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--- blog:2023-02-13 [2023/02/14 06:29] – created pzhou
+++ blog:2023-02-13 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 4: / Line 4: @@
   * What you can do with skein-on-brane, and higher genus open Gromov-Witten invariant.
-===== skein, moduli, Lagrangian =====
+===== skein, moduli of brane =====
- $\gdef\lcal{\mathcal L}$  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.
+ $\gdef\lcal{\mathcal L}$
+==== skein ====
-The genus  $0$  count is well-defined, but higher genus count is valued in 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  $L$ .