• Grading
• Orientation
• Index

Let $\Sigma$ be a Liouville domain, with $\hat \Sigma$ be its Liouville completion. Let $D = [0,1] \times \R$, be the base disk with two punctures. Let $\hat X = \hat \Sigma \times D$. Let $s \in \R$ and $t \in [0,1]$, so $t$ is the Reeb parameter, and $s$ is the gradient flow parameter.

We consider two sets of disjoint exact Lagrangians in $\hat \Sigma$, $\alpha$ and $\alpha'$, conical in the ends.

We choose almost complex structure to the same as the exact one, only perturbing in the interior.

The $k$-fold covering curve of $D$ is called $F$. More precisely, $\dot F$ is the curve with boundary punctures, and covers the boundary punctured domain curves. The projection to Liouville domain $\Sigma$ just maps puncture to intersection points.

Given a holomorphic map $u: S \to M$, with a collection of Lagrangian as boundary condition, what does Maslov index mean? First, we construct a rank $n$ sub-bundle of the symplectic bundle $u^* T\Sigma$. When there are Lagrangians on the boundary, we use the tangent space of the Lagrangian (we don't require Lagrangian to be oriented). What to do at the punctures? We interpolate by $J$. The bottom edge Lagrangian is rotated by $J$ to the top