Gauge theories, the holonomy operator, and the Riemann–Hilbert problem

Abstract

The so-called Riemann–Hilbert problem has arisen and plays a major role in the study of many nonlinear integrable systems, such as the sine–Gordon equation, stationary axial symmetric Einstein equations, etc. Here it is shown how the Riemann–Hilbert problem arises naturally in the study of self-dual Yang–Mills fields in Minkowski space via a simple geometric construction of the holonomy operator on anti-self-dual planes. This Riemann–Hilbert problem is then converted to a linear homogeneous differential equation that is considerably simpler to study than the original problem. Finally it is shown that the nonlinear equation of Yang for self-dual fields is easily understood from the holonomy point of view.