BRANES IN THE MODULI SPACE OF HIGGS BUNDLES

Abstract

In this thesis, we will construct families of branes in the moduli space of Higgs bundles over a Riemann surface. We will give general constructions of hyperkahler and complex Lagrangian submanifolds of two types of manifolds, a cotangent bundle with hyperkahler structure satisfying some properties, and a manifold with an algebraically completely integrable Hamiltonian system. Up to a high codimension subset, the moduli space of Higgs bundles can be identified with the cotangent bundle of the moduli space of stable bundles over the Riemann surface. Also, the moduli space of Higgs bundles admits an algebraically completely integrable Hamiltonian system through the Hitchin system. Hence, we can apply our constructions to the moduli space of Higgs bundles. We will discuss the topology of these hyperkahler and complex Lagrangian submanifolds, and endow them with the appropriate sheaves, turning them into actual branes. We will also discuss some geometric structures on the families of branes constructed. Finally, to visualize the constructions, we will apply them to the moduli space of stable parabolic Higgs bundles of rank 2 over the complex projective curve with 4 marked points.