Higgs bundles, Lagrangians and mirror symmetry

This thesis is dedicated to the study of certain loci of the Higgs bundle moduli space on a compact Riemann surface. Motivated by mirror symmetry, we give a detailed description of the fibres of the $G$-Hitchin fibration containing $G_0$-Higgs bundles for the real forms $G_0 = SU^*(2m)$, $SO^*(4m)$ and $Sp(m,m)$ of $G = SL(2m,\mathbb{C})$, $SO(4m,\mathbb{C})$ and $Sp(4m,\mathbb{C})$, respectively. The spectral curves associated to these fibres are examples of ribbons and our description is done in two different ways, one in term of objects on the reduced scheme associated to the spectral curve, while the other in terms of the (non-reduced) spectral curve. A link is provided between the two approaches. We use this description to give a proposal for the support of the dual BBB-brane inside the moduli space of Higgs bundles for the Langlands dual group of $G$, corroborating the conjectural picture. In the second part of the thesis we discuss Gaiotto Lagrangian subvarieties inside the moduli spaces of $G$-Higgs bundles, where $G$ is a complex reductive group. These Lagrangians are obtained from a symplectic representation of $G$ and we discuss some of its general properties. We then focus our attention to the Gaiotto Lagrangian for the standard representation of the symplectic group. This is an irreducible component of the nilpotent cone for the symplectic Hitchin fibration. We describe this component by using the usual Morse function on the Higgs bundle moduli space restricted to the Lagrangian in question. Lastly, we discuss natural questions and applications of the ideas developed in this thesis. In particular, we say a few words about the hyperholomorphic bundle, how to generalize the Gaiotto Lagrangian to vector bundles which admit many sections and give an analogue of the Gaiotto Lagrangian for the orthogonal group.