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

\(A\)-side Landau-Ginzburg model

SECTIONS
On this side of the mirror, \(Y\) will be a Kähler manifold, along with a holomorphic function \(W: Y \to \CC\) which is: The objects of the Fukaya-Seidel category \(\FS(Y, W)\) are Lagrangian submanifolds \(L \subset Y\) such that \(W|_L\) is a fibration over \(\RR_+\) outside of a compact set.
figure 0.0.1:A Lagrangian submanifold in a Fukaya-Seidel category needs to have prescribed behavior going off to infinity.
To take the Lagrangian intersection Floer cohomology between two such Lagrangians, we need to perturb the Lagrangians at infinity; the perturbation we take will rotate one Lagrangian at infinity. As a vector space, \(\hom(L, K) = \CC(\langle L \cap \phi(K)\rangle)\).
figure 0.0.2:To obtain transversality, the Lagrangian submanifolds in the Fukaya-Seidel category are pushed off one another with respect to the projection \(W: Y \to \CC\).
The differential and product structure on hom-spaces is given by counts of pseudo-holomorphic disks.

1: Comparison to \(B\)-side

Let \(\CritVal(W) \subset \CC\) 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 \(W^{-1}(B(c, \epsilon)) \to B(c, \epsilon)\) to a fibration over \(\CC\).
figure 1.0.1:A small neighborhood in the base of a symplectic LG model can be used to build another LG model.
Define \(\FS_c = \FS(Y_c, W)\).

definition 1.0.2

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 \(\FS_c \to \FS(Y, W)\) by extending Lagrangian submanifolds over the vanishing path using symplectic parallel transport.
figure 1.0.3: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 1.0.4

Consider the symplectic Landau-Ginzburg model whose critical points are arranged as follows:
figure 1.0.5: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 \(\FS(Y, W) = \langle \FS_1, \FS_2, \FS_3 \rangle\).

2: Exceptional collections

Suppose additionally that \(W^{-1}(c)\) has an \(A_1\) singularity (a node) then \[\FS_c = D\Coh(\text{pt}) = D\Vect\] and the generating object is called the vanishing thimble associated with the point. So if all critical points of \(W\) are more (i.e., \(W\) is a Lefschetz fibration), then \(\FS(Y, W)\) admits a full exceptional collection. If \(W\) has isolated singularities, they can be ``Morsified'' by a small perturbation (in the symplectic category). From this, we obtain the following principle: if \((Y, W)\) is a symplectic Landau-Ginzburg model and \(W\) has isolated singularities, then \(\FS(Y, W)\) admits a full exceptional collection. This tells us that many examples arising from mirror symmetry will not have isolated singularities, as their mirror spaces will not have full exceptional collections! However, we still have the following expectations. When \(X\) is Fano, and \((Y, W)\) is its mirror, then we expect \(W\) will have some isolated singularities, and an additional critical value \(c_{\text{Ku}}\) with non-isolated singularities with \[\text{Ku}(X) = \FS_{c_{\text{Ku}}}.\]