Wedge product on deRham complex of a Riemannian manifold $M$ can be pulled back to $H^*(M)$ via explicit homotopy, constructed using Green's operator, to give higher product structures. We prove Fukaya's conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type of $M$. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya.

中文翻译：

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深谷对 Witten 扭曲的 $A_\infty$ 结构的猜想

可以通过使用格林算子构造的显式同伦将黎曼流形 $M$ 的 deRham 复形上的楔积拉回到 $H^*(M)$，以提供更高的积结构。我们证明了 Fukaya 的猜想，该猜想表明，这些更高乘积结构的 Witten 变形具有半经典极限，作为通过计算关于 Morse 函数的梯度流树来定义的算子，这将 deRham 微分的显着 Witten 变形从关于同调的陈述推广到关于实同伦的陈述$M$ 的类型。Fukaya 还提出了将该猜想应用于镜像对称的各种应用。