• Discussion with Vivek
• Discussion with Gus

Let $M$ be a smooth manifold, and $L \In T^*M$ be a Lagrangian graph, i.e $\pi: L \to M$ is 1-to-1, say $L = \Gamma_{df}$ for some smooth function $f: M \to \R$. Then, we can compute $\Hom(M, L)$ using Morse theory. There is a rough matching between holomorphic strips and gradient lines on $M$ and $L$. This was Fukaya.

Let $L \In T^*M$ be a branched cover of $M$. Then, Tobias Ekholm says there is nothing wrong with the branched point, one can still do gradient flow there, from that critical point

What's our problem? We have two extra cotangent fibers. These cotangent fibers are like the universal source and sinks.

• Trouble, if we only modify a little bit around the neighborhood of the fibers, then we should get a string of intersections. We somehow turn the cotangent fiber direction flat using the graph. (Hmm, Fukaya's Morse tree is used by Nadler-Zaslow to turn homs between Lagrangians into homs between sheaves. And somehow these multi-layer Lagrangians, should be turned into sheaves (local systems) as well. The question is: which one?

Path Conormal Lagrangian Here is one crazy idea. Now, what did Nadler-Zaslow do? They created some standard Lagrangians in the cotangent bundle, exterior conormal to the open guys, and ask those guys to hom (as in infinitesimal category) to the given Lagrangian. They are pretty vertical near the boundary. What can we do there? Here is an idea, given a smooth imbedded path, we can take the conormal of the path to turn it into a Lagrangian. Now, let's assume this Lagrangian is really nice.

Read the paper by Kawai-Takei, about algebraic singular perturbation.

Here, they definitely need algebraic spectral curve, and they don't mention at all what is the metric to define the gradient flow.

And they can only do rank two case.

Puzzle: what should be the case.

• Given a differential equation on a curve, the solution should be a local system.
• Now, introduce $\hbar$ to this equation. Send $\hbar \to 0$, get WKB approximate solution, which is a formal series in $\hbar$ and may not converge honestly. (if we only keep the leading term, then it is only an approximation, not exact solution at all. So really, all you have is a formal solution)
• I don't know what do you mean by exact WKB. This is not resurgence of Kontsevich-Soibelman, where their simple resurgence data does not have solution to diff-eq (?no, they do have DQ-mod over $\C[[\hbar] ]$ what do you mean they don't have formal solution?) ok, what I want to say is, they don't tell me how to do 'Borel resummation'. Well somehow they do, say, anytime someone give you a formal $\hbar$ series, you can define the tautological resummation, which is

$$ I(\hbar) = \sum_{k=0}^\infty c_k \hbar^k \leadsto \hat I(u) = \sum_{k=0}^\infty c_k u^k / k! $$ where the relation between the two function is that, suppose $\hat I(u)$ admits analytic continuation, then $$ I(\hbar) = “\int_0^\infty” \hat I(u) e^{-u/\hbar} d(u/\hbar) $$ I am deliberately vague about which direction the integral should be done, since this is a formal integral.

Now, I know I am wrong. Given this holomorphic Lagrangian, we have the holomorphc Legendrian lift, then over every point in hte base, we can have many points in the fiber Legendrian. For simplicity, we have exact Lagrangian. Locally, over a small ball of $p$, if we pick a Legendrian branch $b$ up there, we can construct $e^{f_b(z)/\hbar} I_b(\hbar, z)$, where we have phase function $f_b(z)$ and amplitude $\varphi_b(z)$. Now we can play the KS formalism for each fixed $z$. If you only know the spectral curve locally at that branch (well by analytic continuatino, you should know everything), you can still do WKB and get formal solution. I am not sure if the Borel plane is the actually the $J^0 C$ fiber. If everything works all, everyone consistent, then we do have Borel function $\hat I_b(u, z)$.

We can ask, does the Borel resummed function $\hat I_b(u,z)$, as a function of $u$, satisfies any equation? don't worry. Assume that you have full analytic continuation of $\hat I_b(u,z)$, and you picked a path in the $u$ space, compatible with the $\hbar$ phase choice so that $u/\hbar$ goes to infinity. You then, just integrate that holomorphic function $\hat I_b(u,z)$ along that path, to get an actual honest convergent solution.

OK, say, pointwise, you have many ways to cook up honest convergent $\hbar$ solution. You have local solution space with a canonical lattice labelled by thimbles. maybe, for each generic $\theta \in S^1$ a favorite basis.

Now, the problem is that, how do they talk to each other?

last time we were talking about $\hbar$-differential equation, and constructing solutions. Conceptually, given $\hbar \in \C^*$, given a point on the holomorphic Legendrian, given a path in the fiber avoiding the Legendrian to infinity compatible with the half-space defined by $\hbar$, we may perform some fiber-wise integration. And we may do it locally in a neighborhood of $z$.

Consider stokes ray. Consider the $\hbar$ circle bundle over $C$. Then we have the universal spectral curve on this manifold, induced by the holomorphic Legendrian in $C \times \C$.

Suppose we are on a stokes curve, that means two phases are of the same size for some $\hbar$. Suppose we can connect the two equal-height points by some path. I assume that we have the period (image) lattice, Where we have the $H_1$ lattice, acting by translation.

The question is, how to relate $\hbar$-Stokes curve, to $\hbar$ holomorphic curve. Ah, say $\hbar$ is the twister space parameter? or just $\C^*$ variable.

yes, you can find spectral network is some sort of holomorphic disk weaves.

But, what is the space of solutions?