Boundary conditions and the relationship between Hamiltonian and Lagrangian Floer theories

Question

I am looking for a full account of the relationship between the various versions of Floer theory on a symplectic manifold $M$. If we take the usual Floer equation (Hamiltonian version) \begin{equation*} \frac{\partial u}{\partial s} + J \left( \frac{\partial u}{\partial t} - X_H(u) \right) = 0 \ , \end{equation*} there are two "natural" types of boundary conditions: periodic conditions $u(s,t) = u(s,t+1)$ on the cylinder, or "fixed ends" $u(s,0) \in L_0$ and $u(s,1) \in L_1$ on the strip where the $L_i$ are Lagrangian submanifolds of $M$. One can always find a new complex structure $\tilde{J}$ with map $v : \mathbb{R}^2 \to M$ and transform the Floer equation into the unperturbed Cauchy-Riemann equation \begin{equation*} \frac{\partial v}{\partial s} + \tilde{J} \frac{\partial v}{\partial t} = 0 \ . \end{equation*} The two sets of boundary conditions then become: periodic $v(s,t) = \phi_H(v(s,t+1))$, or "fixed ends" $v(s,0) \in L_0$ and $v(s,1) \in \phi_H(L_1)$ where $\phi_H$ is the time-1 symplectomorphism generated by $X_H$.

I can roughly see how this infers several relationships between both types of Floer theories, but my understanding of all the possible correspondences is far from complete. In particular, fixed-end boundary conditions identify solutions of Hamiltion's equations with intersections of $L_0$ and $\phi_H(L_1)$. Fine, but what can be said about periodic boundary conditions? Is Lagrangian Floer theory of the intersection of the diagonal with the graph of a Hamiltonian diffeomorphism the only version of Lagrangian Floer theory that has a correspondence with periodic boundary conditions? There is also apparently a way to show that, on a compact manifold, intersections of a Lagrangian with are in one-to-one with critical points of Morse function on the Lagrangian and moreover, that Lagrangian Floer theory of the zero section of $T^*M$ is identified with Morse theory on $M$ (or something along those lines).

I have spent quite a bit of time going around in circles with the literature and so if someone could provide me with a nice reference with where these things are explained in detail (or even perhaps willing to describe some themselves) it would be very useful.

Thanks.

Answer 1

These are all standard TQFT structures which reflect the combinatorics of Riemann surfaces. In general, one starts with a compact Riemann surface (possibly with boundary $\bar{S}$ and a finite set of marked points $\Sigma\subset\bar{S}$. There are four kinds of marked points in the set $\Sigma$. First, there is an essential difference between a boundary marked point $\zeta\in\Sigma\cap\partial\bar{S}$ and an interior marked point $\zeta'\in\bar{S}\setminus\Sigma$. Second, $\Sigma=\Sigma^{in}\cup\Sigma^{out}$ is partitioned into inputs and outputs. Formally, a connected component $C\subset\partial\bar{S}\cap\Sigma$ is mapped to a Lagrangian label $L_C$ and a boundary marked point $\zeta\in\bar{S}\cap\Sigma$ is required to converge to an intersection point between two nearby Lagrangian submanifolds $L_{\zeta,0}$ and $L_{\zeta,1}$. On the other hand, every interior marked point is a mark served to break up the symmetry when treating moduli problems or should be mapped to a period orbit of some Hamiltonian vector field $X_H$.

Let me illustrate the above formalism with some elementary examples. When $\bar{S}=\mathbb{D}$ is the closed unit disc, and $\Sigma\subset\partial\mathbb{D}$ consists of two distinct points, we get the Lagrangian Floer cohomology $HF^\ast(L_0,L_1)$, where $L_0$ and $L_1$ are the two Lagrangian labels correspond to two connected components of $\partial\mathbb{D}\setminus\Sigma$. When $\bar{S}=\mathbb{CP}^1$ and $\Sigma\subset\mathbb{CP}^1$ consists of two distinct points, one gets essentially the Hamiltonian Floer cohomology $HF^\ast(\lambda H)$, where $H$ is a Hamiltonian function on some symplectic manifold $M$. Strictly speaking, in the latter case we need to decorate $\bar{S}$ with a closed 1-form $\gamma$ so that $\int_{C_1}\gamma=\int_{C_2}\gamma=\lambda$, where $C_1,C_2\subset\mathbb{CP}^1\setminus\Sigma$ are two small circles centered at the interior marked points. Also, we need to fix choices of directions near the two punctures $\zeta,\zeta'\in\mathbb{CP}^1$ to indicate the difference between an input and an output.

The above formalism can be generalized to families of Riemann surfaces define various operations on Floer cochains, which lead to the so-called Fukaya $A_\infty$ structures, and more generally, the open-closed Seidel maps. The definition of the former one involves only boundary marked points on a disc, but the construction of the latter one involves marked points of both interior and boundary types. The associated Floer equations are similar as the ones defining usual Hamiltonian or Lagrangian Floer homologies.

These Floer theories all in principle all Morse theory associated to some action functional defined on an infinite-dimensonal space, that's why certain assumptions or hard works are needed to make them rigorous. In some cases, these theories can be reduced to usual Morse theories, see for example, the work of Fukaya-Oh (https://www.math.kyoto-u.ac.jp/~fukaya/FO.pdf) in the case of Lagrangian Floer theory. But this is not exactly true in general. For example, when $L_0=L_1=L$ are weakly unobstructed Lagrangian submanifolds of $M$, the work of Biran-Cornea (http://www.dms.umontreal.ca/~cornea/qrel.pdf) shows that one has to deform the usual Morse differential using pearls (disc bubblings connected by Morse trajectories) to make things correct, which makes it similar to the case of quantum cohomologies.