\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
example 0.0.1
Let \(V\subset Q\) be a smooth submanifold. The conormal bundle \(N*V\subset T^*Q\) consists of all covectors \((q, p)\in T^*Q\) with \(q\in V\) and \(p(v)=0\) for all \(v\in TV\).
This is always an \(n\)-dimensional submanifold of \(T^*Q\).
We can choose local coordinates \(q_1, \ldots, q_n, p_1, \ldots p_k\) so that \(V=\{(0, \ldots, 0, q_{k+1}, \ldots, q_n)\;|\; q_i\in \RR\}\).
In these local coordinates, \(N^*V=\{(0, \ldots, q_{k+1}, \ldots, q_n, p_1, \ldots, p_k, 0, \ldots, 0)\}\); which is a Lagrangian linear subspace for the symplectic form \(\sum_{i=1}^n dq_i \wedge dp_i\).
The intersection of the zero section and a conormal bundle is \(Q\cap N^*V=V.\)