Non-Abelian Hodge Theory and Moduli Spaces of Higgs Bundles

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 hyperkähler 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 Poincaré polynomial of the rank 2 moduli space and verify topological mirror symmetry in this case.