Non-Abelian Hodge Theory and Moduli Spaces of Higgs Bundles

TL;DR

This paper introduces non-abelian Hodge theory and explores the geometry of Higgs bundle moduli spaces, highlighting their structures, symmetries, and mirror symmetry phenomena, with explicit computations for rank 2 cases.

Contribution

It develops the moduli theory of Higgs bundles, establishes core correspondences in non-abelian Hodge theory, and investigates mirror symmetry aspects of Hitchin systems.

Findings

Computed the Poincaré polynomial for rank 2 moduli space.

Verified topological mirror symmetry for type A Hitchin systems.

Analyzed the hyperkähler structure and Hitchin fibration properties.

Abstract

This paper provides an introduction to non-abelian Hodge theory and moduli spaces of Higgs bundles on compact Riemann surfaces. We develop the moduli theory of vector bundles and Higgs bundles, establish the main correspondences of non-abelian Hodge theory, and interpret them through the hyperk\"ahler structure on the Hitchin moduli space. We study the Hitchin fibration and its geometric properties, including SYZ mirror symmetry and topological mirror symmetry for type $\mathsf{A}$ Hitchin systems. As an illustration, we compute the Poincar\'e polynomial of the rank 2 moduli space and verify topological mirror symmetry in this case.