ADHM construction of (anti-)self-dual instantons in eight dimensions

Abstract

We study the ADHM construction of (anti-)self-dual instantons in eight dimensions. We propose a general scheme to construct the (anti-)self-dual gauge field configurations F ∧F = ± * 8 F ∧F whose finite topological charges are given by the fourth Chern number. We show that our construction reproduces the known SO(8) one-instanton solution. We also construct multi-instanton solutions of the 't Hooft and the Jackiw-Nohl-Rebbi (JNR) types in the dilute instanton gas approximation. The well-separated configurations of multi-instantons reproduce the correct topological charges with high accuracy. We also show that our construction is generalized to (anti-)self-dual instantons in 4n (n = 3, 4, . . .) dimensions.

ADHM construction of (anti-)self-dual instantons in 4n dimensions

“…We find that, in generally higher dimensions, there is an extra ADHM constraint in addition to the respected four-dimensional one. This situation is same as the eight-dimensional case [18]. One of the most interest things is that the ADHM construction in more higher than twelve dimensions does not require other new constraints.…”

“…However, in the case of n = 2 (i.e. eight dimensions), it was shown that the simple generalization of the 't Hooft type ADHM ansatz is not well-defined as the higher-dimensional ADHM data in [18]. In the following, we will show that the cases in more higher dimensions are same situations as the eight-dimensional case.…”

“…The first ADHM constraint (20) is the natural generalization of the four-dimensional one [3]. On the other hand, the second ADHM constraint (21) is the analogy of the eight-dimensional one [18]. In addition, the Weyl operator requires the non-degeneracy condition: rank ∆ = 2 2n−1 k, and the existence of the inverse E (a) k (a = 1, 2) is guaranteed by this condition (This proof is shown in Appendix B).…”

“…The JNR type ADHM data in four dimensions [21] is well known as multi-instantons data which is different from the 't Hooft type. We can give generalization of the higher-dimensional JNR type ansatz, for instance, an eight-dimensional case in [18]. However, the second ADHM constraint requires the same condition as 't Hooft one (57).…”

“…In this paper, we study an ADHM construction of the 4n-dimensional (n ≥ 2) (anti-)selfdual instantons with the unitary gauge group U(N). The first non-trivial case (n = 2), the eight-dimensional ADHM construction, has been studied in [18]. The 4n-dimensional ADHM construction is generalization of the eight-dimensional one.…”