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 different Lagrangian brane with the same object (here, since it is holomorphic, so we are just going to put local system on $\Sigma$).

What I don't understand is that, I see Neitzke and those WKB people, really have a collection of nice, sophisticated equations, the quantized Lagrangian (where do they come from?) Nadler says there is a unique quantization for a Lagrangian $\Sigma$, sometimes, but why?

Then, there is the so called, non-linear Ricardi equation, which give you higher order correction for $\lambda$, the phase function. That feels wrong, it feels like, we are taking the WKB too seriously, we are trying to provide perturbative (asymptotic) expansion. But why do we even care about those higher terms in $\lambda$? Aren't we trying to come up with non-perturbative.

Hmm, what are we doing? I am a physicists (pretend), and I want to solve equations. I have a parameter $\hbar$. Suppose we have a supermachine, that can tell me precise solutions, for whatever $\hbar$ value (say small enough). What do I want? What am I doing? In terms of relative homology, we have just a monodromy operator. For each angle, we have a prefered basis. There is a change of basis matrix for each 'wall-crossing'. Yes, this is in agreement with the GHKK philosophy, we have a fixed vector space, just different basis. or Bridgeland stability condition, something changes with angles. whatever. We have local flat sections, and global one. The many local one is breaking up the big global oen into small steps.

What DQ-module say? I am quite confused. Given a differential equation with $\hbar$ in it, I can do two things; one is specialize $\hbar$ to a number, say $1$, then solve the equation; the other is to pretend $\hbar$ is a formal small parameter, and obtain formal solution. If you have a sequence of $\hbar \to 0$, and thus a sequence of solutions, one for each $\hbar_i$, such that the seq of solutions converge in some sense. (like fix some macroscopic quantity, like energy, or momentum). Indeed, this is the home for the limit. Any limiting process will register a solution in $\C[[\hbar] ]$. (possibly also the other way around). But this 'remember everything about the leading term order' can be a lossy process, because as we change $\hbar$.

OK, I don't know how to see that, we need to count disk