====== 2023-02-13 ======
Vivek gave a talk, and talked about stuff during dinner. 

  * What you can do with skein-on-brane, and higher genus open Gromov-Witten invariant. 

===== skein, moduli of brane =====
$\gdef\lcal{\mathcal L}$
==== skein ====

 Let $\lcal \In \C$ be a $2 \dim_\R$ Legendrian in a 5 dimensional contact manifold. Let $L_\infty = \lcal \times \R \In \C \times \R$ be corresponding Lagrangian in the symplectization. One can compute curves bounded by $L_\infty$ and Reeb chords. 

The genus $0$ count is well-defined, but higher genus count is valued in skein. (not sure if this is true)

Consider a capping. Liouville manifold $W$ of $C$. And **non-exact** Lagrangian $L$ of $\lcal$. Consider holomorphic curve, bounded by $L$. Maybe also with a Reeb chord at infinity. 

We have an action of skein algebra (on the Legendrian times R), on the skein module associated to the internal Lagrangian.


==== moduli ====
Consider the moduli of rank 1 local system on $\lcal$, $Loc(\lcal)$. Under certain assumption, we have $Loc(\lcal)$ is an algebraic symplectic space. 

For example, if $\lcal$ is a Legendrian torus, then its moduli space is $(\C^*)^2$. 

Let $L$ be a non-exact Lagrangian bounded $\lcal$. Then, restriction $Loc(L) \to Loc(\lcal)$ has image being a holomorphic Lagrangian. 

We can quantize this holomorphic Lagrangian to get a DQ module. 

==== open GW invariant ====
skein algebra acting on the skein module. Certain operator annihilate the module, and that gives some equations.

Then, the solution of certain equation is related to open GW invariant on $L$.