Given a fibration of compact symplectic manifolds and an induced fibration of Lagrangians, one can ask if we can compute the Floer cohomology of the total Lagrangian from information about the base and fibre Lagrangians. The primary example that we have in mind is the manifold of full flags in ℂ3 which fibres as P1→Flag(ℂ3)→P2, and a Lagrangian T3 that fibres over the Clifford torus in P2. It turns out that one can prove the usual transversality and compactness results when the base is a rational symplectic manifold and the fibres are monotone. Assuming that we have a solution to the Maurer-Cartan equation, we then write down a Leray-Serre type spectral sequence which computes the Floer cohomology of the fibered Lagrangian. In the special case that the fibers are Kähler, we derive a formula for the leading-order disk potential.
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
