Darn it, time really flies. No recollection whatsoever what happened last few days. Maybe doesn't matter.

I am recently very interested in the relationship between solving differential equation and doing Floer theory.

a.la Kontsevich-Soibelman

Given a holomorphic symplectic manifold $(M, \omega)$, there are two ways to get a family of categories, $Fuk_\hbar(M) := Fuk(M, Re(\omega/\hbar), B=Im(\omega/\hbar))$ and $DQ_\hbar(M)$. Conjecturally, they are equivalent.

More precisely speaking, we need to encode the 'allowed region' for compatification. Otherwise, the $DQ$-module side has too much freedom.

A few things that makes sense:

• in the setting of spectral curve $\Sigma \In T^*C$ (things still holomorphic). They have spectral networks, shadow of degenerate holomorphic curves.
• In the case for exact WKB, I am totally lost. What are you trying to say? Are you trying to use integrals get equations?

Now, KS gives a conjecture, but as always, there is no ready-made proof strategy. So, what's the situation here?

Our hol symp manifold is $T^*C$, that's for sure. And, then what's our Lagrangian? That's $\Sigma$, hol symp. OK. Then,

• on one side, we consider $DQ_\hbar$-mod, so that in the limit $\hbar \to 0$, has its semi-classical support at $\Sigma$ (what does that mean? taking some associated graded? yes)
• on the other side, we consider $\Sigma$ as a Lagrangian. (that already feels a bit weird. According to KS, one need to rotate $\hbar$, I don't see it here. ).

There are many questions here. First, we can have many $DQ$-module with the same support, and we can have many objects.