Lagrangian Floer (co)homology, Novikov coverings and exact symplectic manifolds

Question

I started reading the book "Lagrangian intersection Floer theory anomaly and obstruction", and there are a couple of details and assumptions in the definition of the Novikov covering that I would like to clarify. (I will follow the notation that they use in the book )

First is why do we need the actual use of the Novikov covering? With this I mean why are we asking for two conditions on that is for the integrals of the symplectic areas to be the same , I guess this makes sense so that the action functional is well-defined, but wouldn't this happen in the universal cover since the symplectic form is closed? I guess I don't understand why we need/want the extra assumption $I_{\mu}(\bar w \# w´)=0$. Will it be for the Maslov index of a point $[l_p,w]$ to be well-defined ? (At least it's the only thing I could see that it's useful for).

Then I am interested in symplying this and consider just exact symplectic manifolds $(M,d\lambda)$. Now here we are able to define the action functional in $\Omega(L_0,L_1)$ and still get that the intersection points are the critical points of this action functional.Just define $A(\gamma)=\int_{0}^{1}\gamma^* \lambda_{can}$.

But then how is the index of a critical point $p$ defined ? We could use the more general definition that they talk about but then we would to pick a path $l_0$ and an homotopy class $[w]$ ,so I am wondering if this will be equal to something simpler and more intuitive ?

My motivation for this simpler case comes from the following: Let's consider our symplectic manifold to be the cotangent bundle and fix the fibers $T^*_{q_0} M, T^*_{q_1}M$. Now suppose we have an Hamiltonian flow $\phi^t$, from an hamiltonian function $H$, such that $\phi^1(T^*_{q_o}M)\pitchfork T^*_{q_1}M$. In the following paper https://arxiv.org/pdf/math/0408280.pdf there is developed an isomorphism between the Floer Homology of the action functional whose critical points are orbits of our hamiltonian vector field and the corresponding Lagrangian functional on $TM$. Then one also defines here the index $\mu_{\Omega}(x)$ where $x$ is an orbit of $X_H$,and is able to prove its the same has the morse index of the action functional of the Lagrangian. And so for example in the case where $H(q,p)=\frac{1}{2}|p|^2$, and we get the Lagrangian $L(q,v)=\frac{1}{2}|v|^2$ we will get that $\mu_{\Omega}(x)$ will give us the index between $x(0)$ and $x(1)$. And so I am wondering if we look at the lagrangian intersection theory of $\phi^1(T^*_{q_0}M)$ and $T^*_{q_1}M$ and we looked at the intersection point that is $x(1)$, we will get that it's index is the same as $\mu_{\Omega}(x)$? If this were true it would be a nice geometrical interpretation of the index relating to geodesics and Jacobi fields.

Any insight is appreciated. Thanks in advance.

Answer 1

The index of a critical point of the action functional cannot be defined. This is where Floer theory departs from the usual Morse theory, and Floer discusses it in the intro to his original paper. I will try to do justice to this huge (but subtle) innovation.

In analogy with finite dimensional Morse theory, you could...

• Treat sections of $TM$ along a critical path $\gamma$ as the tangent space to the path space at $\gamma$: $T_{\gamma}(\Omega(L, L')) = \Gamma(\gamma^{\ast}TM)$. For Lagrangian boundary, restrictions are required on the sections near their endpoints.
• Compute the Hessian for the action functional at $\gamma$, which would split $\Gamma(\gamma^{\ast}TM)$ into positive and negative eigenspaces $E^{\pm}$. Assuming $\gamma$ is non-degenerate, there will be no $0$ eigenspace.
• Interpret $E^{\pm}$ as the tangent spaces to the ascending and descending manifolds for the critical point $\gamma$.

If this were possible, then we may seek to define the index to be the dimension of $E^{-}$ and count trajectories between critical points $\gamma_{0}, \gamma_{1}$ as points in the intersection of the ascending manifold for $\gamma_{0}$ with the descending manifold of $\gamma_{1}$.

This fails for a couple of reasons. First, ascending and descending manifolds are ill defined for analytical reasons (Palais-Smale). Second, both $E^{\pm}$ have infinite dimension. The miracle of Floer homology is that even though the indices of critical points are ill-defined, counting holomorphic strips with fixed ends $\gamma_{j}$ (same equation as one would solve for a Morse flow, but with endpoints determined) determines a Fredholm problem and the Fredholm index can be interpreted as an index difference, which is a finite integer.

The usual Morse homology can be recast in the Floer/Fredholm framework. See Schwarz's "Morse homology" book. But when you're able to express a Floer/Fredholm problems as a Morse problems with a well-defined gradient flow, there must be special circumstances.