So Mauricio asks, what is the 'A-side' window? I thought I know the answer, a cheap one, not in the form of a generator, but just in terms of a subcategory generated by certain classes of objects. And I am not sure if it is right.

so far this is only a dream, but might be a good one.

1. we know Hori-Vafa mirror for $\C^n$. It is $Y=(\C^*)_A^n, W_Y=\sum_i y_i$ where $y_i$ are coordinates on $Y$.

2. we know the mirror for $\C^n / T$ for some (complex) subtorus $T \In (\C^*)_B^n$, whose mirror is $\pi_A: Y \to T^\vee$. Then we do the log unfolding for $\pi_A$, that defines a map $$ \tilde \pi_A: \wt Y \to Lie(T^\vee) $$ We have the pullback super-potential $\wt W_Y$ on $\wt Y$ from $Y$.

3. Somehow, we have an extra factor $Lie(T^\vee)^\vee = Lie(T)$, where the 'sigma' field and 'gamma' contour lives. Doing an integral along $\sigma \in \gamma$ with exponential factor $\exp(i \sigma(\wt \pi - t))$ forces $\wt pi = t$.

4. However, we don't want to just do integral. We want to have A-model objects.