\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
definition 0.0.1
Let \(\pi: X\to \CC\) be a symplectic fibration.
Let \(\gamma:[0, 1]\to \CC\) be path with \(\gamma(0)\) a critical value. Let \(p\) be the critical point above this critical value.
Suppose \(\gamma(t)\) avoids critical values for \(t\neq 0\).
Let \(W^{-1}(\gamma)\) be the collection of fibers above the path \(\gamma\).
Consider the map \((\phi_{\gamma}^{t})^{-1}:W^{-1}(\gamma)\mapsto X_{\gamma(0)}\) given by parallel transport.
Then the thimble of \(\gamma\) from \(p\) is a Lagrangian disk \(D^n_\gamma:= (\phi_{\gamma}^{t})^{-1}(z)\subset W^{-1}(\gamma)\subset X\).
The vanishing cycle of \(\gamma\) is a Lagrangian sphere \(S^{n-1}_\gamma (\phi_{\gamma}^{1})^{-1}(z)\subset X_{\gamma(0)}\), which may also be identified with \( D^n_\gamma\cap \pi^{-1}(\gamma(1))\).