Low-area Floer theory and non-displaceability (Q1729777)

A classical question in symplectic topology, originating from Arnold's conjectures, is to understand whether two given Lagrangian submanifolds are (Hamiltonian) non-displaceable. The nonexistence of the involved Hamiltonian diffeomorphism is sometimes referred to as the Lagrangian rigidity problem, where the Floer theory is the main tool. Most applications of the Floer theory were focused on monotone Lagrangians. Recent developments gave access to nondisplaceability results for non-monotone Lagrangians, those were based on the Floer cohomology with bulk deformations. In the present paper a new version of Floer theory is introduced for non-monotone Lagrangian submanifolds, which uses least-area holomorphic disks with boundary on it. Nondisplaceability theorems are formulated in this new setting, that refer to continuous families of Lagrangian tori in the complex projective plane and other del Pezzo surfaces. New tools to evaluate low-area string invariants enable a number of applications of the main displaceability result. Generalisations to higher dimensions are discussed.