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If you have a closed monotone symplectic manifold $M$, then to any pair of closed monotone Lagrangian submanifolds $L_1$, $L_2$ you can associate (modulo some bubbling assumptions) a $\mathbb{Z}_N$-graded vector space $HF(L_1, L_2)$ (where $N$ is the Maslov number). I think with some additional choices you can even make it $\mathbb{Z}$-gradded (see, for example, Seidel).

Now, there is a well-known mantra saying that "Floer cohomology is infinite-dimensional version of Morse cohomology". This mantra is usually illustrated by comparing the constructions of the complexes for Hamiltonian fixed-point Floer cohomology and Morse cohomology.

My question: is there some expository note in which this mantra is illustrated for Lagrangian intersection Floer theory? So there would be some sort of complex based on intersection points between two submanifolds of complementary dimension and the differential would be given by counting gradient lines between intersection points. I just want to use it in my lectures, and I am too lazy to write it down myself.

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  • $\begingroup$ This is present in Floer's 1988 article The unregularized gradient flow of the symplectic action. The infinite-dimensional manifold is the space of paths with endpoints on the two Lagrangians, and the functional is the symplectic action defined on some covering space (which descends to a $\Bbb R/\Bbb Z$-valued functional on the path-space). $\endgroup$
    – mme
    Commented Oct 4, 2018 at 20:36

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