Involutions and Lagrangian Submanifolds in Higgs-Bundle Moduli Spaces of Curves

In this work, we study geometric properties of moduli spaces of Higgs bundles over Riemann surfaces of genus 2 through the action of natural involutive symmetries. Higgs bundles, introduced by Hitchin in his foundational work on self-duality equations, are pairs consisting of a holomorphic vector bundle and a holomorphic section of its endomorphism bundle tensored with the canonical bundle. The moduli space of such objects carries a natural hyperkähler structure. Every genus-2 Riemann surface is hyperelliptic and therefore admits a canonical involution with six fixed points. This symmetry induces an automorphism of the moduli space of Higgs bundles by pullback. Understanding the geometric properties of fixed-point loci under such automorphisms provides insight into the global structure of these moduli spaces and connects to the theory of real structures in algebraic geometry. According to the results of Schaposnik, the fixed-point locus is Lagrangian with respect to the natural symplectic structure on the moduli space, meaning that the symplectic form vanishes identically on tangent vectors to the fixed-point set. This Lagrangian property connects to mirror symmetry and the geometric Langlands program, where Lagrangian submanifolds correspond to branes and special coherent sheaves. Our main result provides explicit distance estimates relating the hyperkähler distance between a Higgs bundle and its involution image to computable norms of the Higgs field. The proof combines variational methods for path length estimation with the Hitchin–Kobayashi correspondence to control connection variations through the linearized self-duality equations. These distance estimates provide quantitative tools for measuring how involution symmetries deform the hyperkähler geometry, with potential applications to the study of geodesics and metric completeness.