Forschungsseminar: Komplexe Geometrie

16.01.2015 Alexander Ritter (University of Oxford)

Floer theory for negative line bundles via Gromov-Witten invariants

Abstract: I will explain how the Floer theory for negative line bundles over closed symplectic manifolds is determined by Gromov-Witten invariants, which relies on generalizing the Seidel representation to non-compact symplectic manifolds. The analogue of the Seidel element for the obvious circle action on the fibres of the line bundle corresponds to the first Chern class c₁(L) of the line bundle, and the symplectic cohomology turns out to be the quantum cohomology of the total space quotiented by the generalized zero-eigenspace of quantum product by c₁(L). This description can be made rather explicit in the case of toric negative line bundles, in which case the Jacobian ring of the superpotential recovers symplectic cohomology (rather than the quantum cohomology), and the non-zero eigenvalues of the c₁(L) action correspond to Lagrangian tori which often split-generate the Fukaya category.