Morita equivalence of formal Poisson structures

We extend the notion of Morita equivalence of Poisson manifolds to the setting of {\em formal} Poisson structures, i.e., formal power series of bivector fields $π=π_0 + λπ_1 +\cdots$ satisfying the Poisson integrability condition $[π,π]=0$. Our main result gives a complete description of Morita equivalent formal Poisson structures deforming the zero structure ($π_0=0$) in terms of $B$-field transformations, relying on a general study of formal deformations of Poisson morphisms and dual pairs. Combined with previous work on Morita equivalence of star products, our results link the notions of Morita equivalence in Poisson geometry and noncommutative algebra via deformation quantization.

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