Capped Hamiltonian orbits and recapping data

Let $(M,\omega ,J_{\ast })$ be a smooth symplectic manifold with a chosen smooth compatible almost-complex structure. Write $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$. In this item, a smooth loop $x:S^1\to M$ is contractible when it extends to a smooth map $w:D^2\to M$; a specified such extension is a cap. Require $w(e^{2\pi it})=x(t)$, and orient the boundary positively using the standard orientation on $D^2$. The capped loop is denoted $[x,w]$. Two caps of the same loop differ by a recapping sphere $A\in {\pi }_2(M,x(0))$, represented by gluing one disk to the orientation-reversed other disk. Use the gluing orientation for which $w\mathrm{\#}A$ satisfies ${\int }_{D^2}(w\mathrm{\#}A{)}^{\ast }\omega ={\int }_{D^2}{w}^{\ast }\omega +\omega (A)$. A recapping is trivial when its class is zero.

For a smooth time-periodic Hamiltonian $H:\mathbb{R}\mathrm{/}\mathbb{Z}\times M\to \mathbb{R}$, define its vector field by ${\iota }_{X_{H_t}}\omega =-d{H}_t$. A one-periodic Hamiltonian orbit is a smooth loop satisfying $\dot{x}=X_{H_t}(x)$. Let ${\mathrm{\Phi }}_{t,s}$ be the local evolution of this vector field as in Local Hamiltonian evolution preserves the symplectic form, with ${\mathrm{\partial }}_t{\mathrm{\Phi }}_{t,s}=X_{H_t}\circ {\mathrm{\Phi }}_{t,s}$ and ${\mathrm{\Phi }}_{s,s}=\mathrm{id}$; write ${\mathrm{\Phi }}_H^1={\mathrm{\Phi }}_{1,0}$. The orbit is nondegenerate if $D{\mathrm{\Phi }}_H^1(x(0))-I$ is invertible. Salamon uses the opposite convention ${\iota }_{X_h}\omega =dh$; the action formula below is his convention with $h=-H$. Its capped-orbit action is ${\mathcal{A}}_H([x,w])={\int }_0^1{H}_t(x(t))dt-{\int }_{D^2}{w}^{\ast }\omega \mathrm{.}$ Set $\omega (A)={\int }_{S^2}{a}^{\ast }\omega$ for a sphere representative $a$ of $A$. The first Chern number is $c_1(A)=⟨c_1(a^{\ast }TM,a^{\ast }{J}_{\ast }),[S^2]⟩$. For the transition-loop convention specified below, $c_1(A)=-\mathrm{wind}({\det }_{\mathbb{C}}{g}_A)$. These numbers depend only on $A$ because $d\omega =0$ and characteristic classes are homotopy invariant.

For a nondegenerate contractible one-periodic orbit, the derivative of its local Hamiltonian evolution is a symplectic path by Local Hamiltonian evolution preserves the symplectic form. Here $\mathrm{Sp}(2n)$ is the group of linear automorphisms of $({\mathbb{R}}^{2n},{\omega }_0)$ preserving ${\omega }_0={\sum }_{j=1}^{n}d{q}_j\wedge d{p}_j$. Put $g_{J_{\ast }}(v,w)=\omega (v,J_{\ast }w)$, the positive Hermitian metric supplied by the compatible structure. A cap-induced unitary frame along $x$ is the boundary restriction of a chosen smooth unitary trivialization of the complex bundle $(w^{\ast }TM,w^{\ast }{J}_{\ast })$ over the cap disk; pointwise it is a complex-linear isometry from $({\mathbb{C}}^n,⟨\cdot ,\cdot ⟩)$ to $(T_{x(t)}M,J_{\ast },g_{J_{\ast }})$, and hence is also symplectic. Write ${\mathrm{\Phi }}_{t,0}$ for the local Hamiltonian evolution and define the linearized path in this frame by ${\mathrm{\Phi }}_w(t)={\tau }_w(t{)}^{-1}\circ D{\mathrm{\Phi }}_{t,0}(x(0))\circ {\tau }_w(0)$. Then ${\mathrm{\Phi }}_w(0)=I$ and $1$ is not an eigenvalue of ${\mathrm{\Phi }}_w(1)$.

Write $\mathrm{Gr}({\mathrm{\Phi }}_w(t))=\{(v,{\mathrm{\Phi }}_w(t)v):v\in {\mathbb{R}}^{2n}\}$ and $\mathrm{\Delta }=\{(v,v):v\in {\mathbb{R}}^{2n}\}$. Both are Lagrangian in $({\mathbb{R}}^{2n}\oplus {\mathbb{R}}^{2n},-{\omega }_0\oplus {\omega }_0)$. Define the Conley–Zehnder index by ${\mu }_{\mathrm{C}\mathrm{Z}}({\mathrm{\Phi }}_w)={\mu }_{\mathrm{R}\mathrm{S}}(\mathrm{Gr}{\mathrm{\Phi }}_w,\mathrm{\Delta })$, using the Robbin–Salamon Maslov path index. For a crossing time $t_0$, its crossing form on $\mathrm{Gr}{\mathrm{\Phi }}_w(t_0)\cap \mathrm{\Delta }$ is $v\mapsto (-{\omega }_0\oplus {\omega }_0)(v,\dot{v}(t_0))$ for a smooth local section $v(t)\in \mathrm{Gr}{\mathrm{\Phi }}_w(t)$ through $v$; a crossing is regular when this form is nondegenerate. For paths with regular crossings, the index is the sum of crossing-form signatures at interior crossings with half-signatures at endpoint crossings. For arbitrary paths it is the fixed-endpoint homotopy-invariant extension of this rule. The normalization is that a sufficiently short positive rotation in a symplectic two-plane has index $+1$.

For a based unitary loop $g:S^1\to U(n)$, define its Maslov degree to be $\mathrm{wind}({\det }_{\mathbb{C}}g)$, the topological degree of its determinant map. In this convention the loop-shift law is ${\mu }_{\mathrm{C}\mathrm{Z}}(g{\mathrm{\Phi }}_w)-{\mu }_{\mathrm{C}\mathrm{Z}}({\mathrm{\Phi }}_w)=2\mathrm{wind}({\det }_{\mathbb{C}}g)$. If ${\tau }_w(t)$ and ${\tau }_{w\mathrm{\#}A}(t)$ are unitary cap-induced frames of $T_{x(t)}M$ for $w$ and $w\mathrm{\#}A$, choose them to agree at $t=0$ and define $g_A(t)={\tau }_{w\mathrm{\#}A}(t{)}^{-1}{\tau }_w(t)$ along the positively oriented gluing circle. Then $g_A$ is a based loop in $U(n)$, the linearized paths satisfy ${\mathrm{\Phi }}_{w\mathrm{\#}A}(t)=g_A(t){\mathrm{\Phi }}_w(t)$, and the clutching convention is $c_1(A)=-\mathrm{wind}({\det }_{\mathbb{C}}{g}_A)$. Thus the loop shift is $2\mathrm{wind}({\det }_{\mathbb{C}}{g}_A)=-2c_1(A)$. No differential or Floer homology is defined here.

The index does not depend on the chosen unitary trivialization of a fixed cap. After a constant unitary change makes two such frames ${\tau }_w,{\tau }_w^{\mathrm{\prime }}$ agree at $t=0$, their boundary transition $g(t)={\tau }_w^{\mathrm{\prime }}(t{)}^{-1}{\tau }_w(t)$ is a based loop that extends over the disk, so its determinant has winding zero. The corresponding paths satisfy ${\mathrm{\Phi }}_w^{\mathrm{\prime }}(t)=g(t){\mathrm{\Phi }}_w(t)$, and the loop-shift law above gives the same index.

Symplectic and almost-complex curve data

Local Hamiltonian evolution preserves the symplectic form