Peng Zhou

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blog:2023-02-13

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 $L$.

blog/2023-02-13.txt · Last modified: 2023/06/25 15:53 by 127.0.0.1