On the Solution and Ellipticity Properties of the Self-duality equations of Corrigan et al in Eight Dimensions

TL;DR

This paper analyzes the ellipticity and solution structure of self-duality equations in eight-dimensional Yang-Mills theory, revealing their elliptic nature, solution constraints, and proposing an associated action functional.

Contribution

It demonstrates the elliptic and overdetermined elliptic properties of the equations, characterizes solutions via an eigenvalue criterion, and introduces an eight-order action functional.

Findings

Self-duality equations form elliptic and overdetermined elliptic systems.

Solutions depend on at most N arbitrary constants, with N being the gauge group dimension.

An eight-order action functional is proposed where the elliptic set is maximal.

Abstract

We show that the two sets of self-dual Yang-Mills equations in 8-dimensions proposed in (E.Corrigan, C.Devchand, D.B.Fairlie and J.Nuyts, {\it Nuclear Physics} {\bf B214}, 452-464, (1983)) form respectively elliptic and overdetermined elliptic systems under the Coulomb gauge condition. In the overdetermined case, the Yang-Mills fields can depend at most on $N$ arbitrary constants, where $N$ is the dimension of the gauge group. We describe a subvariety ${\cal P}_8 $ of the skew-symmetric $8\times 8$ 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 ${\cal P}_8$. We propose an eight order action for which the elliptic set is a maximum.