\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: Comparision to \(B\)-side

Comparision to \(B\)-side

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let \(\CritVal(W)\subset C\) be the subset of critical values of \(W\). For each \(c\in \CritVal(W)\), we have a ``smaller'' Landau-Ginzburg model \(Y_c\) by trivially extending the \(W^{-1}(B(c, \eps))\to B(c, \eps)\) to a fibration over \(\CC\).
figure 0.0.1:A small neighborhood in the base of a symplectic LG model can be used to build another LG model.
Define \(\FS_c=\FS(Y_v, W)\).

definition 0.0.2

Let \(c \in \CritVal(W)\) be a critical point. A vanishing path for \(c\) is a path \(\gamma: [0, \infty) \to \CC\) such that A vanishing path determines an embedding \(\FS_c\into \FS(Y, W)\) by extending Lagrangian submanifolds over the vanishing path using symplectic parallel transport.
figure 0.0.3:A vanishing path determines a method for extending Lagrangians belonging to the Fukaya-Seidel category near a critical value to the Fukaya-Seidel category of \((Y, W)\).
If we choose ``disjoint'' vanishing paths, we get a semi-orthogonal decomposition. \[\FS(Y, W) = \langle FS_1, \ldots, FS_n\rangle \]

example 0.0.4

Consider the symplectic Landau-Ginzburg model whose critical points are arranged as follows:
figure 0.0.5:A set of vanishing paths for the Lefschetz fibration \(W(z_1, z_2) = z_1 + z_2 + (z_1z_2)^{-1}\)
We obtain a semi-orthogonal decomposition \(\FS(Y, W) = \langle \FS_1, \FS_2, \FS_3 \rangle\).