We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from [35], we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith [2]. We also show that derived Fukaya categories are complete invariants of special isogenous tori.

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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.

Mirror symmetry conjecture identifies the complex geometry of a Calabi $ Yau manifold with the symplectic geometry of its mirror Calabi $ Yau man $ ifold. Using the SYZ mirror transform, we argue that (i) the mirror of an elliptic Calabi $ Yau manifold admits a twin Lagrangian fibration structure and (ii) the mirror of the Fourier $ Mukai transform for dual elliptic fibra $ tions is a symplectic Fourier $ Mukai transform for dual twin Lagrangian fibrations, which is essentially an identity transformation in this case.

We study the geometry of the Grassmannians of symplectic subspaces in a symplectic vector space. We construct symplectic twistor spaces by the symplectic quotient construction and use them to describe the symplectic geometry of the symplectic Grassmannians.