Shifted Poisson Structures on Differentiable Stacks

Abstract

Abstract The purpose of this paper is to investigate $(+1)$-shifted Poisson structures in the context of differential geometry. The relevant notion is that of $(+1)$-shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of $(+1)$-shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (1) We introduce a ${\mathbb{Z}}$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack ${\mathfrak{X}}$. It turns out that $(+1)$-shifted Poisson structures on ${\mathfrak{X}}$ correspond exactly to elements of the Maurer–Cartan moduli set of the corresponding dgla. (2) We introduce the notion of the tangent complex $T_{\mathfrak{X}}$ and the cotangent complex $L_{\mathfrak{X}}$ of a differentiable stack ${\mathfrak{X}}$ in terms of any Lie groupoid $\Gamma{\rightrightarrows } M$ representing ${\mathfrak{X}}$. They correspond to a homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^{\vee } M\rightarrow A^{\vee }[-1]$, respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak{X}}$ defines a morphism ${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$.