The Liouville manifold $T^*S^1$ is said to be "mirror" to the complex variety $C^*$. (see for instance lecture 7 here: http://math.columbia.edu/~topology/Eilenberg_lectures_fall_2016)
This is manifested in the fact that the wrapped Fukaya category of $T^*S^1$ is quasi-equivalent to (a dg enhancement of) the category of coherent sheaves on $C^*.
Given a fiber $F_x \subset T^*S^1$, one can compute that $\operatorname{Hom}_{\operatorname{Fuk}}(F_x, F_x)= \mathbb{C}[x, x^{-1}] = \operatorname{Ext}(\mathcal{O}_{\mathbb{C}^*},\mathcal{O}_{\mathbb{C}^*})$.
So we say that the fibers are "mirror" to the structure sheaf.
Two (related) questions that have been bothering me:
(i) is there a systematic/functorial way to construct a functor from $\operatorname{Fuk}(T^*S^1) \to \operatorname{Coh}(C^*)$? My understanding is that the "Family Floer theory" program constructs this in the setting of compact toric varieties, but this story does not seem to apply here on the nose (e.g. it involves rigid analytic geometry and does not involve wrapped Fukaya categories, at least not in the papers I found).
(ii) the above story seems to rely a lot on the fact that the Fukaya category is defined over $\mathbb{C}^*$. In general, the Fukaya category is defined over a Novikov ring; since $T^*S^1$ is exact, one can also use $\mathbb{C}$-coefficients using a rescaling "trick" described on the top of p7 of these notes: https://arxiv.org/pdf/1301.7056.pdf
Does the story still work with Novikov coefficients? What changes?
