The linear system for self-dual gauge fields in a spacetime of signature 0

Abstract

The overdetermined linear system for the self-dual Yang—Mills (SDYM) equations is examined in a flat four-dimensional space whose metric has signature 0. There are three different domains for the system, and correspondingly three (essentially) different solutions to the linear system for a given gauge field. If the gauge potential is real analytic, two of the solutions patch together to give a holomorphic function in an annular region of projective twistor space. Conversely, an arbitrary holomorphic GL( n, C )-valued function in such a domain can be uniquely factored (on the real lines) to give a solution to SDYM with gauge group U( n). The set of all real analytic u( n)-valued gauge fields can thus be parametrized by the points of a certain double coset space.