TITLE: Generator of a Fukaya category with certain properties
QUESTION [5 upvotes]: There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a compact symplectic manifold where this Lagrangian has the properties that :
1) Simply connected,but not a sphere or a product of spheres  2) The homology of the Floer complex HF*(L,L) is isomorphic as an algebra to H*(L). (Edit: as an ordinary algebra, the higher operations induced by the perturbation lemma can be different.) Extra points if the Floer theory can in a reasonable sense be defined over C. I'd also be really interested to know why this situation can't happen. 
Edit: I know it is kind of rare for a single Lagrangian(impossible?) to generate a Fukaya category. I'd be happy to have a bunch of such Lagrangians L_i where at least one of them was as above as long as Hom(L_i, L_j)=0 for i not equal to j...

REPLY [3 votes]: Even without the condition that $HF(L,L)$ is $A_\infty$-isomorphic to $H^\ast(L)$, your conditions are set up in such a way as to disable the standard tricks. This doesn't rule out the existence of examples, but it does make them hard to find.
One way to produce orthogonal objects in the Fukaya category is to take different spin structures on the same Lagrangian (this works for the Clifford torus in $\mathbb{CP}^{2n}$). Simple connectedness rules this out. So for orthogonality we need Lagrangians that are (at least Floer-theoretically) disjoint.
Free torus-orbits in toric symplectic manifolds give examples of interesting disjoint Lagrangians, but you've doubly disallowed tori! Thanks to Seidel, we know how to go about proving split generation by vanishing cycles of Lefschetz pencils, but these are spheres, and they tend to intersect one another.
We don't have any picture of the Fukaya category of a "generic" symplectic manifold, so it's hard to say how reasonable the conditions you impose really are. For all we know, it could be a common phenomenon that there are few unobstructed Lagrangians, or even none at all.