Solution and ellipticity properties of the self-duality equations of Corrigan et al in eight dimensions

We study the two sets of self-dual Yang-Mills equations in eight dimensions proposed in 1983 by E. Corrigan et al. and show that one of these sets forms an elliptic system under the Coulomb gauge condition, and the other (over-determined) set can have solutions that depend at most on N arbitrary constants, where N is the dimension of the gauge group, hence the global solutions of both systems are finite dimensional. We describe a subvariety P-8 of the skew-symmetric 8 x 8 matrices by an eigenvalue criterion and we show that the solutions of the elliptic equations of Corrigan et al. are among the maximal linear submanifolds of P-8. We propose an eighth-order action for which the elliptic set is a maximum.