  We consider several diffeomorphism invariant field theories of 2- and 3-forms
in six dimensions. They all share the same kinetic term $BdC$, but differ in
the potential term that is added. The theory $BdC$ with no potential term is
topological - it describes no propagating degrees of freedom. We show that the
theory continues to remain topological when either the $BBB$ or $C\hat{C}$
potential term is added. The latter theory can be viewed as a background
independent version of the 6-dimensional Hitchin theory, for its critical
points are complex or para-complex 6-manifolds, but unlike in Hitchin's
construction, one does not need to choose of a background cohomology class to
define the theory. We also show that the dimensional reduction of the
$C\hat{C}$ theory to three dimensions, when reducing on S3, gives 3D gravity.
