• 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.