Higher solutions of Hitchin's self-duality equations

TL;DR

This paper constructs counterexamples to Simpson's 1997 question by using branched Willmore surfaces and the generalized Whitham flow, showing not all real sections produce global solutions of Hitchin's self-duality equations.

Contribution

It provides explicit counterexamples linking Willmore surfaces to solutions of Hitchin's equations, revealing limitations of the global solution correspondence.

Findings

Counterexamples are constructed using branched Willmore surfaces.

Higher solutions are valid on dense subsets but not globally.

Suggests a deeper link between Willmore surfaces and self-duality theory.

Abstract

Solutions of Hitchin's self-duality equations corresponds to special real sections in 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 allow for complex analytic procedure to obtain all solutions of the self-duality equations. The purpose of this paper is to construct counter examples given by certain (branched) Willmore surfaces in $3$-space (with monodromy) via the generalized Whitham flow. Though these higher solutions do not give rise to global solutions of the self-duality equations on the whole Riemann surface $M$, they are solutions on an open dense subset of it. This suggest a deeper connection between Willmore surfaces, i.e., rank $4$ harmonic maps theory, with the rank $2$ self-duality theory.