Pavel Putrov came and give a talk, about Kapustin-Witten equation.

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?

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.

What do I want?