of view. First it is easy to check that one gets the SGA equation by requiring that the saddle point evaluation as $h$ goes to $0$ of the two integrals $$\int e^{\frac{i}{h}[S(p_1,p_2,x)+S(p,p_3,x)-px]}\frac{d^dpd^dx}{(2\pi h)^{d/2}} \quad\textrm{ and }\quad
\int e^{\frac{i}{h}[S(p_2,p_3,x)+S(p_1,p,x)-px]}\frac{d^dpd^dx}{(2\pi h)^{d/2}}$$ be equal. This allows us to provide in Section \[kontform\] a quick but non rigorous proof of the existence of the generating function relying only on the associativity of the Kontsevich star product.

The second way to derive the SGA equation is symplectic reduction. Consider the symplectic groupoid $G$ over $M=\mathbb R^d$ as above. Let us call $\mathcal L_S\subset \overline G\times \overline G\times G$ the Lagrangian submanifold associated to the generating function $S$ (i.e., $\mathcal L_S = {\operatorname{graph}}(dS)$). Now consider the spaces $H(k) = \overline G^k\times G$ and the diagonal $\Delta_{l_1,\dots,l_k}\subset
\overline{H(k)}\times H(l_1)\times \dots\times H(l_k)$, $$\Delta_{l_1,\dots,l_k} = \big\{ (g_1,\dots,g_k,y),(x_{11},\dots,x_{1l_1},g_1),\dots,(x_{k1},\dots,x_{kl_k},g_k)\big\}.$$ This is a coisotropic subspace of $H(k)\times H(l_1)\times\dots\times H(l_k)$. Then one can consider the symplectic reduction by the diagonal $\Delta_{l_1,\dots,l_k}$ which sends Lagrangian submanifolds of $H(k)\times H(l_1)\times\dots\times H(l_k)$ to Lagrangian submanifolds of $H(l_1+\dots+l_k)$. In particular $\mathcal L_S\oplus\mathcal L_S\oplus \mathcal L_I$ ($I(p,x)= px$) is sent to $\mathcal L_1 \subset H(3)$ and $\mathcal L_S\oplus\mathcal L_I\oplus \mathcal L_S$ to $\mathcal L_2\subset H(3)$. One can check 