Lagrangian Floer theory in symplectic fibrations

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 fiber Lagrangians. The primary example that we have in mind is the manifold of full flags in ${\mathbb C}^3$ which fibers as ${\mathbb P}^1 \to {\rm Flag}({\mathbb C}^3) \to {\mathbb P}^2$, and a Lagrangian $T^3$ that fibers over the Clifford torus in ${\mathbb P}^2$. It turns out that one can prove the usual transversality and compactness results when the base is a rational symplectic manifold and the fibers 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 Kaehler, we derive a formula for the leading order disk potential.