SympSnip:

example 0.0.1

Let

C^{*}

be equipped with the symplectic form

ω = \frac{1}{2 π} d (\log r) \land d θ

. Let

L_{r}

be the Lagrangian

L_{r} := {z such that \log | z | = r} .

Let

i_{r} : L \times [r_{0}, r_{1}] \to X

be the isotopy between

L_{r_{0}}

and

L_{r_{1}}

. Let

e \in H_{1} (L_{r})

be the fundamental class of

L

. The amount of flux swept out between these two Lagrangian submanifolds is given by the difference of their

r

-values.

{Flux}_{i_{r}} (e) = \int_{S^{1}} \int_{r_{0}}^{r^{1}} \frac{1}{2 π} d (\log | z |) \land d θ = r_{1} - r_{0} .

For this reason, the value

r

is sometimes called the ``flux coordinate'' of the fibration

C^{*} \to R

. The flux class therefore provides a nice parameterization of the space of Lagrangian submanifolds in this example.