of this article.

\[th:main\] Given a Poisson structure $\alpha$ on $\mathbb R^d$ there exists a unique natural deformation of the trivial generating function such that the first order is precisely $\alpha$. Moreover we have an explicit formula for this deformation $$S_h(p_1,p_2,x) = x(p_1+p_2)+\sum_{n=1}^\infty \frac{h^n}{n!}\sum_{\Gamma\in T_{n,2}}W_\Gamma \hat B_\Gamma(p_1,p_2,x)$$ where $T_{n,2}$ is the set of Kontsevich trees of type $(n,2)$, $W_\Gamma$ is the Kontsevich weight of $\Gamma$ and $\hat B_\Gamma$ is the symbol of the bidifferential operator $B_\Gamma$ associated to $\Gamma$.

Section \[recovering\] explains how to recover the structure maps from the deformed generating function. In Section \[examples\] we present basic examples of formal symplectic groupoids. In particular the main one is in the case of a linear Poisson structure $\alpha^{ij}(x) = \alpha^{ij}_k x^k$, i.e., when one considers the Kirillov–Kostant Poisson structure on the dual $\mathcal G^*$ of a Lie algebra $\mathcal G$. In this case, the generating function of the symplectic groupoid over $\mathcal G^*$ reduces exactly to the Campbell–Baker–Hausdorff formula$$S_h(p_1,p_2,x) = \langle \frac{1}{h}CBH(h p_1,h p_2), x\rangle$$ where $\langle,\rangle$ is the natural pairing between $\mathcal G$ and $\mathcal G^*$.

This basic example suggests to consider the generating function as a generalized CBH formula to the non-linear case and reproves in the 