The Coupled Hitchin-He Equations: Integrable Deformations and Rigidity of the Moduli Space
Haoran He, Qichen He
TL;DR
The paper introduces parameter-geometrization by coupling a second Higgs field $\psi$ to the Hitchin system on a bordered surface, yielding a deformed set of Hitchin-He equations that recover the classical system at $\alpha=0$. It proves local existence and smooth dependence of solutions on $\alpha$, and establishes complete integrability through a Lax pair that absorbs the boundary deformation. A rigidity theorem shows that for small $|\alpha|$ the moduli spaces $\mathcal{M}_\alpha$ are analytically isomorphic to $\mathcal{M}_0$ with the Hitchin fibration preserved, demonstrating that the deformation is a marginal, isomorphism-type perturbation. The work then extends the framework to compact Kähler manifolds via a nonlinear embedding, preserving integrability and the fibration structure. Throughout, a detailed analysis of the linearized operator via Weitzenböck identities and gauge fixing underpins the existence theory and the uniform estimates, enabling a global deformation picture and a conjectural bridge to a deformed geometric Langlands correspondence, supported by a solvable spectral example.
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 $ψ$, recovering the classical system at $α=0$. We prove a unique, smooth solution branch exists near $α=0$ (Theorem A). The system is integrable, admitting a Lax pair (Theorem B). Crucially, the moduli space $\mathcal{M}_α$ is analytically isomorphic to $\mathcal{M}_0$ for small $|α|$, 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 Kähler manifolds (Theorem D).