  We construct a class of quantum field theories depending on the data of a
holomorphic Poisson structure on a piece of the underlying spacetime. The main
technical tool relies on a characterization of deformations and anomalies of
such theories in terms of the Gelfand-Fuchs cohomology of formal Hamiltonian
vector fields. In the case that the Poisson structure is non-degenerate such
theories are topological in a certain weak sense, which we refer to as "de Rham
topological". While the Lie algebra of translations acts in a homotopically
trivial way, we will show that the space of observables of such a theory does
not define an E_n-algebra. Additionally, we will highlight a conjectural
relationship to theories of supergravity in four and five dimensions.
