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.
Sibasish BanerjeeWeyertal 86-90, Department of Mathematics, University of Cologne, 50679, Cologne, Germany; Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, GermanyPietro LonghiInstitute for Theoretical Physics, ETH Zurich, 8093, Zurich, SwitzerlandMauricio Andrés Romo JorqueraYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
Symplectic GeometryAlgebraic GeometryarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.34001
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