I am trying to understand the definition for the category of A-branes for the mirror to the projective line described in this paper by Ballard, Meet homological mirror symmetry (the construction is in section 3.2) and I'm starting to think that I'm missing something. The construction starts like this: the mirror to $\mathbb P ^1$ can be taken to be the Landau-Ginzburg model \begin{align} W:& \mathbb C^* \to \mathbb C \\ & z\mapsto z+\frac1z. \end{align}
We can put a symplectic form on $\mathbb C^*$, defined by $i\frac{dz\wedge d\bar z}{z\bar z}$, and look at the general fiber which, if I understand correctly, should be two points, while the critical points of W are $\pm 1$, and its critical values are $\pm 2$.
It is claimed at this point that we can use the symplectic form to split the tangent space to the domain into the tangent space of the fiber (which has rank 0?) and the symplectic orthogonal (the whole tangent bundle?). From here on I have a very hard time understanding the construction and the notation (up until Definition 3.7 excluded). The upshot is that the category of A-branes on this LG model is equivalent to the category of modules over the path algebra of a certain quiver $Q$, which should appear from considering two Lagrangian submanifolds (called vanishing thimbles associated with the critical points). Can someone help me understand how these Lagrangian are constructed, and/or give me a good pointer to other references on this?