The Coupled Hitchin-He Equations: Integrable Deformations and Rigidity of the Moduli Space

TL;DR

This paper introduces a deformation of the Hitchin system via coupled equations, proves the existence of solutions, demonstrates integrability, and shows the moduli space's rigidity under small deformations, extending results to Kähler manifolds.

Contribution

It develops the parameter-geometrization approach for Hitchin systems, proves integrability and rigidity results, and extends the framework to compact Kähler manifolds.

Findings

Existence of a unique smooth solution branch near zero deformation parameter.

The coupled system is integrable with a Lax pair.

The moduli space remains analytically isomorphic under small deformations.

Abstract

We introduce the \emph{parameter-geometrization} to the Hitchin system, a paradigm embedding deformation parameters into geometry via the coupled Hitchin-He equations on a surface with boundary. A boundary term couples a second Higgs field $\psi$, recovering the classical system at $\alpha=0$. We prove a unique, smooth solution branch exists near $\alpha=0$ (Theorem A). The system is integrable, admitting a Lax pair (Theorem B). Crucially, the moduli space $\mathcal{M}_\alpha$ is analytically isomorphic to $\mathcal{M}_0$ for small $|\alpha|$, preserving the Hitchin fibration -- revealing a deep rigidity where all moduli are controlled by the primary Higgs field (Theorem C). Using the \emph{nonlinear embedding} technique that casts the deformed system into the form of a classical Higgs bundle system, whose integrability and geometry are well-understood, we extends the framework to compact K\"ahler manifolds (Theorem D).