Differences

This shows you the differences between two versions of the page.

--- 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 $L$.