The focus of this project is to understand the geometry of mirror symmetry in the Strominger-Yau-Zaslow (SYZ) setup, in the particular case where the Calabi-Yau manifolds in question are hyperkähler. The hyperkähler structure allows one to study the geometry of mirror symmetry via complex algebraic geometry, a crucial idea in this project.
Our main guiding example is the Hitchin system for complex Lie groups G, given by the moduli space of G-Higgs bundles over a smooth curve, together with the Hitchin fibration, whose generic fibers are abelian subvarieties of Jacobians of so-called spectral curves. This moduli is hyperkähler and fits in the SYZ setup, the mirror being the Hitchin system for the Langlands dual group of G. Indeed, mirror symmetry on Hitchin systems is deeply related with the Geometric Langlands Program.
In a hyperkähler manifold it makes sense to define BBB- and BAA-branes and Kapustin-Witten conjectured that such branes should be exchanged under mirror symmetry, via (extensions of) a Fourier-Mukai transform. Non-trivial examples of these branes are not easily found, but some of them have been, and evidence supporting the KW conjecture was obtained. Most examples were, however, over the smooth locus of the Hitchin fibration. One of the main problems (but not the only one) we intend to address is to develop machinery to deal with mirror symmetry on the singular locus of the Hitchin fibration.