Construction of multi-instantons in eight dimensions

We consider an eight-dimensional local octonionic theory with the seven-sphere playing the role of the gauge group. Duality conditions for two- and four-forms in eight dimensions are related. Dual fields--octonionic instantons--solve an 8D generalization of the Yang-Mills equation. Modifying the ADHM construction of 4D instantons, we find general $k$-instanton 8D solutions which depends on $16k-7$ effective parameters.


Introduction.
The discovery of instantons [1] was an important advance in our understanding of non-perturbative quantum field theory. These objects are (anti-)self-dual ( * F = ±F ) Euclidean solutions to Yang-Mills field equations in 4D. They have lead to a deeper understanding of the QCD vacuum (θ vacuum [2]), and have been conjectured to play a part in the confinement of color charges [3]. Instantons also have a broad significance in mathematics, specifically in the theory of fake R 4 -manifolds [4]. The most general multi-instanton solutions have been constructed [5], and these again played a part in broadening our understanding of gauge theories.
A single instanton solution is spherically symmetric and, in mathematical language, corresponds to the third Hopf map, which is the principal fibre bundle where S 4 is the one-point compactification of R 4 , S 3 ∼ SU (2) is the fibre (gauge group) and S 7 is the total space.
As string theory and M-theory live in higher dimensions, it is of interest to consider higher dimensional analogs of 4D instantons; in particular, there exists a natural generalization of instantons to 8D, where the last Hopf map S 15 S 7 → S 8 resides. The original 4D instanton had gauge group SU (2) embedded in Spin(4) ∼ SU (2) × SU (2), so that the bundle became Spin(5) The analogous single instanton solution in 8D was found in [6], and has a generalized self-duality * F 2 = ±F 2 with the bundle Spin (9) Spin (8) −→ S 8 . The higher dimensional instanton have conformal features similar to those of 4D instantons. The 8D case, and especially its multi-instanton generalization, appears more complicated than its 4D counterpart for the following reasons: 1. The fibre that is twisted with the 4D base space is a three-sphere, but this is a group, while the twisted part of the Spin(8) ∼ S 7 L × S 7 R × G 2 fibre is a seven-sphere. S 7 is the only paralellizable manifold that is not a Lie group, but it does have a close resemblance to a gauge group.
2. As S 7 can be represented by unit octonions, and G 2 is the automorphism group of the octonions, there is a hidden nonassociativity that comes into play.
3. There is only one choice (via the Hodge star) for the form of duality in 4D, but in 8D other possibilities arise, e.g., a tensor form of duality λF µν = 1 2 T µνρσ F ρσ has been studied [7,8]. Attempts [9] to obtain multi-instantons in a Spin(8) gauge theory meet with a number of difficulties. To circumvent these obstructions, we turn to a theory with only S 7 fibre, but to do this, we first need to review the properties of the octonions. Here we will construct multi-instanton solutions in 8D through a generalization of the ADHM procedure, and to do this we must deal with all of the above complications. We will introduce products and operators in a way that nonassociativity is tamed. Next, a new generalized duality is used to provide results that allow us to relate the topologically significant quadratic duality on F 2 to a specific form of tensor duality. We then consider the symmetries of our multi-instanton solutions and show that in 8D the k-instanton S 7 bundles contain 16k − 7 parameters in analogy with the 8k − 3 parameters of the most general 4D k-instanton S 3 bundles.

Octonions
We recall (for a review, see e.g. [10]) that the nonassociative octonionic algebra has the multiplication rule e i e j = −δ ij + f ijk e k , where the f ijk 's are completely anti-symmetric structure constants. The seven-sphere is described by a unit octonion g satisfying g * g = 1. The octonions' nonassociativity complicates construction of the analog of a gauge theory. For example, for imaginary octonionic A and F = dA + A 2 , the corresponding S 7 -gauge transformed quantities are Nonassociativity prevents the terms in the last two lines of (1) from canceling. Using g(g * h) = h, which holds for any octonions g and h, we note that the terms do cancel in LF g , where L is the operator of left octonionic multiplication, L(a 1 . . . a n ) = a 1 (a 2 (a 3 (. . . a n )) . . .).
Any arrangement of parentheses in the argument of L give the same results on the right-hand side of (2). Use of the operator L allows us to perform various operations on the octonions as if they were associative. For simplicity in notations, we omit L in the following. Instead of left octonionic multiplication we could use right multiplication with the same result. From (1) we now find the familiar result F g = g * F g.
For associative A and F , the forms tr F n are closed. To extend this to octonions, which do not admit a matrix representation, we need an octonionic operator with some of the properties of the matrix trace. Consider the operator tr O defined by where differential forms a k are of degrees r k . The operators tr O and d commute and so the forms tr O F n are closed; thus we arrive at the familiar Lie algebra result [11]: where 3 Linear duality Since any pair of imaginary octonions generate a quaternionic subalgebra, we expect to find an octonionic duality condition which is reducible to its quaternionic counterpart. For example, let us define dual octonionic 2-forms according to and determine the tensor f µνρσ from the following two requirements: (i) any 2-form can be written as a sum of its self-dual and anti-self-dual parts, or equivalently, ⋄ 2 = 1; (ii) dxdx * is self-dual and dx * dx is anti-self-dual. Consequently, for octonionic forms we obtain From Eqs. (6) and (7), the components of the ⋄-dual field strength F = 1 2 F µν dx µ dx ν are subject to the following 21 relations: 1 Applied to the quaternions, the above requirements lead to the familiar relations f 0abc = f ab0c = ǫ abc and f abcd = 0. In both the quaternionic and octonionic cases, the components f µνρσ are the matrix elements of the corresponding groups and cosets in the products Spin(4) = S 3 L × S 3 R and Spin(8) = S 7 L × G 2 × S 7 R . Also, the components turn out to coincide with the elements of the torsion and curvature tensors of Spin(4)/Spin(3) and Spin(7)/G 2 respectively (for the latter see [12]). Note the two choices of sign for the curvature tensor f µνρσ in (7) and the two choices of orientation, S 7 L,R = Spin(7) L,R /G 2 . Neither corresponds to the two choices of sign in Eq. (8).
Dual fields satisfy ⋄F = ±F and, in view of the octonionic Bianchi identity DF = 0, they also solve an 8D generalization of the Yang-Mills equation D⋄F = 0. Below we find multi-particle solutions to the duality equations.

Quadratic duality
In addition to the linear form of duality considered above, a quadratic form of duality is also possible in 8D. In the latter case, dual octonionic 4-forms are related via the Hodge star, " * ".
A conformally invariant action I = tr O F 2 * F 2 yields the equation of motion {F, D * F 2 } = 0. The * -dual fields, which are defined by * F 2 = ±F 2 , solve the equation of motion by means of the Bianchi identity DF = 0. In terms of (anti-)self-dual F 2 ± = 1 2 (F 2 ± * F 2 ), the action becomes On the other hand, the topological charge (the forth Chern number) is where we have used F 2 ± F 2 ∓ = 0. It follows from (10) and (11) that the action is bounded from below, I ≥ 384π 4 |n|, 1 While our octonionic duality condition (8) is similar in form to one of the two duality conditions for SO(8) considered in Ref. [7], the latter were not constructed to satisfy either of the two above-mentioned requirements. Consequently, our octonionic instantons are different from the SO(8) solutions in Ref. [8].
with minima achieved when F 2 ± = 0, i.e. for the * -dual fields (9). There are one-particle solutions to the quadratic duality equations (9), and these solutions have a geometric interpretation in terms of the forth Hopf map [6].
It is remarkable but straightforward to verify that ⋄F = ±F implies * F 2 = ∓F 2 . To check this, we need the identity where indices included in braces (brackets) are to be symmetrized (antisymmetrized). We can also view * as a "square" of ⋄. The relation between the linear and quadratic dualities allows us to proceed with the construction of octonionic multi-instantons.

Solution
The ADHM construction [5] gives the most general multi-instanton solutions to the duality equations in four dimensions. We construct octonionic dual fields by a suitable 8D generalization of the ADHM formula. Namely, consider a gauge potential [13] where the k-dimensional vector V and the k × k matrix B have constant octonionic entries. The operator L is suppressed as usual, and the symbol " †" means matrix transposition combined with octonionic conjugation. The corresponding field strength is where For real W , i.e. when and B is symmetric, F involves the expression L(. . . dxdx * . . .). The ⋄-dual of this 2-form is L(. . . ⋄ (dxdx * ) . . .) and, owing to the self-duality of dxdx * , F is ⋄-self-dual itself, but F 2 is * -anti-self-dual. Interchanging x and x * , interchanges self-dual and antiself-dual objects for both dualities.

Instanton number
For the solution obtained above, the gauge potential vanishes at infinity faster than a pure gauge, and has singularities at the instanton locations. A physically acceptable solution results from a suitable gauge transformation.
The singularities are located at eigenvalues {b i } of the k × k matrix B. Expanding around each singularity, we have approximately where A gauge transformation with the gauge function g i = y * i /|y i | removes the singularity at y i = 0 in the potential (15), and leads to Similar to the quaternionic case [14], all singularities inside a finite S 7 can be removed. Inside this S 7 , after using (4), the instanton number becomes where asymptotically (Q 7 ) g ∼ − 1 35 tr O (g * dg) 7 . Since the field strength corresponding to the gauge potential g * dg is zero, we use Stokes's theorem again to replace the integral over the large S 7 by the sum of the integrals over k small spheres S 7 i enclosing singularities b i . Around each singularity, F g looks like the field of a single anti-instanton at the origin, Therefore, the topological charge N and minus the instanton number −k are one and the same.

Parameters
We now count the number of parameters needed to describe a k-instanton. The octonions V and B have, respectively, 8k and 8 1 2 k(k + 1) real parameters. There are 7 1 2 k(k − 1) real equations in (14) constraining V and B. When V is replaced by g * V , where g ∈ S 7 is constant, the potential (12) is gauge transformed, A → g * Ag, eliminating 7 more parameters. Also, a transformation V → V T , B → T −1 BT with real and constant T ∈ O(k), which has 1 2 k(k − 1) parameters, does not change A. Therefore, the number of effective degrees of freedom desrcibing a k-instanton is We do not have a proof that the above construction gives all dual fields, although we suspect it does. At least it does so for the case of a one-instanton [6], which is described by 9 parameters-instanton's scale and location. Perhaps completeness of the construction can be proved by using octonionic projective spaces [15] and generalized twistors in analogy with the 4D case ( [5,13]). Other multi-instanton solutions are subsets of our solutions. For example, one can generalize Witten's and 't Hooft's [16] 4D multi-instanton solutions to 8D. The single 8D instanton has entered string theory and produced a solitonic member of the brane scan (for a review, see [17]). We hope our general construction will facilitate further applications to string and M-theory, and perhaps in pure mathematics.