An Introduction to Hitchin Systems

Abstract: This thesis aims to introduce the tools that are needed to understand — and historically led to the inception of — Hitchin systems. We lay out a basic theory of Hamiltonian systems on symplectic manifolds, and prove the Liouville-Arnold theorem, which states that integrable Hamiltonian systems admit coordinates in which their solution is basically linear. We then introduce a theory of holomorphic structures on complex vector bundles, following a 1983 paper by Atiyah and Bott on the Yang-Mills equations in proving a one-to-one correspondence between such structures and the Dolbeault operator. Finally, the thesis gives an overview of Nigel Hitchin’s 1987 paper Stable Bundles and Integrable Systems.