The main references are [], [].
Let \((X, \omega)\) be a \(2n\) dimensional symplectic manifold. Fix an
almost compatible complex structure \(J\) on \(X\).
[Novikov Field]
The Novikov field with \(\mathbb C\)-coefficients is the ring of formal power series
\[\Lambda_{\geq 0}:= \left\{\sum_{i=0}^\infty a_i T^{\lambda_i}\st a_i\in \CC, \lambda \in \RR, \lim_{i\to\infty} \lambda_i = \infty\right\}.
\]
See also: the Novikov Ring
[Lagrangian Intersection Floer groups]
Let \((X, \omega)\) be a symplectic manifold. Let \(L_0, L_1\subset (X, \omega)\) be Lagrangian submanifolds whose intersection in \(X\) is transverse. The Lagrangiain intersection Floer complex is the vector space
\[\CF(L_0, L_1):= \bigoplus_{q\in L_0\cap L_1} \Lambda \langle p\rangle \]
equipped with an endomorphism \(\partial: \CF(L_0, L_1)\to \CF(L_0, L_1)\) whose structure coefficients are given by counts of psuedoholomorphic disks. For \(p,q\in L_0\cap L_1\) and \(\beta\in H_2(X, L_1\cup L_2, \ZZ)\), consider the moduli space of strips:
\[\mathcal M(p, q, \beta):= \left\{u: \RR\times [0, 1]\to X \middle | \begin{array}{cc} u(s, t)\to p & \text{ as } s\to-\infty \\ u(s, t)\to q & \text{ as } s\to \infty \\ u(s, 0) \in L_0, u(s, 1)\in L_1 \\ [u]=\beta \end{array}\right. / \RR\]
We work over \(\Lambda\) to ensure convergence of the differential.
Under some extra assumptions, \((m^1)^2=0\), so we get some well-defined cohomology groups \(\HF(L_0, L_1)\), called the Lagrangian intersection Floer cohomology groups of \(L_0, L_1\).
[Invariance of Lagrangian intersection Floer cohomology]
Under appropriate conditions, the Lagrangian intersection Floer cohomology is independent of choices made in its definition and the Hamiltonian isotopy class of \(L_0\) or \(L_1\).
[PSS isomorphism]
If \(L\) bounds no psuedoholomorphic disks, then \(\HF(L, L)\simeq H^\bullet(L, \Lambda)\).
[Fukaya Category]
The Fukaya category \(\Fuk(X)\) is an \(\Lambda\)-linear \(A_\infty\) category whose
morphisms are generated by the Lagrangian Intersection Floer groups \(\CF(L_0, L_1 )\)
compositions \(m^2\) are given by counts of pseudoholomorphic triangles.
We also have some higher maps \(m^k: \hom(L_{k-1}, L_k)\tensor \cdots \hom(L_0, L_1)\to \hom(L_0, L_k)[2-k]\) given by counting holomorphic \(k+1\)-gons.
With extra assumptions on the symplectic manifold \(X\) and Lagrangians \(L\), we can equip \(\Fuk(X)\) with a \(\ZZ\)-grading.
definition 0.0.1
A Liouville domain is a pair \((X,\lambda)\), where
\(X\) is a \(2n\)-manifold with boundary \(\partial X\) and,
\(\lambda\in \Omega^1(X, \RR)\) is a one form on \(\Omega^1(X, \RR)\) so that \(\omega=d\lambda\) is a symplectic form for \(X\).
To this data we can associate a Liouville vector field \(Z\) defined by the property \(\iota_Z\omega= \lambda\).
We require that this vector field transversely points outward along \(\partial X\).
We call \(X\) a Liouville manifold if \(X\) is non-compact and obtained from a Liouville manifold \(X_0\) by attaching the \snap{symplectization of its contact boundary}{def:symplectization}
example 0.0.2
Let \(Q\) be a smooth \(n\)-dimensional manifold.
We now describe a canonical symplectic form on the cotangent bundle, \(T^*Q\).
At every point \(q\in Q\), there exists chart \(q\in U\subset Q\) which we can parameterize with coordinates \((q_1, \ldots, q_n)\).
The cotangent bundle \(T^*U\) inherits coordinates \((q_1, p_1, q_2, p_2, \ldots, q_n, p_n)\), where the \(p_i\) linearly parameterize the fibers of the cotangent bundle in the direction of the basis element \(dq_i\).
0
In these coordinates, the canonical symplectic form on this chart is:
\[\omega=\sum_{i=1}^n dq_i \wedge dp_i=-d(p dq).\]
example 0.0.3
Let \(Q\) be a manifold. The cotangent bundle \(X=T^*Q\) is an exact symplectic manifold; call the primitive \(\lambda_{can}=\sum_{i=1}^np_idq_i\). In local coordinates, the Liouville vector field is \(Z= \sum_{i=1}^n p_i \partial p_i\). The flow of this vector field acts by scalar multiplication on the fibers of \(T^*Q\),
\begin{align*}\phi^t: T^*Q\to T^*Q && (q, p) \mapsto (q, tp)\end{align*}
which inflates the symplectic form.
Now equip \(Q\) with a metric \(g\). The metric determines a unit sphere bundle \(S^*Q\subset T^*Q\), which is transverse to \(V\). Letting \(\alpha=\lambda|_{S^*Q}\) makes \((S^*Q, \alpha)\) a contact manifold. The time \(t\)-flow on \((S^*Q, \alpha)\) corresponds to the time \(t\) geodesic flow on the base ((geodesics and symplectic cohohomology of the cotangent bundle)). It is known ([Fet52]) that every closed manifold has at least 1 closed geodesic, so all of these examples of contact manifolds have a Reeb orbit.
could not find file def_wrappedFukayaCategory
[Wrapped Floer cohomology in cotangent bundle of the circle]
To compute the wrapped Floer cohomlogy of \(L:=T^*_0S^1\subset T^*S^1\) with itself, we first wrap the Lagrangian submanifold with the Hamiltonian \(p^2: T^*S^1\to \RR\). This gives the red ``spiral'', \(L^w\) drawn in the picture below. The intersections between \(L, L^w\) are in bijection with the integers. After fixing an intersection point \((`` e'')\), we (suggestively) identify the Lagrangian intersection Floer cochains with the polynomial algebra. \(\Hom(L, L)\simeq \CC[x, x^{-1}]\).
figure 0.0.4:Computation of the wrapped Floer cohomology of the cotangent fiber with itself in \(T^*S^1\). In fact, this identification sends the Floer theoretic product (given by counts of holomorphic polygons) to the product structre on \(\CC[x, x^{-1}]\).
This provides us with a method for understanding the Fukaya category of a non-compact manifold; we will additionally want to incorporate the data of a potential so taht we can study the Fukaya category of a symplectic Landau-Ginzberg model.
[exact symplectic fibration]
Let \(B\) be a symplectic manifold with boundary; for example \((B=D^2\subset \CC)\).
An exact symplectic fibration is a smooth proper fiber bundle \(\pi: E|to B\) such that:
\(E\) is a manifold with codimension two corners, and a decomposition of the boundary into a vertical and horizontal component \(\partial E= \partial_h E\cup \partial_v E\);
We have differential forms \(\omega\in \Omega^2(E), \theta\in \Omega^1(E)\) such that \(E_z=\pi^{-1}(z)\) is a Liouville domain with symplectic form \(\omega_z=\omega|_{E_z}\) whose primitive \(\theta_z=\theta|_{E_z}\);
The fibration \(\pi\) is ``trivial'' near the horizontal boundary.
[exact Landau-Ginzberg model]
An exact LG model is an exact symplectic manifold \((E, \omega=d\theta)\) that carries a proper smooth map \(\pi: E\to D^2\) with a compatible almost complex structure \(J\) making \(\pi\) holomorphic. Furthermore, we require the locus of critical points \(\Crit(\pi)\) to be disjoint from \(\partial_h E\) and \(\pi\) is a exact symplectic fibration away from the \(\Crit(\pi)\).
[Lefschetz fibration]
We will call a exact LG model \((E, \pi)\) a Lefschetz fibration if \(\Crit(\pi)\) is a fibite set of points, every fiber has at most one critical point, and \(\pi\) is ``Morse'' in the sense that it is locally modelled on the fibration \(\pi(z_1, \ldots, z_n)= z_1^1+\cdots + z_n^2\).
[my first Lefschetz fibration]
Let \(B=D^2\susbet \CC\), let \(E=left\{ z\in \CC^n \st \left|\sum_{i=1}^n z_i^2\right|\leq 1, \|z\|\leq Tright\}\) for some \(T>1\), then we have a Lefschetz fibration
\begin{align*}
\pi: E\to &B\\
z\mapsto & z_1^2+\cdots + z_n^2
\end{align*}
[a non-Lefschetz fibration]
Consider the map
\begin{align*}
\CC^3\to & \CC\\
(x, y, z) \mapsto & xyz.
\end{align*}
This is an example mirror to the pair-of-pants.
[Lefschetz fibration from a pencil]
Given a smooth projective variety \(X\), an ample line bundle \(L\to X\), pick two sections \(\sigma_0, \sigma_1\in H^0(L)\) to give a Lefschetz pencil. From the data of a Lefschetz pencil we can define an exact Lefschetz fibration
\[E:=X\setminus Y_\infty, \pi(x)=\frac{\sigma_0(x)}{\sigma_1(z)}\]
where \(Y_t=\{x\in X \st \sigma_1(x)/\sigma_0(x)=z\}\),
Then \(\pi: E\to \CC\) is a Lefschetz fibration.
The Lagrangian intersection Floer cohomology of a pair of Lagrangian thimbles
\(L_i, L_j\) in a Lefschetz fibration,
can be computed in the fiber as
\[\hom_{\FS(Y, W)}(L_i, L_j)= \left\{\begin{array}{cc}\hom_{\Fuk(Y_t)}(V_i, V_j) & \text{ if } ij
\end{array}\right.
\]
The Lagrangian \(\langle L_1, \ldots, L_k\rangle\) form a full exceptional collection.
Here, fix a Lefschetz fibration \(W: Y\to \CC\) with critical values \(\lambda_1, \ldots, \lambda_k\in \CC\). Pick a
collection of vanishing paths for the \(\lambda_j\). This gives us Lagrangian thimbles \(L_1, \cdots, L_k\), and vanishing cycles \(V_1, \ldots, V_k\) (which are Lagrangians inside some fixed fiber \(W^{-1}(t)\), where \(t\gg 0\). )
We now have a full exceptional collections. We can modify a set of vanishing paths by ``twisting'' them in the base of the Lefschetz fibration.
\begin{tikzpicture}
\begin{scope}[]
\fill[gray!20] (8,2.5) rectangle (2,-1.5);
\node at (5,-0.5) {\(\times\)};
\node at (5,0.5) {\(\times\)};
\draw (5,-0.5) -- (8,-0.5) ;
\draw (5,0.5) -- (8,0.5);
\node at (6.5,-1) {\(\gamma_1\)};
\node at (6.5,1) {\(\gamma_2\)};
\end{scope}
\begin{scope}[shift={(8,0)}]
\fill[gray!20] (8,2.5) rectangle (2,-1.5);
\node at (5,-0.5) {\(\times\)};
\node at (5,0.5) {\(\times\)};
\draw (5,0.5) -- (8,0.5);
\node at (7.5,2) {\(\tilde\gamma_1\)};
\node at (7.5,0) {\(\gamma_2\)};
\end{scope}
\draw (13,-0.5) .. controls (13.5,-0.5) and (14.5,-0.5) .. (14.5,0) .. controls (14.5,0.5) and (13.5,0) .. (13,0) .. controls (12.5,0) and (12.5,0.5) .. (12.5,1) .. controls (12.5,1.5) and (14,1.5) .. (14.5,1.5) .. controls (15,1.5) and (15.5,1.5) .. (16,1.5);
\end{tikzpicture}
@inproceedings{seidel2001vanishing,
title={Vanishing cycles and mutation},
author={Seidel, Paul},
booktitle={European Congress of Xathematics: Barcelona, July 10--14, 2000 Volume II},
pages={65--85},
year={2001},
organization={Springer}
}
@article{seidel2001more,
title={More about vanishing cycles and mutation},
author={Seidel, Paul},
journal={Symplectic geometry and mirror symmetry (Seoul, 2000)},
pages={429--465},
year={2001}
}
