Capped Lagrangian Hamiltonian chords and action

Let $(X,\omega )$ be symplectic and let $L_0,L_1\subset X$ be compact Lagrangians. Fix a connected component $\mathcal{P}$ of the smooth path space of paths $x:[0,1]\to X$ with $x(0)\in L_0$ and $x(1)\in L_1$, together with one reference path $x_{\ast }$. A capping of $x\in \mathcal{P}$ is a smooth homotopy $w:[0,1{]}_s\times [0,1{]}_t\to X$ from $x_{\ast }$ to $x$ such that $w(s,0)\in L_0$ and $w(s,1)\in L_1$. Capping classes retain the relative homotopy class of this homotopy; gluing two caps with opposite orientations gives the corresponding relative two-cycle. A capped chord is a Hamiltonian chord together with such a capping class.

For a smooth Hamiltonian $H:[0,1]\times X\to \mathbb{R}$ use ${\iota }_{X_{H_t}}\omega =-d{H}_t$. On the capping cover define ${\mathcal{A}}_H([x,w])={\int }_0^1{H}_t(x(t))dt-{\int }_{[0,1{]}^2}{w}^{\ast }\omega \mathrm{.}$ Changing the reference path changes this action by a constant on the component, and changing the capping changes it by the negative symplectic area of the glued relative two-cycle. The action differential on path space is $d{\mathcal{A}}_H(x)[\xi ]=-{\int }_0^1\omega (\xi (t),\dot{x}(t)-X_{H_t}(x(t)))dt,$ for variations with $\xi (0)\in T_{x(0)}{L}_0$ and $\xi (1)\in T_{x(1)}{L}_1$. To obtain this formula, vary the cap with boundary variation $\xi$ at $s=1$. Stokes' formula gives the area variation ${\int }_0^1\omega (\xi ,\dot{x})dt$: the reference-path edge is fixed, and each remaining boundary term pairs two tangent vectors to one of the Lagrangians, so vanishes. The Hamiltonian term varies by ${\int }_0^{1}d{H}_t(\xi )dt={\int }_0^1\omega (\xi ,X_{H_t})dt$ under ${\iota }_{X_{H_t}}\omega =-d{H}_t$. Subtracting the area variation gives the displayed differential. This is an action and capping interface; it asserts no compactness theorem or existence of a Floer differential.

Symplectic and almost-complex curve data

Lagrangian intersection and Hamiltonian chord data