Higher solutions of Hitchin’s self-duality equations

Solutions of Hitchin’s self-duality equations correspond to special real sections of the Deligne–Hitchin moduli space—twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would in principle allow for complex analytic procedures to obtain all solutions of the self-duality equations. The purpose of this article is to construct counter examples given by certain (branched) Willmore surfaces in three-space (with monodromy) via the generalized Whitham flow. Though these sections do not give rise to global solutions of the self-duality equations on the whole Riemann surface M, they induce solutions on an open and dense subset of it. This suggest a connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, with the rank 2 self-duality theory.