Shifted Poisson structures on differentiable stacks

Abstract

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 $\mathbb{Z}$-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 $\mathfrak{X}$. It turns out that shifted $(+1)$ 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 tangent complex $T_{\mathfrak{X}}$ and 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 homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\to TM$ and $T^{\vee }M\to A^{\vee }[-1]$, respectively. We prove that a $(+1)$-shifted Poisson structure on a differentiable stack $\mathfrak{X}$, defines a morphism $L_{\mathfrak{X}}[1]\to T_{\mathfrak{X}}$.

49 pages; corrected misprints, added references. Section 4 and 5 underwent major rewriting. Examples are placed in a separate section. To be published in International Mathematics Research Notices