The Floer Equation is Elliptic

Question

Let $(M,\omega)$ be a symplectic manifold and $H \in C^\infty(M \times \mathbb{S}^1)$. Furthermore, let $J$ be an $\omega$-compatible almost complex structure on $M$. The Floer equation is the following partial differential equation:

$$\partial_s u + J(\partial_tu - X_{H_t} \circ u)=0$$

where $X_{H_t}$ is the unique vector field defined by $i_{X_{H_t}}\omega = -dH_t$ and $u \colon \mathbb{R} \times \mathbb{S}^1 \to M$, where $(s,t) \mapsto u(s,t)$. I think this is a nonlinear first-order partial differential equation. In studying solutions to this equation, one applies elliptic regularity. I do know what a second order elliptic equation is and I checked the definition on Wikipedia for the other cases than $2$, but I am confused what exactly the terminology elliptic means and how we can see this explicitely in the case of the Floer equation.

Answer 1

First of all, ellipticity is defined in terms of the principal symbol of an operator, and the Hamiltonian term is zeroth order in the derivatives of u, so let's assume wlog that we're taking about J-holomorphic maps. Moreover, in local coordinates, the difference between J and your favourite constant-coefficient complex structure also doesn't depend on derivatives of u, so the Floer equation is a zeroth order perturbation of the usual Cauchy-Riemann equation, which is elliptic in the sense explained by Denis Serre above (i.e. it implies harmonicity of the components).

Here's an alternative way to see that pseudoholomorphic maps satisfy an elliptic second order equation. Consider the $L^2$-energy $\int|du|^2 dvol$ of your map u with respect to the almost Kaehler metric given by J and the symplectic form. The critical points of this are called harmonic maps and satisfy the (second order) harmonic map equation (usual harmonicity of components if the metric is Euclidean). Pseudoholomorphic maps are not only critical points, they're global minimisers of the energy: the energy can be rewritten as $\int\omega$ (constant) plus a term which vanishes if and only if u is pseudoholomorphic.

In the instanton/soliton literature, I think this is called the Bogomolny trick (writing your action as a topological term plus a term which vanishes if and only if a first order equation vanishes).

Finally, it is possible to write the Floer equation as a pseudoholomorphic map equation into a larger dimensional manifold with an almost complex structure depending on J and H (I think this is due to Gromov, and is explained somewhere in McDuff-Salamon "J-holomorphic curves and symplectic topology"), so ellipticity of the Floer equation follows from that of the pseudoholomorphic map equation.

Answer 2

This is actually a system of first-order PDEs, of $2n$ (the dimension of $M$) equations. To see that it is elliptic, let us consider the symplest case of $M={\mathbb R}^{2n}$ with the standard symplectic structure, $H\equiv0$ and $$J=\begin{pmatrix} 0_n & I_n \\ -I_n & 0_n \end{pmatrix}.$$ then the system writes blockwise as $$\partial_sv+\partial_tw=0,\qquad\partial_sw-\partial_tv=0.$$ Eliminating $w$, for instance, yields $$(\partial_{ss}+\partial_{tt})v=0,$$ which is elliptic.