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--- blog:2023-03-23 [2023/03/23 23:59] – [About comparing exact WKB] pzhou
+++ blog:2023-03-23 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 38: / Line 38: @@
 Now, the problem is that, how do they talk to each other?
+==== exact WKB and holomorphic disks ====
+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?