YMinstanton.dvi CUBO A Mathematical Journal Vol.12, No¯ 03, (99–120). October 2010 Self-Dual and Anti-Self-Dual Solutions of Discrete Yang-Mills Equations on a Double Complex VOLODYMYR SUSHCH Koszalin University of Technology, Sniadeckich 2, 75-453 Koszalin, Poland email: volodymyr.sushch@tu.koszalin.pl ABSTRACT We study a discrete model of the SU (2) Yang-Mills equations on a combinatorial analog of R 4. Self-dual and anti-self-dual solutions of discrete Yang-Mills equations are constructed. To obtain these solutions we use both techniques of a double complex and the quaternionic approach. Interesting analogies between instanton, anti-instanton solutions of discrete and continual self-dual, anti-self-dual equations are also discussed. RESUMEN Estudiamos el modelo discreto de las ecuaciones de Yang-Mills SU (2) sobre un análogo combinatório de R4. Soluciones auto-dual y anti-auto-dual para las ecuaciones discretas de Yang-Mills son construidas. Para obtener estas soluciones usamos las técnicas de doble complejo y abordage cuaternionico. Interesantes analogías entre soluciones instantones y anti-instantones de ecuaciones discretas y continuas auto-dual y anti-auto-dual son discu- tidas. Key words and phrases: Yang-Mills equations, self-dual and anti-self-dual equations, in- stantons and anti-instantons, difference equations. Math. Subj. Class.: 81T13, 39A12. 100 Volodymyr Sushch CUBO 12, 3 (2010) 1 Introduction It is well known that the self-dual and anti-self-dual connections are the absolute minima of the Lagrangian for a 4-dimensional non-abelian gauge theory. The first self-dual solution - the one instanton - to the SU(2) Yang-Mills equations on R4 was obtained by Belavin et al [3]. Later other more general multi-instanton solutions were described in [5, 11]. Since then numerous extensions have been made. Classical references are the books by Atiyah [1], Freed and Uhlenbeck [8]. In this paper we study a discrete analog of the SU(2) Yang-Mills equations on a combi- natorial analog of R4. The ideas presented here are strongly influenced by book of Dezin [6]. We develop discrete models of some objects in differential geometry, including the Hodge star operator, the differential and the covariant exterior differential operator, in such a way that they preserve the geometric structure of their continual analogs. We continue the investiga- tions which were originated in [7, 19, 20, 21]. The purpose of this paper is to construct the self-dual and anti-self-dual solutions of discrete SU(2) Yang-Mills equations which imitate the corresponding solutions of continual theory. The geometrical discretisation techniques used here extend those introduced in [6] and [19]. A combinatorial model of R4 based on the use of the double complex construction is taken from [21]. There are many other approaches to the discretisation of Yang-Mills theories. Numerous papers have been written on this subject. See, for example, [2, 4, 9, 10, 12, 13, 15, 18, 16] and the references therein. Most of them are based on the lattice discretisation scheme. However, in the case of the lattice formulation there are difficulties in keeping geometrical properties of an origin gauge theory. An alternative geometrical discretisation scheme of a field theory can be found in [17]. The paper is organized as follows. In Section 2 we review some basic facts of the SU(2) Yang-Mills theory on R4. We begin by recalling the connection between the Lie group SU(2) and the space of quaternions. Finally, we write down the basic instanton and anti-instanton solutions in quaternionic form. The notations here are compiled from [1] and [14]. Section 3 contains a brief summary of definitions and properties due to the double com- plex construction. We repeat here the relevant material from [21]. This article is also the main reference for this section. In particular, we introduce discrete matrix-valued forms (analog of differential forms) and define analogs of the main continual operations on them. In Section 4 using the quaternionic approach we present the discrete Yang-Mills equa- tions. We write out components of the discrete curvature 2-form in quaternionic form. The discrete self-dual and anti-self-dual equations are described. We try to be as close to con- CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 101 tinual SU(2) Yang-Mills theory as possible. Hence we discuss conditions when the discrete curvature will be su(2)-valued. Finally, Section 5 is devoted to self-dual and anti-self-dual solutions of the discrete Yang- Mills equations. We construct these solutions as discrete quaternionic 1-forms and discuss some analogies with continual instanton and anti-instanton solutions. 2 Quaternions and SU (2)-Connection In this section we briefly recall some well known settings of the smooth Yang-Mills theory in Euclidean 4-dimensional space (see, for example, [14]). We begin with a brief review of some preliminaries about quaternions. The quaternions are formed from real numbers by adjoining three symbols i, j, k and an arbitrary quaternion x can be written as x = x1 + x2i + x3j + x4k, (2.1) where x1, x2, x3, x4 ∈ R. The symbols i, j, k satisfy the following identities i 2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. (2.2) It is clear that the space of quaternions is isomorphic to R4. By analogy with the complex numbers x1 is called the real part of x and x2i + x3j + x4k is called the imaginary part. In further we will write Im x = x2i + x3j + x4k. The conjugate quaternion of x is defined by x̄ = x1 − x2i − x3j − x4k. Then the norm |x| of a quaternion can be introduced as follows |x| 2 = xx̄ = x 2 1 + x 2 2 + x 2 3 + x 2 4. (2.3) If x 6= 0, then it has a unique inverse x−1 given by x −1 = x̄/|x| 2 . (2.4) The algebra of quaternions can be represented as a sub-algebra of the 2 × 2 complex matrices M(2,C). We identify the quaternion (2.1) with a matrix f (x) ∈ M(2,C) by setting f (x) = ( x1 + x2 i x3 + x4 i −x3 + x4 i x1 − x2 i ) . (2.5) 102 Volodymyr Sushch CUBO 12, 3 (2010) Here i is the imaginary unit. It is well known that the unit quaternions, i.e., they have norm |x| = 1, form a group and this group is isomorphic to SU(2). The following 2 × 2 complex matrices i = ( i 0 0 −i ) , j = ( 0 1 −1 0 ) , k = ( 0 i i 0 ) (2.6) realize a representation of the Lie algebra su(2) of the group SU(2). Note that multiplying by −i these tree matrices we obtain the standard Pauli matrices. Matrices (2.6) correspond to the units i, j, k given by (2.2). Thus the Lie algebra su(2) can be viewed as the pure imaginary quaternions with basis i, j, k. Let now A be an SU(2)-connection. This means that A is an su(2)-valued 1-form and we can write A = ∑ µ Aµ(x)dx µ , (2.7) where Aµ(x) ∈ su(2) and x = (x1, ..., x4 ) is a point of R 4. The connection A is also called a gauge potential. Define a gauge transformation by a function g(x) taking value in SU(2). Then the gauge potential A must transform like A → g −1 A g + g −1 d g. (2.8) Let us define the curvature 2-form F by F = d A + A ∧ A, (2.9) where ∧ denotes the exterior multiplication. Consider also the covariant exterior differential operator d A given by d A Ω = dΩ+ A ∧Ω+ (−1) p+1 Ω∧ A, (2.10) where Ω is a su(2)-valued p-form. The Yang-Mills action S can be expressed in terms of the 2-forms F and ∗F as S = −tr ˆ R4 F ∧∗F, (2.11) where ∗ is the Hodge star operator. In R4 the operator ∗2 is either an involution or anti- involution, i.e., ∗2 = ±1. The Yang-Mills Lagrangian L = −tr(F ∧ ∗F) is invariant under the gauge transformation (2.8). By the physical requirement it is clear that the action S should be finite. Hence the curvature F should be square integrable. This means that F → 0 as |x| → ∞. CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 103 Consequently, we must describe the boundary condition at infinity for the connection A. By virtue of gauge freedom (2.8) we have A ∼ g −1 d g as |x| → ∞, (2.12) where ∼ implies asymptotic behaviour. Here and subsequently we do not specify the rate of decay. Written in terms of the covariant exterior differential operator d A the Euler-Lagrange equations for the extrema of (2.11) have the form d A F = 0, d A ∗ F = 0. (2.13) These equations are the Yang-Mills equations. The first equation of (2.13) is known also as the Bianchi identity. In 4-dimensional Yang-Mills theories the following equations F = ∗F, F = −∗ F (2.14) are called self-dual and anti-self-dual respectively. These equations are first-order non-linear equations for the potential A which imply the second-order Yang-Mills equations (2.13). So- lutions of (2.14) – the self-dual and anti-self-dual connections – are called also instantons and anti-instantons [8]. It is known that the self-dual and anti-self-dual connections are the absolute minima of the action S. The connection 1-form A can be defined also as taking values in the space of pure imagi- nary quaternions. To express A in quaternion form we consider the quaternion differential dx = dx1 + dx2i + dx3j + dx4k and the conjugate quaternion of dx dx̄ = dx1 − dx2i − dx3j − dx4 k. Let f (x) be a function of the quaternion variable x with quaternion values. Then we can write A as A = Im( f (x)dx), (2.15) where f (x) = f1(x) + f2(x)i + f3(x)j + f4(x)k. Using the rules of multiplication (2.2) we have A1(x) = f2(x)i + f3(x)j + f4(x)k, A2(x) = f1(x)i + f4(x)j − f3(x)k, 104 Volodymyr Sushch CUBO 12, 3 (2010) A3(x) = −f4(x)i + f1(x)j + f2(x)k, A4(x) = f3(x)i − f2(x)j + f1(x)k. Using (2.15) we can rewrite (2.9) as follows F = Im(d f (x) ∧ dx + f (x)dx ∧ f (x)dx). (2.16) Note that calculation of the imaginary part of f (x)dx and computing its curvature commute. Let us take the following expression for f (x): f (x) = x̄ 1 +|x|2 . (2.17) Then the connection 1-form A is defined by A = Im { x̄dx 1 +|x|2 } . (2.18) The explicit components Aµ can be written as A1(x) = −x2i − x3j − x4k 1 +|x|2 , A2(x) = x1i − x4j + x3k 1 +|x|2 , A3(x) = x4i + x1j − x2k 1 +|x|2 , A4(x) = −x3i + x2j + x1k 1 +|x|2 . (2.19) Putting (2.17) in (2.16) we get the pure imaginary expression F = dx̄ ∧ dx (1 +|x|2)2 . (2.20) It is easy to show that the 2-form dx̄ ∧ dx is anti-self-dual. Hence F is anti-self-dual too and the connection (2.18) describes an anti-instanton . See for details [1]. Similarly, if we take A = Im { xdx̄ 1 +|x|2 } , (2.21) then we obtain the self-dual 2-form F = dx ∧ dx (1 +|x|2)2 . (2.22) Thus the curvature is self-dual and (2.21) describes an instanton . 3 Double Complex We will need the double complex construction described in [21]. In with section for the con- venience of the reader we repeat the relevant material from [21] without proofs, thus making our presentation self-contained. CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 105 Let the tensor product C(4) = C ⊗ C ⊗ C ⊗ C of an 1-dimensional complex C be a combi- natorial model of Euclidean space R4 (see for details also [6]). The 1-dimensional complex C is defined in the following way. Let C0 denotes the real linear space of 0-dimensional chains generated by basis elements x j (points), j ∈ Z. It is convenient to introduce the shift operators τ,σ in the set of indices by τ j = j + 1, σ j = j − 1. (3.1) We denote the open interval (x j , xτ j ) by e j . We’ll regards the set {e j } as a set of basis ele- ments of the real linear space C1 of 1-dimensional chains. Then the 1-dimensional complex (combinatorial real line) is the direct sum of the introduced spaces C = C0 ⊕C1. The boundary operator ∂ on the basis elements of C is given by ∂x j = 0, ∂e j = xτ j − x j . (3.2) The definition is extended to arbitrary chains by linearity. Multiplying the basis elements x j , e j in various ways we obtain basis elements of C(4). Let s ( p) k , where k = (k1, k2 , k3, k4) and ki ∈ Z, be an arbitrary basis element of C(4). Then a p-dimensional chain is given by c p = ∑ k ∑ p c k ( p) s ( p) k , c k ( p) ∈ R. (3.3) We suppose that the superscript ( p) contains the whole requisite information about the quan- tity and places of 1-dimensional elements e j in s ( p) k . For example, the 1-dimensional basis elements ei k of C(4) can be written as e 1 k = e k1 ⊗ xk2 ⊗ xk3 ⊗ xk4 , e 2 k = xk1 ⊗ e k2 ⊗ xk3 ⊗ xk4 , e 3 k = xk1 ⊗ xk2 ⊗ e k3 ⊗ xk4 , e 4 k = xk1 ⊗ xk2 ⊗ xk3 ⊗ e k4 (3.4) and for the 2-dimensional basis elements ε i j k we have ε 12 k = e k1 ⊗ e k2 ⊗ xk3 ⊗ e k4 , ε 23 k = xk1 ⊗ e k2 ⊗ e k3 ⊗ xk4 , ε 13 k = e k1 ⊗ xk2 ⊗ e k3 ⊗ xk4 , ε 24 k = xk1 ⊗ e k2 ⊗ xk3 ⊗ e k4 , ε 14 k = e k1 ⊗ xk2 ⊗ xk3 ⊗ e k4 , ε 34 k = xk1 ⊗ xk2 ⊗ e k3 ⊗ e k4 . (3.5) Using (3.2) we define the boundary operator ∂ on chains of C(4) in the following way: if c p , cq are chains of the indicated dimension, belonging to the complexes being multiplied, then ∂(c p ⊗ cq ) = ∂c p ⊗ cq + (−1) p c p ⊗∂cq . (3.6) For example, for the basis element ε24 k we have ∂ε 24 k = ∂(xk1 ⊗ e k2 ) ⊗ xk3 ⊗ e k4 − xk1 ⊗ e k2 ⊗∂(xk3 ⊗ e k4 ) 106 Volodymyr Sushch CUBO 12, 3 (2010) = ∂xk1 ⊗ e k2 ⊗ xk3 ⊗ e k4 + xk1 ⊗∂e k2 ⊗ xk3 ⊗ e k4 − xk1 ⊗ e k2 ⊗∂xk3 ⊗ e k4 − xk1 ⊗ e k2 ⊗ xk3 ⊗∂e k4 = xk1 ⊗ xτk2 ⊗ xk3 ⊗ e k4 − xk1 ⊗ xk2 ⊗ xk3 ⊗ e k4 − xk1 ⊗ xk2 ⊗ xk3 ⊗ xτk4 + xk1 ⊗ xk2 ⊗ xk3 ⊗ xk4 . For convenience we also introduce the shift operators τi and σi which act in the set of indices k = (k1, k2, k3 , k4), ki ∈ Z, as τi k = (k1, ...τki , ...k4 ), σi k = (k1, ...σki , ...k4 ), (3.7) where τ and σ are given by (3.1). Let us introduce the construction of a double complex. Together with the complex C(4) we consider its double, namely the complex C̃(4) of exactly the same structure. Define the one-to-one correspondence ∗ : C(4) → C̃(4), ∗ : C̃(4) → C(4) (3.8) in the following way. Let s ( p) k be an arbitrary p-dimensional basis element of C(4), i.e., the product s ( p) k = sk1 ⊗ sk2 ⊗ sk3 ⊗ sk4 contains exactly p of 1-dimensional elements e ki and 4 − p of 0-dimensional elements xki , p = 0, 1, 2, 3, 4, ki ∈ Z. Then ∗ : s ( p) k → ±s̃ (4−p) k , ∗ : s̃ (4−p) k → ±s ( p) k , (3.9) where s̃ (4−p) k = ∗sk1 ⊗∗sk2 ⊗∗sk3 ⊗∗sk4 and ∗ski = ẽ ki if ski = xki and ∗ski = x̃ki if ski = e ki . In the first of mapping (3.9) we take "+" if the permutation (( p), (4 − p)) of (1, 2, 3, 4) is even and "−" if the permutation (( p), (4 − p)) is odd. Recall that in symbol ( p) the number of basis element is contained. For example, for the 2-dimensional basis element ε13 k = e k1 ⊗ xk2 ⊗ e k3 ⊗ xk4 we have ∗ε 13 k = −ε̃24 k since the permutation (1, 3, 2, 4) is odd. The mapping ∗ : s̃ (4−p) k → ±s ( p) k is defined by analogy. Proposition 3.1. Let cr ∈ C(4) be an r-dimensional chain (3.3). Then we have ∗∗ cr = (−1) r(4−r) cr . (3.10) Proof. See [21]. Now we consider a dual object of the complex C(4). Let K (4) be a cochain complex with gl(2,C)-valued coefficients, where gl(2,C) is the Lie algebra of the group GL(2,C). Re- call that gl(2,C) consists of all complex 2 × 2 matrices M(2,C) with bracket operation [·,·]. CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 107 We suppose that the complex K (4), which is a conjugate of C(4), has a similar structure: K (4) = K ⊗ K ⊗ K ⊗ K , where K is a conjugate of the 1-dimensional complex C. Basis ele- ments of K can be written as x j , e j . Then an arbitrary basis element of K (4) is given by sk ( p) = sk1 ⊗ sk2 ⊗ sk3 ⊗ sk4 , where sk j is either xk j or ek j . For example, we denote the 1-, 2- dimensional basis elements of K (4) by ek i , εk i j respectively, cf. (3.4), (3.5). For a p-dimensional cochain ϕ ∈ K (4) we have ϕ = ∑ k ∑ p ϕ ( p) k s k ( p) , (3.11) where ϕ ( p) k ∈ gl(2,C). We will call cochains forms, emphasizing their relationship with the corresponding continual objects, differential forms. We define the pairing operation < · , · > for arbitrary basis elements εk ∈ C(4), s k ∈ K (4) by the rule < εk, as k >= { 0, εk 6= sk a, εk = sk , a ∈ gl(2,C). (3.12) Here for simplicity the superscript ( p) is omitted. The operation (3.12) is linearly extended to cochains. The operation ∂ (3.6) induces the dual operation d c on K (4) in the following way: < ∂εk, as k >=< εk, ad c s k > . (3.13) For example, if ϕ = ∑ k ϕk x k, where xk = xk1 ⊗ xk2 ⊗ xk3 ⊗ xk4 , is a 0-form, then d c ϕ = ∑ k 4 ∑ i=1 (∆iϕk)e k i , (3.14) where ∆iϕk = ϕτi k −ϕk and e k i is the 1-dimensional basis elements of K (4). The coboundary operator d c is an analog of the exterior differentiation operator. Now we describe a cochain product on the forms of K (4). See [6] for details. We denote this product by ∪. In terms of the homology theory this is the so-called Whitney product. First we introduce the ∪-product on the chains of the 1-dimensional complex K . For the basis elements of K the ∪-product is defined as follows x j ∪ x j = x j , e j ∪ x τ j = e j , x j ∪ e j = e j , j ∈ Z, supposing the product to be zero in all other case. To arbitrary forms the ∪-product be ex- tended linearly. Let us introduce an r-dimensional complex K (r), r = 1, 2, 3, in an obvious notation. Let sk ( p) be an arbitrary p-dimensional basis element of K (r). It is convenient to write the basis element of K (r + 1) in the form sk ( p) ⊗ s j , where sk ( p) is a basis element of K (r) 108 Volodymyr Sushch CUBO 12, 3 (2010) and s j is either e j or x j , j ∈ Z. Then, supposing that the ∪-product in K (r) has been defined, we introduce it for basis elements of K (r + 1) by the rule (s k ( p) ⊗ s j ) ∪ (s k (q) ⊗ s µ ) = Q( j, q)(s k ( p) ∪ s k (q) ) ⊗ (s j ∪ s µ ), (3.15) where the signum function Q( j, q) is equal to −1 if the dimension of both elements s j , sk (q) is odd and to +1 otherwise. The extension of the ∪-product to arbitrary forms of K (r + 1) is linear. Note that the coefficients of forms multiply as matrices. Proposition 3.2. Let ϕ and ψ be arbitrary forms of K (4). Then d c (ϕ∪ψ) = d c ϕ∪ψ+ (−1) p ϕ∪ d c ψ, (3.16) where p is the dimension of a form ϕ. The proof of Proposition 3.2 is totally analogous to one in [6, p. 147] for the case of discrete forms with real coefficients. The complex of the cochains K̃ (4) over the double complex C̃(4) with the operator d c de- fined in it by (3.13) has the same structure as K (4). The operation (3.8) induces the respective mapping ∗ : K (4) → K̃ (4), ∗ : K̃ (4) → K (4) by the rule: < c̃, ∗ϕ >=< ∗c̃, ϕ >, < c, ∗ψ̃ >=< ∗c, ψ̃ >, (3.17) where c ∈ C(4), c̃ ∈ C̃(4), ϕ ∈ K (4), ψ̃ ∈ K̃ (4). Hence for the basic elements of K (4) or K̃ (4) we have relations (3.9). It is obviously that Proposition 3.1 is true for any r-dimensional cochain cr ∈ K (4). So we have ∗∗ϕ = (−1) r(4−r) ϕ for any discrete r-form ϕ on K (4) and note that the same relation holds for the Hodge star operator. Thus this operator is a combinatorial analog of the Hodge star operator. Let us introduce the following operation ι̃ : K (4) → K̃ (4), ι̃ : K̃ (4) → K (4) by setting ι̃s k ( p) = s̃ k ( p) , ι̃s̃ k ( p) = s k ( p) , (3.18) where sk ( p) and s̃k ( p) are basis elements of K (4) and K̃ (4). Hence for a p-form ϕ ∈ K (4) we have ι̃ϕ = ϕ̃. Recall that the coefficients of ϕ̃ ∈ K̃ (4) and ϕ ∈ K (4) are the same. CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 109 Proposition 3.3. The following hold ι̃ 2 = I d, ι̃∗ = ∗ι̃, ι̃d c = d c ι̃, (3.19) ι̃(ϕ∪ψ) = ι̃ϕ∪ ι̃ψ, where ϕ, ψ ∈ K (4). Proposition 3.4. Let h be a discrete 0-form. Then for an arbitrary p-form ϕ ∈ K (4) we have ι̃∗ (h ∪ϕ) = h ∪ ι̃ ∗ϕ. (3.20) Proof. See [21]. Note that the definition of inner product in the double complex and a discrete analog of the Yang-Mills actions (2.11) can be found in [21]. 4 Quaternions and Discrete Forms Let us consider a discrete 0-form with coefficients belonging to M(2,C). We put f = ∑ k f k x k , (4.1) where xk = xk1⊗xk2⊗xk3 ⊗xk4 is the 0-dimensional basis element of K (4), k = (k1, k2, k3, k4 ), ki ∈ Z. Suppose that the matrices f k ∈ M(2,C) look like (2.5), i. e. f k = ( f 1 k + f 2 k i f 3 k + f 4 k i −f 3 k + f 4 k i f 1 k − f 2 k i ) , (4.2) where f s k ∈ R, s = 1, 2, 3, 4. Then f k in quaternionic form can be expressed as f k = f 1 k + f 2 k i + f 3 k j + f 4 k k. (4.3) Hence the form (4.1) can be considered as a discrete form with quaternionic coefficients. We will call it simply the quaternionic form when no confusion can arise. In a proper way we define the quaternionic 0-form f̄ with coefficients f̄ k regarded as the conjugate quaternions of f k. Let f −1 be the quaternionic form, where f −1 k is given by (2.4). Then we have f ∪ f −1 = ∑ k f k f −1 k x k = ∑ k x k . (4.4) Proposition 4.1. Let f be a discrete 0-form and f 6= 0. Then we have d c f ∪ f −1 = −f ∪ d c f −1 . (4.5) 110 Volodymyr Sushch CUBO 12, 3 (2010) Proof. By definition (3.14) and according to (4.4), we have d c ( f ∪ f −1) = 0. Using Proposi- tion 3.2 we immediately obtain (4.5). Let us denote by e the following quaternionic 1-form e = ∑ k e k = ∑ k (e k 1 + e k 2i + e k 3j + e k 4 k), (4.6) where ek i is the 1-dimensional basis elements of K (4). Let A ∈ K (4) be a discrete 1-form. We define the discrete SU(2)-connection A to be A = ∑ k 4 ∑ i=1 A i k e k i , (4.7) where Ai k ∈ su(2) and k = (k1, k2, k3, k4 ), ki ∈ Z. Using (4.3) and (4.6) we write (4.7) in quaternionic form as A = Im( f ∪ e) = Im ( ∑ k f k e k ) . (4.8) Then the Ai k are given by A 1 k = f 2 k i + f 3 k j + f 4 k k, A 2 k = f 1 k i + f 4 k j − f 3 k k, A 3 k = −f 4 k i + f 1 k j + f 2 k k, A 4 k = f 3 k i − f 2 k j + f 1 k k. (4.9) Define the quaternionic 0-form x by x = ∑ k κx k , κ = k1 + k2i + k3j + k4k, (4.10) where ki ∈ Z. It is easy to check that d c x = e. (4.11) Therefore we can rewrite (4.8) as A = Im( f ∪ d c x). (4.12) Let g be a quaternionic 0-form (4.1) with the components of unit norm, i.e., |gk| = 1 for any k. It means that the corresponding discrete form is SU(2)-valued. We now define a gauge transformation for the discrete potential A which is analogous to (2.8). This is A → g −1 ∪ A ∪ g + g −1 ∪ d c g, (4.13) where A is given by (4.8) or (4.12). Note that the gauge transformed discrete form A is su(2)-valued too. It is not so obviously as in the continual case but follows immediately from the definition of ∪-multiplication and formula (3.16). More generally, if we assume that the gauge transformation g is an arbitrary quaternionic 0-form, then we take the imaginary part CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 111 of g−1 ∪ A ∪ g+ g−1 ∪ d c g in (4.13). For a deeper discussion of gauge invariant discrete models of the Yang-Mills theory we refer the reader to [19, 21]. An arbitrary discrete 2-form F ∈ K (4) can be written as follows F = ∑ k ∑ i< j F i j k ε k i j , (4.14) where F i j k ∈ gl(2,C), εk i j is the 2-dimensional basis element of K (4) and 1 ≤ i, j ≤ 4, k = (k1, k2, k3, k4 ), ki ∈ Z. Let F is given by F = d c A + A ∪ A. (4.15) Combining (4.7) and (4.15) and using (3.12), (3.13) and (3.15), we obtain F i j k = ∆i A j k −∆j A i k + A i k A j τi k − A j k A i τj k , (4.16) where ∆i A j k = A j τi k − A j k and τi k is given by (3.7). Let us define a discrete analog of the exterior covariant differentiation operator (2.10) as follows d c A Ω = d c Ω+ A ∪Ω+ (−1) p+1 Ω∪ A, (4.17) where Ω is an arbitrary p-form of K (4) looking like (3.11). Then a discrete analog of Equations (2.13) can be written as d c A F = 0, d c A ∗ ι̃F = 0, (4.18) where ι̃ is given by (3.18). It is easy to check that the combinatorial Bianchi identity: d c F + A ∪ F − F ∪ A = 0 (4.19) holds for the discrete curvature form (4.15) (cf. (2.13)). Remark 4.2. In the continual case the curvature form F (2.9) takes values in the algebra su(2) for any su(2)-valued connection form A. Unfortunately, this is not true in the discrete case because, generally speaking, the components Ai k A j τi k − A j k Ai τj k of the form A∪ A (see (4.16)) do not belong to su(2). To define an su(2)-valued discrete analog of the curvature 2-form we use the quaternionic form of A (4.8) and put in (4.15). Then the discrete curvature form F is given by F = Im{d c f ∪ e + ( f ∪ e) ∪ ( f ∪ e)}. (4.20) It should be noted that in the discrete case calculation of the imaginary part of f ∪ e and computing its curvature do not commute. 112 Volodymyr Sushch CUBO 12, 3 (2010) Proposition 4.3. If A = Im(x−1 ∪ d c x), where x is given by (4.10), then F = 0. Proof. Using (4.5) and putting f = x−1 in (4.20) we get F = Im(d c (x −1 ∪ d c x) + (x −1 ∪ d c x) ∪ (x −1 ∪ d c x) = Im(d c x −1 ∪ d c x − d c x −1 ∪ x ∪ x −1 ∪ d c x). According to (4.4) the form x ∪ x−1 has unit components. Hence d c x −1 ∪ x ∪ x −1 ∪ d c x = d c x −1 ∪ d c x. We now write down the components of (4.14) using quaternions. Putting (4.9) in (4.16) we find that F 12 k = (∆1 f 1 k −∆2 f 2 k − f 3 k f 3 τ1 k − f 4 k f 4 τ1 k − f 3 k f 3 τ2 k − f 4 k f 4 τ2 k )i + (∆1 f 4 k −∆2 f 3 k + f 2 k f 3 τ1 k + f 4 k f 1 τ1 k + f 1 k f 4 τ2 k + f 3 k f 2 τ2 k )j + (−∆1 f 3 k −∆2 f 4 k + f 2 k f 4 τ1 k − f 3 k f 1 τ1 k − f 1 k f 3 τ2 k + f 4 k f 2 τ2 k )k − f 2 k f 1 τ1 k − f 3 k f 4 τ1 k + f 4 k f 3 τ1 k + f 1 k f 2 τ2 k + f 4 k f 3 τ2 k − f 3 k f 4 τ2 k , F 13 k = (−∆1 f 4 k −∆3 f 2 k + f 3 k f 2 τ1 k − f 4 k f 1 τ1 k − f 1 k f 4 τ3 k + f 2 k f 3 τ3 k )i + (∆1 f 1 k −∆3 f 3 k − f 2 k f 2 τ1 k − f 4 k f 4 τ1 k − f 4 k f 4 τ3 k − f 2 k f 2 τ3 k )j + (∆1 f 2 k −∆3 f 4 k + f 2 k f 1 τ1 k + f 3 k f 4 τ1 k + f 4 k f 3 τ3 k + f 1 k f 2 τ3 k )k + f 2 k f 4 τ1 k − f 3 k f 1 τ1 k − f 4 k f 2 τ1 k − f 4 k f 2 τ3 k + f 1 k f 3 τ3 k + f 2 k f 4 τ3 k , F 14 k = (∆1 f 3 k −∆4 f 2 k + f 3 k f 1 τ1 k + f 4 k f 2 τ1 k + f 2 k f 4 τ4 k + f 1 k f 3 τ4 k )i + (−∆1 f 2 k −∆4 f 3 k − f 2 k f 1 τ1 k + f 4 k f 3 τ1 k + f 3 k f 4 τ4 k − f 1 k f 2 τ4 k )j + (∆1 f 1 k −∆4 f 4 k − f 2 k f 2 τ1 k − f 3 k f 3 τ1 k − f 3 k f 3 τ4 k − f 2 k f 2 τ4 k )k − f 2 k f 3 τ1 k + f 3 k f 2 τ1 k − f 4 k f 1 τ1 k + f 3 k f 2 τ4 k − f 2 k f 3 τ4 k + f 1 k f 4 τ4 k , F 23 k = (−∆2 f 4 k −∆3 f 1 k + f 4 k f 2 τ2 k + f 3 k f 1 τ2 k + f 1 k f 3 τ3 k + f 2 k f 4 τ3 k )i + (∆2 f 1 k −∆3 f 4 k − f 1 k f 2 τ2 k + f 3 k f 4 τ2 k + f 4 k f 3 τ3 k − f 2 k f 1 τ3 k )j + (∆2 f 2 k +∆3 f 3 k + f 1 k f 1 τ2 k + f 4 k f 4 τ2 k + f 4 k f 4 τ3 k + f 1 k f 1 τ3 k )k + f 1 k f 4 τ2 k − f 4 k f 1 τ2 k + f 3 k f 2 τ2 k − f 4 k f 1 τ3 k + f 1 k f 4 τ3 k − f 2 k f 3 τ3 k , F 24 k = (∆2 f 3 k −∆4 f 1 k + f 4 k f 1 τ2 k − f 3 k f 2 τ2 k − f 2 k f 3 τ4 k + f 1 k f 4 τ4 k )i CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 113 + (−∆2 f 2 k −∆4 f 4 k − f 1 k f 1 τ2 k − f 3 k f 3 τ2 k − f 3 k f 3 τ4 k − f 1 k f 1 τ4 k )j + (∆2 f 1 k +∆4 f 3 k − f 1 k f 2 τ2 k − f 4 k f 3 τ2 k − f 3 k f 4 τ4 k − f 2 k f 1 τ4 k )k − f 1 k f 3 τ2 k + f 4 k f 2 τ2 k + f 3 k f 1 τ2 k + f 3 k f 1 τ4 k − f 2 k f 4 τ4 k − f 1 k f 3 τ4 k , F 34 k = (∆3 f 3 k +∆4 f 4 k + f 1 k f 1 τ3 k + f 2 k f 2 τ3 k + f 2 k f 2 τ4 k + f 1 k f 1 τ4 k )i + (−∆3 f 2 k −∆4 f 1 k + f 4 k f 1 τ3 k + f 2 k f 3 τ3 k + f 3 k f 2 τ4 k + f 1 k f 4 τ4 k )j + (∆3 f 1 k −∆4 f 2 k + f 4 k f 2 τ3 k − f 1 k f 3 τ3 k − f 3 k f 1 τ4 k + f 2 k f 4 τ4 k )k + f 4 k f 3 τ3 k + f 1 k f 2 τ3 k − f 2 k f 1 τ3 k − f 3 k f 4 τ4 k − f 2 k f 1 τ4 k + f 1 k f 2 τ4 k . To obtain the components of (4.20) we must take the imaginary part of these equations. Proposition 4.4. The discrete curvature 2-form F (4.15) is su(2)-valued if and only if −f 2 k f 1 τ1 k − f 3 k f 4 τ1 k + f 4 k f 3 τ1 k + f 1 k f 2 τ2 k + f 4 k f 3 τ2 k − f 3 k f 4 τ2 k = 0, f 2 k f 4 τ1 k − f 3 k f 1 τ1 k − f 4 k f 2 τ1 k − f 4 k f 2 τ3 k + f 1 k f 3 τ3 k + f 2 k f 4 τ3 k = 0, −f 2 k f 3 τ1 k + f 3 k f 2 τ1 k − f 4 k f 1 τ1 k + f 3 k f 2 τ4 k − f 2 k f 3 τ4 k + f 1 k f 4 τ4 k = 0, f 1 k f 4 τ2 k − f 4 k f 1 τ2 k + f 3 k f 2 τ2 k − f 4 k f 1 τ3 k + f 1 k f 4 τ3 k − f 2 k f 3 τ3 k = 0, −f 1 k f 3 τ2 k + f 4 k f 2 τ2 k + f 3 k f 1 τ2 k + f 3 k f 1 τ4 k − f 2 k f 4 τ4 k − f 1 k f 3 τ4 k = 0, f 4 k f 3 τ3 k + f 1 k f 2 τ3 k − f 2 k f 1 τ3 k − f 3 k f 4 τ4 k − f 2 k f 1 τ4 k + f 1 k f 2 τ4 k = 0. Proof. From the above it follows immediately. Proposition 4.5. Let e is given by (4.6). Then the 2-form e ∪ ē is self-dual, i.e., e ∪ ē = ∗ι̃(e ∪ ē), (4.21) and ē ∪ e is anti-self-dual, i.e., ē ∪ e = −∗ ι̃(ē ∪ e). (4.22) Proof. Denote e i = ∑ k e k i , εi j = ∑ k ε k i j . Recall that ek i and εk i j are the 1-dimensional and 2-dimensional basic elements of K (4) (see also (3.4) and (3.5)). From this by (3.15) we obtain e i ∪ e j = εi j and e j ∪ e i = −εi j for all i < j. Then we have e ∪ ē = (e1 + e2i + e3j + e4k) ∪ (e1 − e2i − e3j − e4k) = −2{(e1 ∪ e2 + e3 ∪ e4)i + (e1 ∪ e3 − e2 ∪ e4)j + (e1 ∪ e4 + e2 ∪ e3)k} = −2{(ε12 +ε34)i + (ε13 −ε24)j + (ε14 +ε23)k}. 114 Volodymyr Sushch CUBO 12, 3 (2010) Using (3.17) and (3.19) we get ∗ι̃(e ∪ ē) = −2ι̃{(ε̃34 + ε̃12)i + (−ε̃24 + ε̃13)j + (ε̃23 + ε̃14)k} = e ∪ ē. In the same way we obtain (4.22). Corollary 4.6. For any quaternionic 0-form f the form f ∪ e ∪ ē is self-dual and f ∪ ē ∪ e is anti-self-dual. Proof. This follows immediately from (3.20). Discrete self-dual and anti-self-dual equations (discrete analogs of Equations (2.13)) are defined by F = ι̃∗ F, F = −ι̃∗ F, (4.23) where F is the discrete curvature form (4.4). Using (4.5), by the definitions of ι̃ and ∗, the first equation (self-dual) of (4.23) can be rewritten as follows F 12 k = F 34 k , F 13 k = −F 24 k , F 14 k = F 23 k . (4.24) By analogue with the continual case solutions of (4.23) (or (4.24)) are called instantons and anti-instantons respectively. 5 Discrete Instanton and Anti-Instanton In further analogy with the continual case consider the discrete SU(2)-connection A. Let A be the quaternionic 1-form (4.8), where the components of f are given by f k = κ̄ 1 +|κ|2 , (5.1) where κ = k1 + k2i + k3j + k4k, ki ∈ Z. Putting the last in (4.9) we obtain A 1 k = −k2i − k3j − k4k 1 +|κ|2 , A 2 k = k1i − k4j + k3k 1 +|κ|2 , A 3 k = k4i + k1j − k2k 1 +|κ|2 , A 4 k = −k3i + k2j + k1k 1 +|κ|2 . (5.2) It is convenient to denote Mi = 1 (1 +|κ|2)(1 +|τiκ| 2) , i = 1, 2, 3, 4. (5.3) Recall that the shift operator τi is given by (3.7). Substituting (5.2) in (4.16) and using (5.3) we find that F 12 k = {M1(1 + k 2 2 − k 2 1 − k1) + M2(1 + k 2 1 − k 2 2 − k2)}i CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 115 + {M1(k4 k1 + k2 k3) − M2(k3 k2 + k4 k1)}j + {M1(k2 k4 − k1 k3) + M2(k1 k3 − k2 k4)}k + M1(k1 k2 + k2) − M2(k1 k2 + k1), F 13 k = {M1(k2 k3 − k1 k4) + M3(k1 k4 − k2 k3)}i + {M1(1 + k 2 3 − k 2 1 − k1) + M3(1 + k 2 1 − k 2 3 − k3)}j + {M1(k1 k2 + k3 k4) − M3(k3 k4 + k1 k2)}k + M1(k1 k3 + k3) − M3(k1 k3 + k1), F 14 k = {M1(k1 k3 + k2 k4) − M4(k2 k4 + k1 k3)}i + {M1(k3 k4 − k1 k2) + M4(k1 k2 − k3 k4)}j + {M1(1 + k 2 4 − k 2 1 − k1) + M4(1 + k 2 1 − k 2 4 − k4)}k + M1(k1 k4 + k4) − M4(k1 k4 + k1), F 23 k = {−M2(k2 k4 + k1 k3) + M3(k1 k3 + k2 k4)}i + {M2(k3 k4 − k1 k2) + M3(k1 k2 − k3 k4)}j − {M2(1 + k 2 3 − k 2 2 − k2) + M3(1 + k 2 2 − k 2 3 − k3)}k + M2(k2 k3 + k3) − M3(k2 k3 + k2), F 24 k = {M2(k2 k3 − k4 k1) + M4(k1 k4 − k2 k3)}i + {M2(1 + k 2 4 − k 2 2 − k2) + M4(1 + k 2 2 − k 2 4 − k4)}j − {M2(k1 k2 + k3 k4) − M4(k3 k4 + k1 k2)}k + M2(k2 k4 + k4) − M4(k2 k4 + k2), F 34 k = −{M3(1 + k 2 4 − k 2 3 − k3) + M4(1 + k 2 3 − k 2 4 − k4)}i + {M3(−k2 k3 − k1 k4) + M4(k1 k4 + k2 k3)}j + {M3(k2 k4 − k1 k3) + M4(k1 k3 − k2 k4)}k + M3(k3 k4 + k4) − M4(k3 k4 + k3). Proposition 5.1. The 2-form F with components F i j k above is su(2)-valued if and only if k1 = k2 = k3 = k4. (5.4) Proof. From Proposition 4.4 F is su(2)-valued if and only if Mi (ki k j + k j ) − M j (ki k j + ki ) = 0 for any ki ∈ Z, i, j = 1, 2, 3, 4 and i < j. It follows immediately (5.4). 116 Volodymyr Sushch CUBO 12, 3 (2010) Thus, the su(2)-valued discrete curvature 2-form F can be written in the quaternionic form as follows F = ∑ k, ki=µ Mµ(2 − 2µ){(ε k 12 −ε k 34)i + (ε k 13 +ε k 24)j + (ε k 14 −ε k 23)k}. (5.5) From (5.2) here we have Mµ = 1 2(1+4µ2)(1+µ+2µ2) . Since ki = µ, in (5.5) we can write ε µ i j instead of εk i j . If we consider the 0-form ω = ∑ µ Mµ(1 −µ)x µ , µ ∈ Z (5.6) and use the following relation (see the proof of Proposition 4.5) ē ∪ e = 2{(ε12 −ε34)i + (ε13 +ε24)j + (ε14 −ε23)k}, then F can be written as F = ω∪ ē ∪ e. (5.7) In view of Corollary 4.6 F is anti-self-dual, i.e., F = −ι̃∗ F. Thus under condition (5.4) A with components (5.1) describes an anti-instanton. In the same manner we can see that the following quaternionic 1-form A = Im( f ∪ ē), (5.8) where f has the components f k = κ 1 +|κ|2 , (5.9) leads to an instanton solution of (4.24). Indeed, substituting (5.8) and (5.9) in (4.16) we now obtain F 12 k = {−M1(1 + k 2 2 − k 2 1 − k1) − M2(1 + k 2 1 − k 2 2 − k2)}i + {M1(k4 k1 − k2 k3) + M2(k3 k2 − k4 k1)}j + {M1(−k2 k4 − k1 k3) + M2(k1 k3 + k2 k4)}k + M1(k1 k2 + k2) − M2(k1 k2 + k1), F 13 k = {M1(−k2 k3 − k1 k4) + M3(k1 k4 + k2 k3)}i − {M1(1 + k 2 3 − k 2 1 − k1) + M3(1 + k 2 1 − k 2 3 − k3)}j + {M1(k1 k2 − k3 k4) + M3(k3 k4 − k1 k2)}k CUBO 12, 3 (2010) Self-Dual and Anti-Self-Dual Solutions ... 117 + M1(k1 k3 + k3) − M3(k1 k3 + k1), F 14 k = {M1(k1 k3 − k2 k4) + M4(k2 k4 − k1 k3)}i + {M1(−k3 k4 − k1 k2) + M4(k1 k2 + k3 k4)}j − {M1(1 + k 2 4 − k 2 1 − k1) + M4(1 + k 2 1 − k 2 4 − k4)}k + M1(k1 k4 + k4) − M4(k1 k4 + k1), F 23 k = {M2(−k2 k4 + k1 k3) + M3(−k1 k3 + k2 k4)}i + {M2(k3 k4 + k1 k2) − M3(k1 k2 + k3 k4)}j − {M2(1 + k 2 3 − k 2 2 − k2) + M3(1 + k 2 2 − k 2 3 − k3)}k + M2(k2 k3 + k3) − M3(k2 k3 + k2), F 24 k = {M2(k2 k3 + k4 k1) − M4(k1 k4 + k2 k3)}i + {M2(1 + k 2 4 − k 2 2 − k2) + M4(1 + k 2 2 − k 2 4 − k4)}j + {M2(k1 k2 − k3 k4) + M4(k3 k4 − k1 k2)}k + M2(k2 k4 + k4) − M4(k2 k4 + k2), F 34 k = −{M3(1 + k 2 4 − k 2 3 − k3) + M4(1 + k 2 3 − k 2 4 − k4)}i + {M3(−k2 k3 + k1 k4) + M4(−k1 k4 + k2 k3)}j + {M3(k2 k4 + k1 k3) − M4(k1 k3 + k2 k4)}k + M3(k3 k4 + k4) − M4(k3 k4 + k3). Again, under condition (5.4) we can write F as F = ∑ µ Mµ(2µ− 2){(ε µ 12 +ε µ 34 )i + (ε µ 13 −ε µ 24 )j + (ε µ 14 +ε µ 23 )k}, where µ ∈ Z. Therefore F = ω∪ e ∪ ē, (5.10) where ω is given by (5.6). Thus the discrete curvature form (5.10) is self-dual and we can say that (5.8) describes an instanton. Now to complete the analogy with the continual case we describe more precisely how the anti-instanton given by (5.1) behaves as |κ| → ∞. It is clear that f k is asymptotically κ̄ |κ|2 = κ−1. Then A ∼ Im(x −1 ∪ d c x) as |κ| → ∞. (5.11) Here x is given by (4.10). By virtue of Proposition 4.3 the discrete curvature F = 0 at infinity. 118 Volodymyr Sushch CUBO 12, 3 (2010) Proposition 5.2. The anti-instanton (5.1) has the same form at ∞ as it has near 0. Proof. Introduce the quaternionic 0-form y = ∑ k yk x k , where yk = 1 κ and remind κ = k1 + k2i + k3j + k4k. Clearly, y = x −1. We first compute x ∪ f ∪ e ∪ x−1, where f is given by (5.1). To do this, take (4.10), (4.11) and use the ∪-product definition. We have x ∪ f ∪ e = ( ∑ k κx k ) ∪ ( ∑ k κ̄ 1 +|κ|2 x k ) ∪ e = ( ∑ k |κ|2 1 +|κ|2 x k ) ∪ e = e − ( ∑ k 1 1 +|κ|2 x k ) ∪ e = d c x − ( ∑ k 1 1 +|κ|2 x k ) ∪ d c x. From this by (4.5) we get x ∪ f ∪ e ∪ x −1 = −x ∪ d c x −1 + ( ∑ k κ 1 +|κ|2 x k ) ∪ d c x −1 . (5.12) Now gauge transform the form f ∪ e by the gauge transformation g = x−1. We must take the imaginary part of (4.13). This yields by (5.12) Im( g −1 ∪ f ∪ e ∪ g + g −1 ∪ d c g) = Im (( ∑ k κ 1 +|κ|2 x k ) ∪ d c x −1 ) = Im (( ∑ k ȳk 1 +|yk| 2 x k ) ∪ d c y ) . Hence the gauge transformed anti-instanton A has precisely the form (5.11) near y = 0. The same conclusion can be drawn for the instanton (5.8). In the continual theory Proposition 5.2 shows that the anti-instanton (or instanton) ex- tends to the 4-sphere S4. This follows from the fact that S4 can be obtained from R4 by adding the point at infinity, i.e., S4 ≃ R4 ∪ {∞}. 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