Metrics and Special Kahler Geometry on the Moduli Spaces of Higgs Bundles and Hitchin Systems

Abstract

The notions of Hitchin systems and Higgs bundles (also called Higgs pairs) were introduced by N. Hitchin in 1987. They rapidly formed a subject lying on the crossroads of representation theory, symplectic geometry, and algebraic geometry. In this research area, the main objects that attract mathematicians' attention are the moduli spaces of Higgs bundles Mn,d (the moduli space of Higgs bundles is a space parameterizing the collection of all Higgs bundles). These moduli spaces have many good properties that make them interesting objects worthy of study. For example, they are symplectic manifolds, e.g. the moduli space of the Higgs bundles with rank one and degree zero is the cotangent bundle of the Jacobian variety of a Riemann surface. These moduli spaces are also equipped with Riemannian hyperkähler metrics ghk which can not be written explicitly in general, but they can be approximated by another metric, gsf which is called the semi-at metric. Roughly, we can say that ghk = gsf + correction. In fact, Gaiotto, Moore and Neitzke conjectured what the "corrections" should be in 2011[GMN13]. To find the semi-at metric of Mn,d, we first investigate the integrable system (Hitchin system) h :Mn,d → B where the map h is known as the Hitchin map. The Lagrangian fibers of the Hitchin systems are Jacobian (or Prym) varieties of spectral curves of a given compact Riemann surface. The semi-flat metric of Mn,d is induced by a special Kähler metric on B through the Lagrangian fibration. In my thesis, I will compute these metrics as explicitly as possible and give close observations to them...