Let $(M,\omega)$ be a symplectic manifold and $L \subseteq M$ be a compact Lagrangian in M. My question is what is a geometric/natural representative for the unit of the Floer cohomology $HF(L,L)$? Or what is a naive reason why one can expect that the Floer cohomology is unital in good cases?
I know that there is technicality about when one can use what as the coefficient ring but let's just assume $(M,\omega)$ and $L$ satisfy reasonable assumptions and come with an extra structure, a perturbation, etc.. From my understanding, the Floer cochain complexes is freely generated by points in the intersection $L \cap \phi^1(L)$ where $\phi^1$ is a time-$1$ Hamiltonian isotopy. The product structure $$\mu^2 : CF(L,\phi^1(L) ) \otimes CF(\phi^1(L),\phi^2(L)) \rightarrow CF(L, \phi^2(L))$$ is given by counting $\deg 0$ pseudo-holomorphic discs having three marketed points with conditions correspond to the points.
My question is how can one expect this definition produce a unital ring and how can one know which points should give the unit? Any reference or answers with any assumption are welcome.