Lagrangian Circle Actions

TL;DR

This paper studies Hamiltonian diffeomorphism paths that preserve a monotone Lagrangian, computes related Seidel elements, and demonstrates their minimality in Lagrangian Hofer length, with applications to uniruledness and quantum cohomology.

Contribution

It introduces methods to compute the Lagrangian Seidel element for specific Hamiltonian paths and shows these paths minimize the Lagrangian Hofer length, with applications to symplectic topology.

Findings

Computed the leading term of the Lagrangian Seidel element.

Proved certain Hamiltonian paths minimize Lagrangian Hofer length.

Applied results to Lagrangian uniruledness and quantum cohomology of real Lagrangians.

Abstract

We consider paths of Hamiltonian diffeomorphism preserving a given compact monotone Lagrangian in a symplectic manifold that extend to an $S^1$--Hamiltonian action. We compute the leading term of the associated Lagrangian Seidel element. We show that such paths minimize the Lagrangian Hofer length. Finally we apply these computations to Lagrangian uniruledness and to give a nice presentation of the Quantum cohomology of real lagrangians in Fano symplectic toric manifolds.