Symplectic Flatness and Twisted Primitive Cohomology

Abstract

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.

To appear on The Journal of Geometric Analysis

References in corpus (1)

• Odd sphere bundles, symplectic manifolds, and their intersection theory

Cited by in corpus (2)

• Mapping Cone Connections and their Yang-Mills Functional
• Mapping cone Thom forms