Symplectic Flatness and Twisted Primitive Cohomology

TL;DR

This paper introduces symplectic flatness for connections on symplectic manifolds, enabling twisted cohomology theories and revealing connections to Yang-Mills theory with new vanishing results.

Contribution

It defines symplectic flatness for connections, links it to twisted $A_ abla$-algebra structures, and studies the resulting cohomologies with new vanishing theorems.

Findings

Symplectic flat connections form a special subclass of Yang-Mills connections.

Twisted cohomologies can be computed and exhibit vanishing properties.

The framework generalizes existing differential form complexes on symplectic manifolds.

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.