving derivatives (or shifts). It was first systematically used in [24] to investigate integrable quadrilateral equations and later developed in [43,46] to more general cases. We will construct the SU(2) SDYM equation together with its explicit solutions.
The paper is organized as follows. In Sec.2 we recall Yang's formulation for the self-duality of the Yang-Mills equation. Then, in Sec.3 we make use of the Cauchy matrix approach to construct a SDYM equation together with its solutions. These solutions will be elaborated in Sec.4 by imposing constraints so that they meet self-duality of SU(2) gauge group. Finally, concluding remarks are given in Sec.5. There are three appendixes, which are devoted to proving Lemma 2, presenting solutions of the Sylvester equation
Equation (2.9) (or its alternative form) is usually the equation that was solved by using integrable methods, see [21,25,26,29]. In SU(2) case, once J is obtained, D can be recovered by Yang's R-gauge [44] as the following. Writing J in the form where f is real and e = −g * . Then D takes the form where U ∈ SU (2). Hence B χ 's are recovered from (2.8) and so are F µν 's from (2.1). In this paper we will develop the Cauchy matrix approach to construct equation (2.9) together with its explicit solution J.

The Cauchy matrix approach to the SDYM equation
In order to construct the SU(2) SDYM equation (2.9), we start from the Sylvester equation where {x n } are independent infinite complex variables, K, M , A, r, s are block matrices in the form of In addition, we assume K 1 and K 2 are invertible and do not share any eigenvalues so that the Sylvester equation (3.1) has a unique solution M for given K, r, s [30]. Define an infinite matrix S = ( where I specially denotes the 2N -th order identity matrix. The above settings are the same as those for deriving the Ablowitz-Kaup-Newell-Segur (AKNS) system [47], except here in (3.2) we have introduced variable x 0 and its dispersion relation. Thus we may make use of the results already obtained in [47]. However, the variable x 0 does play a useful role in our procedure of deriving the SDYM equation (2.9). For S (i,j) defined in (3.4) where K, M , r, s are governed by the Sylvester equation (3.1), there exists a recursive relation independent of dispersion relations, see Proposition 2 in [47] and cf. [43].
where K, M , r, s obey the Sylvester equation (3.1). In particular, when s = 1, it reads With respect to the dispersion relation (3.2), S (i,j) evolves as the following.
Formulae (3.8a) and (3.8c) have been derived in [47] and (3.8b) can be obtained similarly. For the completeness of the paper, a proof of Lemma 2 is given in Appendix A.
Now we come to the first main result of this paper.
Next, we prove (3.10) for positive n. Formula (3.8a) gives rise to (with i = j = 0) where we have made use of (3.11) and separated the first term from the summation. Then, in light of definition of v, replacing the index l with l + 1 and using (3.11) once again, we arrive at which is (3.10) with n ≥ 1. The case of n less than −1 can be proved similarly from (3.8c). In details, we have Thus, we have proved (3.10) for all n ∈ Z.
Considering the compatibility (u xn ) xm = (u xm ) xn , we immediately arrive at the following.
where n, m ∈ Z.
As a by-product of (3.18), u satisfies a potential SDYM equation (cf. [20]) Note that equation (3.18) differs from the SDYM equation (2.9) by a sign "−". In next section, we will recover (2.9) from (3.18) by imposing certain reductions. Besides, apart from {x n } with dispersion relation (3.2), we may introduce for {y m } such that (3.20) The resulting equation is which is in a same form as (2.9). In next section, we will work on (3.18) and reduce it to (2.9).

Solutions to the SU(2) SDYM equation
When the gauge group is of SU (2), J in the SDYM equation (2.9) should be a positive-definite Hermitian matrix with |J| = 1 [27,44]. In the following we will look for a Hermitian matrix v with |v| = 1. This will be able to be achieved by imposing some constraints on K and {x n }. The Sylvester equation (3.1) can be written as a more explicit form For given invertible K 1 and K 2 that do not share any eigenvalues, M 1 and M 2 can be uniquely solved. In practice, since S (i,j) is invariant with respect to K 1 , K 2 and any matrices similar to them [47] (cf. [43,46]), we can always consider K 1 and K 2 to be their canonical forms, say Γ and Λ. Solutions M i together with r i and s i can be explicitly presented. One may refer [47] or Appendix B of the present paper. It turns out that these solutions can be presented via the following form: and these elements satisfy (symmetric or commutative) relations With the above notations, we are able to investigate symmetric property of the infinite matrix S, i.e. the relations between S (i,j) and S (j,i) . i.e. S (i,j) T = −σ 2 S (j,i) σ 2 .
Proof. Making use of expressions (4.2) and relations (4.3), from (3.5) we find The third relation s can be proved in a similar way.
With this lemma we are able to prove |v| = 1. In fact, the relation (3.13) yields Then, by Lemma 3 we may replace s Next, we make v to be a Hermitian matrix by imposing constraints. First, we introduce z n . = x n = ξ n + iη n ,z n . = (−1) n+1 x −n = ξ n − iη n , n = 1, 2, · · · (4.5) where ξ n , η n ∈ R. This indicatesz n = z * n . Then we take m = −n − 1. The resulting equation which coincides with the form (2.9). We next introduce further constraints by N 2 = N 1 and to the Cauchy matrix scheme (3.1,3.2,3.3). Equation (4.5) implies ∂ ξn = ∂ zn + ∂z n , ∂ ηn = i(∂ zn − ∂z n ), (4.8) and the dispersion relation (3.2) is equivalently written in terms of ξ n and η n as and We are able to take 1 in light of the uniqueness of solutions of the Sylvester equations (4.1).
and v * 4 = v 4 in a similar way way, and All these together indicate v = v † when we take δ = 1, i.e. v is a Hermitian matrix when δ = 1. We end up the section with the following summarization. where z n = ξ n + iη n ,z n = z * n = ξ n − iη n , n = 1, 2, · · · , (4.12) ξ n and η n are real, and Here, K 1 ∈ C N ×N , K 1 and −(K * 1 ) −1 do not share any eigenvalues, M 1 , r 1 and s 1 are determined by the system for n = 1, 2, · · · . Solution v satisfies v = v † and |v| = 1. v is piecewisely positive-definite or negative-definite, depending on the domains where v is positive or negative.
Explicit solutions of the system (4.14) can be formulated from Appendix B. Some examples of solution v will be listed in Appendix C.

Concluding remarks
In this paper we have constructed explicit solution v for the SDYM equation (4.6) by using a direct method, namely, the Cauchy matrix approach. We started with the Sylvester equation (3.1) together with the dispersion relation (3.2) with respect to infinite coordinates {x n }, made use of the recursive relation (3.7), proved u and v satisfy the key equation (3.10) that gives rise to the equation (3.18). Then, we introduced real independent coordinates ξ n and η n , and imposed constraints on z n ,z n and K 1 and K 2 . Finally, we obtained solution v that solves the SDYM equation (4.6). Solutions have been presented via Theorem 3.
Our Cauchy matrix approach is based on the scheme for the AKNS system, cf. [47]. Compared with [47], here we introduced the auxiliary variable x 0 , frequently made use of the recursive relation (3.7), and finally we were able to construct the key equation (3.10). We also discussed symmetric relation of S (i,j) and S (j,i) . All these elaborations enabled us to finally prove the property v = v † and |v| = 1, and obtain exact solutions to the SDYM equation (4.6).
There are similar direct approaches to construct equation (3.10), e.g., the one based on bidifferential graded algebra [14,15], where it is assumed there exists a function v to satisfy the equation (3.10). In our approach, v is clearly defined by I 2 − S (−1,0) , which provides an explicit solution to the SDYM equation (4.6) In our scheme, after introducing z n andz n by (4.5), we took m = −n − 1. Instead of doing that, if we take m = n − 1 in equation (3.18), we have an equation and by redefining x n by ix n , it becomes [21] (v x n+1 v −1 ) x n−1 + (v xn v −1 ) xn = 0, (n ∈ Z).

(5.2)
Manakov and Zakharov constructed its solutions using its Lax pair [21]. Since the equation depends only on 3 independent variables, its solutions may generate monopoles (see [28]). Solutions obtained in [21] are different from ours. In fact, the two constraints on n and m hold simultaneously only when n = 0. In this case, our results can be applied and the equation (5.2) will be written as where we have made use of relation (3.15). Note that this equation is a 2D equation and was recently derived in [34] by means of a similar direct method. With regard to the methods of solving the SDYM equation, most of them are direct and constructive. The approach in [7] can be thought of a Darboux-Bäcklund transformation employing the Lax pair of the SDYM equation. Solutions obtained are algebraic type, i.e. rational solutions in terms of polynomials of independent of variables. The bilinear approach given in [29] (cf. [26]) is to reformulate the bilinear relations satisfied by the functions e, f, g in (2.12) to a new set of bilinear equations involving nine τ functions in Hankelians. In principle, they provide algebraic solutions as well. Our solutions are expressed in terms of exponential functions, which are in expression formally similar to those obtained via Darboux transformation [25]. However, our solutions are different from those in [25], as J is a solution of the SU(2) SDYM equation but it is not necessary J ∈ SU(2), cf. [25]. In addition, our expression for v is more explicit and includes solutions generated by possible canonical forms of K 1 (not only diagonal form or a Jordan form, but also any combinations of them).
There are several further investigations related to approach and results of the present paper. For example, extend the approach to the SU(N ) SDYM equation and noncommutative case, e.g. [16,17,32]. Besides, it is well known that as a 4D integrable system, the SDYM equation allows various reductions to lower dimensional integrable equations [1,11,22,23,38]. That would be interesting to understand how reductions play roles in generating solutions to lower dimensional integrable equations. In addition, Ward used to discuss discretisation of the SDYM equation [37]. Since the Cauchy matrix approach originated from solving discrete integrable systems [24], it would be interesting to have a discrete analogue of the SDYM equation from this approach.