TITLE: Comparison between Hamiltonian Floer cohomology and Lagrangian Floer cohomology of the diagonal
QUESTION [15 upvotes]: Let X be a compact symplectic manifold with a form $\omega$. And $X \times X$ is equipped with the symplectic form $(\omega,-\omega)$. The diagonal $\Delta:X \mapsto X \times X$ is a Lagrangian submanifold. So, in this question, Hochschild (co)homology of Fukaya categories and (quantum) (co)homology. Tim Perutz says "PSS is a canonical ring isomorphism from QH∗(X) to the Hamiltonian Floer cohomology of X, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal." I have no doubt that this second assertion is straightforward, since I have consulted a couple of references and no one spells this out. But, I don't quite see it. I believe what I am missing is the relationship between holomorphic strips in $X\times X$ and holomorphic cylinders in X. (Edit: I would also like to understand the comparison of the product structures too) 
Edit: here is a rough geometric idea which might have something to do with the truth. I want to assume that my Hamiltonian is time independent and that all orbits are actually fixed points. Given a map of a strip into $X\times X$ the two projections give us two strips into X. The idea is to glue the strips together to form a cylinder which is a map into X. Of course, this doesn't take into account issues of compactifications and so on... Anyways, if someone would be happy to spell it out I would appreciate it.

REPLY [9 votes]: Michael's answer describes the sort of thing I had in mind when I brusquely dismissed this as "straightforward" in the earlier, big-picture-style answer that Daniel quoted. (Michael, thanks for explaining.) 
This isomorphism intertwines the product structures: think of cutting a pair of pants into two identical pieces.
I'd like to add a cautionary note. Before you can compare Hamiltonian Floer homology to the Lagrangian Floer self-homology of the diagonal, you have to define both groups. The difficulties in doing so are not identical. In the Lagrangian case, disc-bubbling potentially obstructs the construction.
For particularly simple manifolds - monotone ones, where $c_1$ is a positive multiple of $[\omega]$ when evaluated on spherical homology classes - both groups have reasonably straightforward definitions. This is a setting where the isomorphism really works.
For Calabi-Yau manifolds, the definition of Hamiltonian Floer homology is still not too complicated (it involves understanding transversality of evaluation on moduli spaces of somewhere-injective holomorphic spheres). But Lagrangian Floer homology of a Maslov-index-zero Lagrangian in a Calabi-Yau is hard to construct. So far as I know, one generally has to work rationally and to compute obstructions in the sense of Fukaya-Oh-Ohta-Ono. Presumably, in the case of the diagonal these obstructions vanish. 
So there is a definite reason for preferring the Hamiltonian Floer homology model when trying to formulate things precisely: it bypasses the obstructions, and (for Calabi-Yau manifolds) the virtual transversality theory.