In the last presentation I discussed deformation theory in a cohomological language and how, under suitable conditions, a deformation of Kähler structures on a fixed symplectic manifold induces an infinitesimal one for holomorphic sections of a fixed pre-quantum line bundle. This is the first step in Hitchin's construction of his famous connection: given a family of Kähler structures, the spaces of holomorphic sections of the pre-quantum line bundle form a vector bundle over the parametrising space (up to replacing it with a sufficiently large power), and the assignment above can be regarded as a parallel transport equation.

In this second part, I will continue following Hitchin's original paper to summarise the most relevant properties of the moduli spaces of flat connections over a surface and go over the proof of projective flatness of the connection for these spaces with their natural family of Kähler structures parametrised by Teichmüller space.