I'm sorry in advance for such a wordy question! Someone in-the-know could probably skip ahead to "QUESTION" below. If no one bites, I will try to shorten it.. thanks!

Let $(M,\omega)$ be a compact, monotone symplectic manifold, and fix two Lagrangians $L_1,L_2\subset M$ for which the Lagrangian intersection Floer homology $HF(L_1,L_2)$ is well-defined.

Suppose $x,y \in L_1\cap L_2$ are two generators for $CF(L_1,L_2)$, and let $\mathcal{M}(x,y)$ denote the moduli space of parameterized holomorphic strips (disks) from $x$ to $y$, with boundary on $L_1$ and $L_2$. By fixing boundary basepoints $b_i$ and interior basepoints $p_j$ on a model strip, we get evaluation maps $ev_{b_i}$ and $ev_{p_j}$ from $\mathcal{M}(x,y)$ to $M$, which send a map in $\mathcal{M}(x,y)$ to its image at the chosen basepoint.

Here's my situation: I've fixed various chains {$c_i$} in $L_1$, $L_2$, and $M$; they're not exactly closed, but their boundaries are subject to certain constraints so that IF I can assume the images of $ev_{b_i}$ and $ev_{p_j}$ "behave like" chains in $L_i$ and $M$, respectively, with boundary given in the obvious way by either the boundary of the strips themselves, or by gluing of strips a la Gromov compactness, then by intersecting with the {$c_i$}, I get numbers which, while they are not independent of my choice of chains (due to the boundaries of the chains), can be put together to get a map $CF(L_1,L_2)\to CF(L_1,L_2)$ whose dependency on the chains reveals it to be a chain map.

This is analogous to a way one might hope to get a "quantum cap product" $\cap: QH(M)\to \text{End}(HF(L_1,L_2))$, where we use just one interior marked point $p_0$, and we replace my {$c_i$} with a single chain $c_0$ which is actually closed. However, I haven't found a way to fit my construction exactly into the quantum cap product construction.

QUESTION: is there a detailed account of such a "pseudocycle open Gromov-Witten theory" which I could hopefully generalize to my case? It is mentioned on page 29 of

http://www-math.mit.edu/~auroux/papers/slagmirror.pdf

but Seidel's paper, referenced in the above, does not give a pseudo-cycle definition, as Auroux uses, but rather a definition in terms of $HF(id)$. Are there any potentially impassable pitfalls in generalizing the pseudo-cycle apparatus for closed Gromov-Witten theory?

SIDE QUESTION: It seems to me that if one naively follows a psuedo-cycle recipe for the quantum cap product $\cap: QH(M)\to \text{End}(HF(L_1,L_2))$, then one runs into difficulties if the class $[c]\in QH(M)$ intersects $L_1$ and $L_2$, namely, the invariant could change if one changes $c$ by a homology, which intersects the boundary of $ev(\mathcal{M}(x,y))$ in a piece which comes from the boundary of the strips, and not the gluing-boundary (the latter is accounted for by the differential in $(HF$)). If anyone understands what I mean, can you explain why this isn't a problem.