Generator of a Fukaya category with certain properties 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...
 A: Even the condition that you have a collection of Lagrangians which are categorically orthogonal and each with $HF^\ast(L)=H^\ast(L)$  as an  $A_{\infty}$ algebra is unreasonable:  There could a priori be symplectic manifolds with such Fukaya categories, but at the present state of knowledge, it is unlikely that we would be able to prove it since all methods for proving that a certain collection of Lagrangians generate the Fukaya category ultimately pass through a split-generation result for the diagonal (even the one used in Seidel's book can be interpreted in that language).  On the other hand, the category you describe does not have such a resolution (you can see this by noting its Hochschild cohomology is a direct sum of homologies of free loop spaces and hence is of finite homological dimension).
A: 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.
