\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
example 0.0.1
Consider the symplectic manifold \(X=T^*L\).
Consider a closed one form \(\eta \in \Omega^1(L, \RR)\).
Use this to create a Lagrangian isotopy
\begin{align*}
\li_t: L\times [0, 1]\to& T^*L && (q,t) \mapsto& t\cdot \eta_p
\end{align*}
between the zero section and the graph of \(\eta\).
To each loop \(\gamma\in L\), look at the cylinder \(\li_t\circ \gamma:S^1\times I\to X\).
This can be explicitly parameterized by
\begin{align*}
c: S^1\times I \to& T^*L&&
(\theta, t) \mapsto& (\gamma(\theta), t\cdot \eta_{\gamma(\theta)}).
\end{align*}
The cotangent bundle has an exact symplectic form, \(\omega=d\lambda\), so computing the flux can be simplified by using Stoke's theorem:
\begin{align*}
\Flux_{\li_t}(\gamma)=&\int_{c}\omega=\int_{c}d\lambda \\
=&\int_{S^1\times \{1\}}c^*\lambda - \int_{S^1\times\{0\}}c^*\lambda \\
=&\int_{S^1}\gamma^*\eta=\eta(\gamma)
\end{align*}
So \([\Flux_{\li_t}]=[\eta]\in \Omega(L)\).