This talk was given by Danil The main reference is [Sei01b, Sei01a]. Let $(X, \omega)$ be a $2n$ dimensional symplectic manifold. Fix a compatible almost complex structure $J$ on $X$. [Novikov ring] The Novikov ring with $\mathbb C$-coefficients is the ring of formal sums \[\Lambda_{\geq 0}:= \left\{\sum_{i=0}^\infty a_i T^{\lambda_i}\st a_i\in \CC, \lambda_i \in \RR_{\geq 0}, \lim_{i\to\infty} \lambda_i = \infty\right\}. \] 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 Lagrangian intersection Floer complex is the vector space \[\CF(L_0, L_1):= \bigoplus_{p\in L_0\cap L_1} \Lambda \langle p\rangle \] equipped with an endomorphism $\partial: \CF(L_0, L_1)\to \CF(L_0, L_1)$ whose structure coefficients are given by counts of pseudoholomorphic strips. For $p,q\in L_0\cap L_1$ and $\beta\in H_2(X, L_0\cup L_1, \ZZ)$, 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] Under hypotheses excluding obstruction and disk-bubbling phenomena, for example in the exact setting, $\HF(L, L)\simeq H^\bullet(L, \Lambda)$. [Fukaya Category] The Fukaya category $\Fuk(X)$ is an $\Lambda$-linear $A_\infty$ category whose

objects are suitably decorated Lagrangian submanifolds
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 $dq_i$. ¹ In these coordinates, the canonical symplectic form on this chart is: \[\omega=\sum_{i=1}^n dp_i \wedge dq_i=d\left(\sum_{i=1}^n p_i\,dq_i\right).\]

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, e^t p)\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 $Z$. 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.

definition 0.0.4

Let $X$ be a Liouville manifold. The wrapped Fukaya category $\mathcal W(X)$ is an $A_\infty$ category whose objects are exact, properly embedded Lagrangian submanifolds which are cylindrical at infinity, together with the usual brane data. Its morphism spaces are wrapped Floer complexes \[ CW^\bullet(L_0,L_1):=\varinjlim_H CF^\bullet(L_0,\phi_H^1(L_1)), \] where the limit is taken over admissible Hamiltonians which wrap at infinity. The higher products are defined by counts of inhomogeneous pseudoholomorphic polygons. [Wrapped Floer cohomology in cotangent bundle of the circle] To compute the wrapped Floer cohomology 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 Laurent polynomial algebra. $\Hom(L, L)\simeq \CC[x, x^{-1}]$.

Floer theory of zero section in $T^*S^1$ — figure 0.0.5: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 structure 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 that 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 $\omega_z=\omega|_{E_z}$ and 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 an exact symplectic fibration away from $\Crit(\pi)$. [Lefschetz fibration] We will call a exact LG model $(E, \pi)$ a Lefschetz fibration if $\Crit(\pi)$ is a finite 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 $\pi(z_1, \ldots, z_n)= z_1^2+\cdots + z_n^2$. [my first Lefschetz fibration] Let $B=D^2\subset \CC$, let $E=\left\{ z\in \CC^n \st \left|\sum_{i=1}^n z_i^2\right|\leq 1, \|z\|\leq T\right\}$ 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 $X$ and an ample line bundle $L\to X$, choose a Lefschetz pencil generated by sections $\sigma_0, \sigma_1\in H^0(L)$. From the data of a Lefschetz pencil we can define an exact Lefschetz fibration \[E:=X\setminus Y_\infty, \pi(x)=\frac{\sigma_0(x)}{\sigma_1(x)}\] where $Y_t=\{x\in X \st \sigma_0(x)/\sigma_1(x)=t\}$, 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 \[\hom_{\FS(Y, W)}(L_i, L_j)= \left\{\begin{array}{cc}\hom_{\Fuk(Y_t)}(V_i, V_j) & \text{ if } i<j \\ \CC\cdot x_i & \text{ if } i=j\\ 0 & \text{ if } i>j \end{array}\right. \] The ordered Lagrangian thimbles associated to a distinguished basis of vanishing paths form a full exceptional collection in the directed Fukaya-Seidel category. 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 collection. We can modify a set of vanishing paths by ``twisting'' them in the base of the Lefschetz fibration.

mutation of vanishing paths — figure 0.0.6:we can mutate vanishing cycles in the base of an LG fibration

@inproceedings{seidel2001vanishing, title={Vanishing cycles and mutation}, author={Seidel, Paul}, booktitle={European Congress of Mathematics: 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} }

Connections

Uses

Liouville domaindefinition / def:liouvilleDomain
symplectic structure on cotangent bundleexample / def:symplecticCotangentBundle
Liouville structure on the cotangent bundleexample / exm:cotangentBundleIsLiouville
wrapped Fukaya categorydefinition / def:wrappedFukayaCategory
Floer theory of zero section in $T^*S^1$figure / fig:wrappedFloerCylinder
mutation of vanishing pathsfigure / fig:mutationOfVanishingPaths
symplectic manifolddefinition / def:symplecticManifold
compatible almost complex structuredefinition / def:compatibleAlmostComplexSTructure
Novikov Ringdefinition / def:novikovRing
$A_\infty$ categorydefinition / def:aInfinityCategory

Used in

No direct links found.

References

[Fet52]	Abram Il'ich Fet. Variational problems on closed manifolds. Matematicheskii Sbornik, 72(2):271--316, 1952.
[Sei01a]	Paul Seidel. More about vanishing cycles and mutation. Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pages 429--465, 2001.
[Sei01b]	Paul Seidel. Vanishing cycles and mutation. In European Congress of Mathematics: Barcelona, July 10--14, 2000 Volume II, pages 65--85. Springer, 2001.