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 CĈ potential term is added. The latter theory can be viewed as a background independent version of the 6-dimensional Hitchin

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... , 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 CĈ theory to three dimensions, when reducing on S3, gives 3D gravity.