  We introduce the concept of a Fock bundle, a smooth principal bundle over a
surface equipped with a special kind of adjoint-valued 1-form, as a new tool
for studying character varieties of surface groups. Although similar to Higgs
bundles, the crucial difference is that no complex structure is fixed on the
underlying surface. Fock bundles are the gauge-theoretic realization of higher
complex structures. We construct a canonical connection to a Fock bundle
equipped with compatible symmetric pairing and hermitian structure. The space
of flat Fock bundles maps to the character variety of the split real form.
Determining the hermitian structure such that this connection is flat gives a
non-linear PDE similar to Hitchin's equation. We explicitly construct solutions
for Fock bundles in the Fuchsian locus. Ellipticity of the relevant linear
operator provides a map from a neighborhood of the Fuchsian locus in the space
of higher complex structures modulo higher diffeomorphisms to a neighborhood of
the Fuchsian locus in the Hitchin component.
