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--- blog:2023-03-06 [2023/03/07 05:53] – created pzhou
+++ blog:2023-03-06 [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 ===== Witten's path integral =====
+What does Witten want to do? Analyltic continuation of Chern-Simons.
+We want to get finite dimensional approximation. Intersection of holomorphic Lagrangian in a Hitchin system. Then, we do Floer theory. Or, should I do holomorphic Floer theory?
+What do they have, and what do they want? What is this Z-hat business?
+===== About cluster mutation and skein relation =====
+I need some more input from Mathias, maybe.
+What's the problem? OK, you found the interpolation Lagrangian, that connects the torus. The whole space have geometry like $S^1 \times I \times I$. It is a non-exact Lagrangian with two exact Legendrian ends, both Legendrian are torus.
+One puzzle is that: if this is the whole picture, then the story is trivial. The setup is too local to have interesting invariant.
+What did STW teach me:
+  * There are some co-oriented circles on the surface, where we attach disks.
+  * There are ways to mutate these disks, and know how the other circles mutate.
+  * The cluster transformation rule, seems to be about cluster x-variable? since the co-orientation get changed. Here, we do need to compare a bit with the old surface, so let's compare.
+What is the 'augmentation variety'? What is the character variety? The input is some smooth Legendrian $L$ in some contact manifold $M$. Suppose $M = T^\infty X$, and $L = F^\infty$, for some Lagrangian $F$,like, we had fillings. Then, we can ask how does linking disk to $L$ wrap to each other with $L$ as the stop? That's the Legendrian DGA for $L$, in the Weinstein-filled contact manifold.
+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.