Let \((M^{2n},\omega)\) be a symplectic manifold, \(\varphi\) is a symplectic morphism if \(\varphi^*\omega = \omega\).
Also, \(\omega : TM\cong T^*M, v\mapsto \omega(v,\cdot)\).
\(H:M\to \mathbb{R}\) 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 \(N, T^*N\) naturally have symplectic form (natural in mechanics and representation theory). A Hamiltonian function
\[H : T^*\mathbb{R}^3\to \mathbb{R}\quad (x,p)\mapsto \frac{1}{2m}|p|^2+V(x,p)\]
zero section and cotangent fibre of \(T_p^*N\) 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
\[\phi_t^H(T^*_pN) \cap T^*N_q\]
we care about the Lagrangians and their intersections.
The idea is given two lagrangians \(L,L'\) I want to associate these chain complexes
\[CF(L,L',k) := \left(\bigoplus_{p\in L\cap L'} k \cdot p, d\right)\]
In very nice cases, the cohomology of the complex \(HF(L,L)\) is the same as the cohomology of \(L\).
We want hamiltonian invariance
\[ HF(\phi_t^H(L), \phi_t^H(L')) \cong HF(L,L')\]
Differentials counts pseudo-holomorphic disks with boundary on \(L,L'\).
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.
\(\mathrm{Hom}(L,L') = HF(L,L')\)
Composition is given by counting triangles
\[\HF(L,L')\otimes \HF(L',L'')\to \HF(L,L''): \quad (p,q)\mapsto \Delta_{p,q',q}\]
Fukaya category: The objects in the Fukaya category are the Lagrangians, and the Hom spaces are the Floer chain complexes.