Symplectic Flatness and Twisted Primitive Cohomology

We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_{\mathrm{\infty }}$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_{\mathrm{\infty }}$-algebra. The symplectic flatness condition arises from twisting the $A_{\mathrm{\infty }}$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.