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--- blog:2023-03-06 [2023/03/07 06:32] – pzhou
+++ blog:2023-03-06 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 25: / Line 25: @@
 On the other hand, we can pass to the open neighborhood of the Legendrian. This is GPS restriction of sheaf. On the co-core level, we have the co-variant inclusion, so we have $A_0 \to A$, where $A_0$ is the endormorphisms of cocores inside the small guy, and $A$ is the endomorphism of cocore in the big guy. And, we get $A-mod \to A_0-mod$, hence ** the functor induces map on moduli space **.
+Let's be concrete, suppose we are in $S^5$. The augmentation variety is certainly very complicated, since Legendrian DGA was. A rank-1 module of the Legendrian DGA probably comes a simple Lagrangian filling. And you could have different ones.
+How does Legendrian weave mutation work? Let's think globally first. These are explained in Schrader-Shen-Zaslow.
+Now, what do I want? These boundaries of Legendrian disks labels generators in the Lagrangian disks.
+In terms of coordinates on local system on the Legendrian surface, these X variables are the holonomy of the $\C^*$ local systems.
+The paper of SSZ is so rich in details, it is hard to read. (but great!) read section 4.2, which is about quantization, and section, which is purely about cluster algebra.
+Read section 7.2 of Casal-Zaslow.
+Read Thm5.13 in STWZ. Why we have classical mutation formula? Where does bipartite graph come from?
+very much confused. even if we have a non-exact lagrangian with Legendrian ends, what can you say.
+where does classical cluster transformation come from? What does it mean? It tells you which local system go to which local system.
+Why? Quantum torus is about U(1)-skein. Can we do skein-quantum torus? Basically, from a genus $g$ surface. Previously we had 2g cycles, hence a $2g$-dimensional complex torus.