Author: Eiko

Time: 2025-03-07 17:43:09 - 2025-03-07 17:43:09 (UTC)

Introduction and Setup

Let $(M^{2n},\omega )$ be a symplectic manifold, $\phi$ is a symplectic morphism if ${\phi }^{\ast }\omega =\omega$.

Also, $\omega :TM\cong T^{\ast }M,v\mapsto \omega (v,\cdot )$.

$H:M\to \mathbb{R}$ Hamiltonian functions, associate a vector field by $dH=-{\iota }_{X_H}\omega$, this gives a one parater family of diffeomorphisms and are actually symplectic morphisms.

A Lagrangian submanifold $L$ is an $n$ dimensional submanifold of $M$ such that $\omega {|}_L=0$.

Why do we care about Lagrangian submanifolds? Consider $N,T^{\ast }N$ naturally have symplectic form (natural in mechanics and representation theory). A Hamiltonian function

$$H:T^{\ast }{\mathbb{R}}^3\to \mathbb{R}(x,p)\mapsto \frac{1}{2m}|p{|}^2+V(x,p)$$

zero section and cotangent fibre of $T_p^{\ast }N$ are examples of Lagrangian submanifolds.

if we flow from $p\to q$ in $N$, the possible paths corresponds to the intersection points of the time t folw of the cotangent fibfre of

$${\varphi }_t^H(T_p^{\ast }N)\cap T^{\ast }{N}_q$$

we care about the Lagrangians and their intersections.

Lagrangian Floer Homology

The idea is given two lagrangians $L,L^{\prime }$ I want to associate these chain complexes

$$CF(L,L^{\prime },k):=(\underset{p\in L\cap L^{\prime }}{⨁}k\cdot p,d)$$

In very nice cases, the cohomology of the complex $HF(L,L)$ is the same as the cohomology of $L$.

• We want hamiltonian invariance

  $$HF({\varphi }_t^H(L),{\varphi }_t^H(L^{\prime }))\cong HF(L,L^{\prime })$$

Differentials counts pseudo-holomorphic disks with boundary on $L,L^{\prime }$.

Define the space of almost complex structures that compatible with symplectic forms $\{J\in \mathrm{End}(TM):J^2=-\mathrm{id},\omega (J\cdot ,\cdot )=\omega (\cdot ,\cdot ),\omega (J\alpha ,\alpha )>0\}$.

The space is contractible.

Remarks

• $\mathrm{Hom}(L,L^{\prime })=HF(L,L^{\prime })$

• Composition is given by counting triangles

  $$\text{HF}(L,L^{\prime })\otimes \text{HF}(L^{\prime },L^{″})\to \text{HF}(L,L^{″}):(p,q)\mapsto {\mathrm{\Delta }}_{p,q^{\prime },q}$$

• Fukaya category: The objects in the Fukaya category are the Lagrangians, and the Hom spaces are the Floer chain complexes.