The frame bundle X of a hyperbolic 3-manifold M is naturally a complex manifold with trivial canonical bundle. From this it is
possible to give a holomorphic interpretation of flat vector bundles over M. In one direction, one can interpret a flat
SL(2,C)-bundles over M a defining an alternative complex structure on X. Alternatively, one can fix the holomorphic structure on X, then the
pull-back of flat complex vector bundles on M are naturally holomorphic vector bundles on X. This procedure embeds the Chern-Simons
theory of M in the holomorphic Chern-Simons theory of X and one may hope to use techniques from one side to solve problems in the
other. For example, given a flat bundle over M with a Hermitian metric, the pull-back metric in the holomorphic bundle is
Hermitian-Einstein if and only if the original metric is harmonic. From here we intend to investigate if the theory of harmonic maps can
shed light on the behaviour of Hermitian-Einstein connections over X.