Shifted Poisson structures on differentiable stacks

The purpose of this paper is to investigate shifted (+1) Poisson structures in context of differential geometry. The relevant notion is shifted (+1) Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of (+1) Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests: (1) We introduce a Z-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under Morita equivalence of Lie groupoids, thus can be considered as polyvector fields on the corresponding differentiable stack X. It turns out that shifted (+1) Poisson structures on X correspond exactly to elements of the Maurer-Cartan moduli set of the corresponding dgla. (2) We introduce the notion of tangent complex TX and cotangent complex LX of a differentiable stack X in terms of any Lie groupoid Γ⇉M representing X. They correspond to homotopy class of 2-term homotopy Γ-modules A[1]→TM and T∨M→A∨[−1], respectively. We prove that a (+1)-shifted Poisson structure on a differentiable stack X, defines a morphism LX[1]→TX. We rely on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids.