that $\mathcal L_1=\mathcal  L_2$ iff $S$ satisfies the SGA equation. In fact we have here, hidden in the background, a structure of an operad, the Lagrangian operad (see [@Cattaneo2002]).

Now consider $M= \mathbb R^d$ with the zero Poisson structure. The symplectic groupoid $G_0$ over it is the cotangent bundle ($G_0 = \mathbb{R}^{*d}\times\mathbb{R}^d$). The source map and the target map $s,t:G_0\rightarrow \mathbb R^d$ are identified with the cotangent bundle projection. The inclusion $\epsilon:\mathbb R^d \rightarrow G_0$ is defined by $\epsilon(x) = (0,x)$, the inverse map $i:G_0\rightarrow G_0$ by $i(p,x)=(-p,x)$ and the product is the fiber wise addition, i.e., $(p_1,x)\bullet(p_2,x) =
(p_1+p_2,x)$. The product space $G_0^{(m)}$ can be seen as the graph of the differential of the function $S_0(p_1,p_2,x)= x(p_1+p_2)$. It is easy to check that $S_0$ satisfies the SGA equation. We investigate deformations of this trivial generating function. Let us be more precise.

A **deformation** of the trivial generating function is a formal power series in $h$, $S_h = S_0 +hS_1+h^2S_2\dots$, obeying the SGA equation and such that $S_0(p_1,p_2,x)=x(p_1+p_2)$.

Such a deformation is called **natural** if

1.  $S_n(p,q,x)$ are polynomial in $q$ and $q$

2.  $S_n(\lambda q,\lambda p,x) = \lambda^{n+1} S_n(p,q,x)$

3.  $S_n(p,0,x)=S_n(0,p,x)=0$

4.  $S_n^i(p,p) = 0$, where $S_n^i$ is the 