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--- blog:2023-03-21 [2023/03/22 05:35] – [Generalized Riemann-Hilbert Correspondence] pzhou
+++ blog:2023-03-21 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 28: / Line 28: @@
 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), then, what is my dream? My dream
+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