The moduli space of G-Higgs bundles carries a natural Hyperkahler structure, through which we can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes) with respect to each structure. Notably, these A and B-branes have gained significant attention in string theory.
We shall begin the talk by first introducing G-Higgs bundles for reductive Lie groups and the associated Hitchin fibration, and sketching how to realize Langlands duality through spectral data. We shall then look at particular types of branes (BAA-branes) which correspond to very interesting geometric objects, hyperholomorphic bundles (BBB-branes).
The presentation will be introductory and my goal is simply to sketch some of the ideas relating these very interesting areas.