We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-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.
Zhengyang XueUniversity of Science and Technology of ChinaYinhua XiaUniversity of Science and Technology of ChinaChen LiState Key Laboratory of AerodynamicsXianxu YuanState Key Laboratory of Aerodynamics
Numerical Analysis and Scientific Computingmathscidoc:2208.25003