Lagrangian intersection and Hamiltonian chord data

Let $(M,\omega )$ be a smooth symplectic manifold and let $i_j:L_j\hookrightarrow M$ for $j=0,1$ be compact smooth embedded Lagrangian submanifolds. The pair $(L_0,L_1)$ is transverse if $T_p{L}_0+T_p{L}_1=T_{p}M$ for every $p\in L_0\cap L_1$. In the unperturbed intersection model, the candidate morphism generators are the points of $L_0\cap L_1$.

Let $H:[0,1]\times M\to \mathbb{R}$ be smooth, write $H_t(p)=H(t,p)$, and define $X_{H_t}$ by ${\iota }_{X_{H_t}}\omega =-d_M{H}_t$. A Hamiltonian chord from $L_0$ to $L_1$ is a smooth map $\gamma :[0,1]\to M$ satisfying $\dot{\gamma }(t)=X_{H_t}(\gamma (t))$, $\gamma (0)\in L_0$, and $\gamma (1)\in L_1$. It is nondegenerate when the time-one local evolution ${\mathrm{\Phi }}_H^1$ is defined near $\gamma (0)$ and $D{\mathrm{\Phi }}_H^1(\gamma (0))(T_{\gamma (0)}{L}_0)+T_{\gamma (1)}{L}_1=T_{\gamma (1)}M\mathrm{.}$ In the Hamiltonian-chord model, these chords are the candidate morphism generators. This is a definition of generator data only; it proves no finiteness, compactness, transversality theorem, differential identity, or Floer homology statement.

Symplectic form and symplectic manifold

Isotropic, coisotropic, symplectic, and Lagrangian submanifolds