Let $L$ be a Lagrangian submanifold of a closed symplectic manifold $X$. What I gather from Costello (see specifically $\S$2.5 there), is that one expects to have a morphism of closed TCFT's

$\tag{1}\label{map} \Phi \to \Psi$

where $\Phi$ is a closed TCFT derived from the unital CY $A_\infty$ algebra $\mathcal A$ of $L$. It sends the generator object to the reduced Hochschild chain complex of $\mathcal A$,

$\Phi(1) = \underline{CC}_* \mathcal A$.

$\Psi$ is the closed TCFT derived from Gromov-Witten theory of $X$,

$\Psi(1) = C_*(X)$.

($\ref{map}$) means that for every pair of non-negative integers $a,b$ there should be a (strictly commutative) square of chain maps $\require{AMScd}$ \begin{CD} \tag{2}\label{square} \mathcal M(b,a) \otimes \Phi(a) @>>> \Phi(b)\\ @V V V @VV V\\ \mathcal M(b,a) \otimes \Psi(a) @>>> \Psi(b) \end{CD}

where $\mathcal M (b,a)$ is a certain chain model for the hom-spaces in Segal's category.

On the other hand, for the case of the coproduct, which corresponds to the point class $[pt]$ in the pair of pants moduli, $[pt] \in H(\mathcal M (2,1))$, Sheridan (see in particular Figure 2 there) constructs a certain subset $\mathcal C$ of the moduli spaces of disks with two interior markings and any number of boundary markings which seems to be a homotopy "filling in" the square \begin{CD} \mathcal \Phi(1) @>>> \Phi(2)\\ @V V V @VV V\\ \mathcal \Psi(1) @>>> \Psi(2) \end{CD}

More precisely, Sheridan uses $\partial \mathcal C$ to prove that the dual, closed-open map is a map of algebras (after taking homology), but it seems that the same $\mathcal C$ can also be interpreted as a homotopy for the square above.

Is there a recipe for generalizing the construction of the subset $\mathcal C$, to demonstrate the commutativity of other squares obtained from ($\ref{square}$)?

I'd be content with a heuristic argument, that would at least help to see the construction of $\mathcal C$ as derived naturally from general geometric considerations, or even just a few other examples of what $\mathcal C$ should be replaced with for other classes in $H(\mathcal M (b,a))$.


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