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

## TL;DR

This paper proves Fukaya's conjecture by connecting Witten's deformed equivariant de Rham complexes with Morse theoretical $A_
ablafty$ structures involving gradient trees, advancing understanding of equivariant symplectic cohomology.

## Contribution

It extends and proves Fukaya's conjecture, linking Witten's deformation of equivariant complexes to Morse theory and gradient trees, revealing new structures in equivariant symplectic cohomology.

## Key findings

- Established the equivalence between Witten's deformed complexes and Morse $A_
ablafty$ complexes.
- Introduced counting of gradient trees with jumping to define new $A_
ablafty$ structures.
- Connected equivariant de Rham complexes with symplectic cohomology via Morse theory.

## Abstract

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.