Equivariant Lagrangian displacements

TL;DR

This paper develops a Borel equivariant quantum cohomology framework to show that certain monotone Lagrangians in symplectic vector spaces cannot be displaced by specific Hamiltonian isotopies, revealing new symplectic invariants.

Contribution

It introduces a Borel equivariant version of quantum cohomology and demonstrates its effectiveness in detecting equivariant displacements of Lagrangians.

Findings

Equivariant quantum cohomology is sensitive to equivariant displacements.

Certain monotone Lagrangians cannot be displaced by commuting Hamiltonian isotopies.

The Floer--Euler class appears as a key term in the equivariant differential.

Abstract

This paper proves that certain monotone Lagrangians in the standard symplectic vector space cannot be displaced by a Hamiltonian isotopy which commutes with the antipodal map. The method of proof is to develop a Borel equivariant version of the quantum cohomology of Biran and Cornea, and prove it is sensitive to equivariant displacements. The Floer--Euler class of Biran and Khanevsky appears as a term in the equivariant differential in certain cases.