I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as long!)

I'm taking the following set-up directly from page 29 of http://www-math.mit.edu/~auroux/papers/slagmirror.pdf

Let $(M,\omega, J)$ denote a compact symplectic manifold and compatible almost complex structure, and let $L$ be a compact, oriented Lagrangian submanifold of $M$.

Given a class $\beta \in \pi_2 (X, L)$, denote by $\mathcal{M}(L, \beta)$ the space of parameterized $J$-holomorphic maps from $(D , \partial D )$ to $(X, L)$ representing the class $\beta$ (parameterized means we don't quotient by automorphisms of the disc).

For "marked points" $\pm 1 \in \partial D$, and $0\in D$, let $$ev_{\beta,\pm 1} : \mathcal{M}(L, \beta)\to L$$ and $$ev_{\beta,0} : \mathcal{M}(L, \beta)\to M$$ denote the corresponding evaluation maps (sending a $J$-holomorphic disk to its image at the point in question).

Auroux says that up to introducing suitable perturbations (of the Cauchy Riemann equations, I presume) we can assume $\mathcal{M}(L, \beta)$ carries a fundamental chain, and that the evaluation maps can be chosen to be transverse to fixed chains in $L$ and $X$.

QUESTION: Is there any detailed write up of the above claim? (The new work by FOOO?) I'm actually interested in a slightly more general case, where we consider moduli spaces of disks between two different Lagrangians, but I assume the same arguments will work in both cases (given proper assumptions on the Lagrangians and symplectic manifolds in question). Any reference would be appreciated (with any caveats about the reference, if relevant).

Note: Many of the applications of the above constructions can be accomplished with an alternative construction, which examines moduli spaces of disks with extra $S^1$ boundary components, required to lie on generic orbits of a generic Hamiltonian vector field, rather than basepoints required to lie on chains (which is the above method). I think this second approach is technically easier, but I haven't been able to fit what I need into it, unfortunately. But I do have a related QUESTION about this approach: as far as I can see, it is only a substitute to the chain-level approach in the case of interior marked points. Is there an analogous construction providing an alternative to boundary marked points?

Thanks!

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as long!)

I'm taking the following set-up directly from page 29 of http://www-math.mit.edu/~auroux/papers/slagmirror.pdf

Let $(M,\omega, J)$ denote a compact symplectic manifold and compatible almost complex structure, and let $L$ be a compact, oriented Lagrangian submanifold of $M$.

Given a class $\beta \in \pi_2 (X, L)$, denote by $\mathcal{M}(L, \beta)$ the space of parameterized $J$-holomorphic maps from $(D , \partial D )$ to $(X, L)$ representing the class $\beta$ (parameterized means we don't quotient by automorphisms of the disc).

For "marked points" $\pm 1 \in \partial D$, and $0\in D$, let $$ev_{\beta,\pm 1} : \mathcal{M}(L, \beta)\to L$$ and $$ev_{\beta,0} : \mathcal{M}(L, \beta)\to M$$ denote the corresponding evaluation maps (sending a $J$-holomorphic disk to its image at the point in question).

Auroux says that up to introducing suitable perturbations (of the Cauchy Riemann equations, I presume) we can assume $\mathcal{M}(L, \beta)$ carries a fundamental chain, and that the evaluation maps can be chosen to be transverse to fixed chains in $L$ and $X$.

QUESTION: Is there any detailed write up of the above claim? (The new work by FOOO?) I'm actually interested in a slightly more general case, where we consider moduli spaces of disks between two different Lagrangians, but I assume the same arguments will work in both cases (given proper assumptions on the Lagrangians and symplectic manifolds in question). Any reference would be appreciated (with any caveats about the reference, if relevant).

EDIT: let me stress that I'm interested in the simplest cases, eg monotone symplectic manifolds with monotone Lagrangians.

Note: Many of the applications of the above constructions can be accomplished with an alternative construction, which examines moduli spaces of disks with extra $S^1$ boundary components, required to lie on generic orbits of a generic Hamiltonian vector field, rather than basepoints required to lie on chains (which is the above method). I think this second approach is technically easier, but I haven't been able to fit what I need into it, unfortunately. But I do have a related QUESTION about this approach: as far as I can see, it is only a substitute to the chain-level approach in the case of interior marked points. Is there an analogous construction providing an alternative to boundary marked points?

Thanks!

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.

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as long!)

I'm taking the following set-up directly from page 29 of http://www-math.mit.edu/~auroux/papers/slagmirror.pdf

Let $(M,\omega, J)$ denote a compact symplectic manifold and compatible almost complex structure, and let $L$ be a compact, oriented Lagrangian submanifold of $M$.

Given a class $\beta \in \pi_2 (X, L)$, denote by $\mathcal{M}(L, \beta)$ the space of parameterized $J$-holomorphic maps from $(D , \partial D )$ to $(X, L)$ representing the class $\beta$ (parameterized means we don't quotient by automorphisms of the disc).

For "marked points" $\pm 1 \in \partial D$, and $0\in D$, let $$ev_{\beta,\pm 1} : \mathcal{M}(L, \beta)\to L$$ and $$ev_{\beta,0} : \mathcal{M}(L, \beta)\to M$$ denote the corresponding evaluation maps (sending a $J$-holomorphic disk to its image at the point in question).

Auroux says that up to introducing suitable perturbations (of the Cauchy Riemann equations, I presume) we can assume $\mathcal{M}(L, \beta)$ carries a fundamental chain, and that the evaluation maps can be chosen to be transverse to fixed chains in $L$ and $X$.

QUESTION: Is there any detailed write up of the above claim? (The new work by FOOO?) I'm actually interested in a slightly more general case, where we consider moduli spaces of disks between two different Lagrangians, but I assume the same arguments will work in both cases (given proper assumptions on the Lagrangians and symplectic manifolds in question). Any reference would be appreciated (with any caveats about the reference, if relevant).

Note: Many of the applications of the above constructions can be accomplished with an alternative construction, which examines moduli spaces of disks with extra $S^1$ boundary components, required to lie on generic orbits of a generic Hamiltonian vector field, rather than basepoints required to lie on chains (which is the above method). I think this second approach is technically easier, but I haven't been able to fit what I need into it, unfortunately. But I do have a related QUESTION about this approach: as far as I can see, it is only a substitute to the chain-level approach in the case of interior marked points. Is there an analogous construction providing an alternative to boundary marked points?

Thanks!