\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
conventions for Symplectic topology
\begin{convention}
We take the following conventions:
- The canonical 1-form on the cotangent bundle is \(\lambda_{can}=\sum_{i=1}^n p_i dq_i\).
- We will use \(\jmath=\sqrt{-1}\).
- A compatible almost complex structure satisfies \(\omega(v, Jv)>0\).
- The Hamiltonian vector field is the vector field which satisfies:
\[\iota_{V_H}\omega = - dH\]
- The symplectic structure on the cotangent bundle is \(d\lambda_{can}=\sum_i dp_i\wedge dq_i\).
- The complex plane \(\CC\) is identified with the cotangent bundle by \((q, p)\mapsto q-\jmath p\)
- The Floer action functional is
\[A_{H_t}(\gamma):=-\int_\gamma \lambda+\int_0^1 H_t(\gamma(t))\,dt\]
giving us the Floer equation:
\[\partial_su +J_t(\partial_tu-X_{H_t})=0.\]
The input end of a Floer trajectory is the limit as \(s\to\infty\), and the output end is the end as \(s\to-\infty\). In this convention, Floer trajectories are gradient flow lines of the Floer action. The differential on symplectic cohomology is given by counting downward flow lines.
From this it follows that :
- The standard almost complex structure on \(\CC\) is compatible with the canonical symplectic form on \(T^*\RR\), as \(\omega(\partial_p, J\partial_p) = \omega(\partial_p, \partial_q) = 1\). The standard symplectic form on \(\CC\) is \(d d^c\left( \frac{1}{2}(x^2+y^2)\right)\).
- The Hamiltonian vector field is related to the gradient by
\[\iota_{J\grad_g H}\omega(v)=\omega(J\grad_gH, v)= -\omega(v,J\grad_gH)=-g(v, \grad_gH)=-dH(v).\]
- The symplectization of a contact manifold \((M, \alpha)\) is \(\RR\times M\). It is equipped with the symplectic form \(d(\exp(r) \alpha)= \exp(r) (dr \wedge \alpha + d\alpha)\). The Reeb vector field agrees with the Hamiltonian vector field of \(\exp(r)\) as
\[ \iota_{R_\alpha}d(\exp(r)\alpha)=-\exp(r) dr = -d(\exp(r)).\]
\end{convention}