A geometric refinement of a theorem of Chekanov

We prove a conjecture of Barraud and Cornea in the monotone setting, refining a result of Chekanov on the Hofer distance between two Hamiltonian isotopic Lagrangian submanifolds.Comment: 19 pages, 7 figures. Second version, to appear in Journal of Symplectic Geometry. We changed the statement of Theorem 1.2 to take into account sphere bubbling and non transversal Lagrangian

Categorification of Seidel's representation

Two natural symplectic constructions, the Lagrangian suspension and Seidel's quantum representation of the fundamental group of the group of Hamiltonian diffeomorphisms, Ham(M), with (M,\omega) a monotone symplectic manifold, admit categorifications as actions of the fundamental groupoid \Pi(Ham(M)) on a cobordism category recently introduced in \cite{Bi-Co:cob2} and, respectively, on a monotone variant of the derived Fukaya category. We show that the functor constructed in \cite{Bi-Co:cob2} that maps the cobordism category to the derived Fukaya category is equivariant with respect to these actions.Comment: 32 pages, 4 figures. Updated to agree with the published version. To appear in Israel Journal of Mathematic

Quantum Reidemeister torsion, open Gromov–Witten invariants and a spectral sequence of OH

We adapt classical Reidemeister torsion to monotone Lagrangian submanifolds using the pearl complex of Biran and Cornea. The definition involves generic choices of data and we identify a class of Lagrangians for which this torsion is invariant and can be computed in terms of genus zero open Gromov–Witten invariants. This class is defined by a vanishing property of a spectral sequence of Oh in Lagrangian Floer theory