Form field theory on Calabi-Yau threefold

We show how the natural Abelian duality of 2 and 3-form gravity theories in 7-dimensional manifold $CY_3\times S^1$, leads to an S-duality between 2 and 3-form theories on simply connected $CY_3$. The massless sector of the 2-form field theory on $CY_3$ corresponds to the A model string field theory. We discuss the complex structure independence of the 2-form theory for a general K\"{a}hler manifold and derive the holomorphic anomaly equations for simply connected $CY_3$