Displacement of polydisks and Lagrangian Floer theory

TL;DR

This paper corrects a key proof in Lagrangian Floer theory, establishes a new lower bound for polydisk displacement energy, and introduces a novel approach using Floer cohomology torsion thresholds in toric manifolds.

Contribution

It corrects an error in a foundational theorem and provides a stronger, more general lower bound for polydisk displacement energy using Floer cohomology techniques.

Findings

Corrected a proof in Floer theory literature.

Established a new lower bound for polydisk displacement energy.

Used torsion thresholds of Floer cohomology in toric manifolds.

Abstract

There are two purposes of the present article. One is to correct an error in the proof of Theorem 6.1.25 in \cite{fooo:book}, from which Theorem J \cite{fooo:book} follows. In the course of doing so, we also obtain a new lower bound of the displacement energy of polydisks in general dimension. The results of the present article are motivated by the recent preprint of Hind \cite{hind} where the 4 dimensional case is studied. Our proof is different from Hind's even in the 4 dimensional case and provides stronger result, and relies on the study of torsion thresholds of Floer cohomology of Lagrangian torus fiber in simple toric manifolds associated to the polydisks.