Fukaya's conjecture on $S^1$-equivariant de Rham complex

Ziming Nikolas Ma The Chinese University of Hong Kong

Differential Geometry mathscidoc:1901.10003

Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and prove Fukaya's conjecture relating this Witten's deformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$ complexes defined by counting gradient trees with jumping which are closely related to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed by Siedel.
Fukaya's conjecture, Witten deformation, Morse category, $\mathbb{S}^1$-equivariant
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  • 18 pages, 4 figures
  title={Fukaya's conjecture on $S^1$-equivariant de Rham complex},
  author={Ziming Nikolas Ma},
Ziming Nikolas Ma. Fukaya's conjecture on $S^1$-equivariant de Rham complex. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190128225450139042194.
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