Symplectic Flatness and Twisted Primitive Cohomology

Li-Sheng Tseng University of California, Irvine Jiawei Zhou Yau Mathematical Sciences Center, Tsinghua University

Differential Geometry Symplectic Geometry Algebraic Topology and General Topology mathscidoc:2209.10001

The Journal of Geometric Analysis, 32, (282), 2022.8
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.
Symplectic geometry , Flat bundles , A-infinity algebra
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  title={Symplectic Flatness and Twisted Primitive Cohomology},
  author={Li-Sheng Tseng, and Jiawei Zhou},
  booktitle={The Journal of Geometric Analysis},
Li-Sheng Tseng, and Jiawei Zhou. Symplectic Flatness and Twisted Primitive Cohomology. 2022. Vol. 32. In The Journal of Geometric Analysis.
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