Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential Gerstenhaber-Batalin-Vilkovisky (dGBV) algebra. From rational/real homotopy theory we know that the dGA $(\Omega^{*}(M),d,\wedge)$ contains all real homotopy information of $M$. My question is:

are there some symplectic topological/geometric information contained in the dGBV $(\Omega^{*}(M),d,d^{\star})$?