  Unitary representations of the fundamental group of a Kahler manifold
correspond to polystable vector bundles (with vanishing Chern classes).
Semisimple linear representations correspond to polystable Higgs bundles. In
this paper we find the objects corresponding to affine representations: the
linear part gives a Higgs bundle and the translation part corresponds to an
element of a generalized de Rham cohomology.
