The coupling of Poisson sigma models to topological backgrounds

We extend the coupling to the topological backgrounds, recently worked out for the 2-dimensional bf-model, to the most general poisson sigma models. the coupling involves the choice of a casimir function on the target manifold and modifies the brst transformations. this in turn induces a change in the brst cohomology of the resulting theory. the observables of the coupled theory are analyzed and their geometrical interpretation is given. we finally couple the theory to 2-dimensional topological gravity: this is the first step to study a topological string theory in propagation on a poisson manifold. as an application, we show that the gauge-fixed vectorial supersymmetry of the poisson sigma models has a natural explanation in terms of the theory coupled to topological gravity.