Real interpretations of Discontinuities in Floer homology

Question

This question is motivated by the answer in this question (you don't have to read it to understand the following).

I am not that proficient in calculating Floer homology, and I held back on answering and just commenting on the question because of the following.

Let $L$ be the equator on $S^2$ then it is obvious that no "Hamiltonian" symplectomorphism can take $L$ to another Lagrangian not intersecting $L$. Indeed, $L$ cuts $S^2$ into two equally sized pieces and it is not difficult from there.

It is quite a different thing to assert that the intersection Floer homology $FH(L,L)$ is non-trivial, which is used in the solution to the problem (together with a Kunneth formula).

The thing that initially bugged me about this was that arbitrarily close to $L$ there are Lagrangians whose Floer homology intersection is $0$ both with $L$ and themselves. I have later come to realize that I myself when considering exact Lagrangians in cotangent bundles have encountered this discontinuity frequently, and should really not be surprised. Here the exactness is very important for non-triviality. However, for me this comes up in a different way because I am considering finite reductions of the loop spaces using Chaperons broken geodesics (as Viterbo does).

I, however, realized that the discontinuity does not seem that bad and this leads me to my question:

Question: Is there a natural Floer homology with real coefficients, which resolves these discontinuities in the sense that the differentials are continuous.

A little more motivation: In the above case it seems that with coefficients in the Novikov ring $\mathbb{Z}[t]$ we can realize the Floer complex with two generators, and then for all Lagrangians not dividing $S^2$ into two equally volumed parts the differential is an isomorphism. But if the Lagrangians are both of that particular type the differential is $0$. This could be described continuously with real coefficients as:

$ d \colon \mathbb{R}[t] \to \mathbb{R}[t]$

where $d$ is the differential from the $\mathbb{Z}[t]$ coefficient complex multiplied by the scalar $(A_1-A_2)^2+(A_1'-A_2')^2$ where $A_i$ are the two volumes for the first Lagrangian and $A_i'$ are the volumes for the second Lagrangian.

Answer 1

The brief answer is yes, using ideas from Novikov homology.

Here's an example of the discontinuity and how it can be fixed. Take $L=S^1\times y$ as a Lagrangian in standard symplectic $T^2=S^1\times S^1$. Then $HF_\ast(L,L)$ (which means $HF_\ast(L,\phi L)$ for $\phi$ Hamiltonian) is isomorphic to $H_\ast(L)$, while $HF(L,S^1\times y')=0$ for $y' \neq y$. But if we couple the Floer complex to a local system $l$ on the space of paths from one Lagrangian to another which restricts to a non-trivial local system $\alpha$ on $L$, then we have $HF_\ast(L,S^1\times y';l)=0$ for $y'=y$ (because there are no intersections) and $HF_\ast(L,L;l)\cong H_\ast(L;\alpha) = 0$ as well.

Now here's some detail.

There's a loose principle in Floer theory which says that if you have Floer modules $HF(a_t)$ for data $a_t$ depending smoothly on a parameter $t$, all defined over the same coefficient ring, then you can expect them to vary continuously provided a certain vector space $K_t$ does not vary with $t$. Let $C$ be the ambient configuration space of the theory. Then we have an index (or spectral flow) class $I\in H^1(C;\mathbb{Z})$, and for each $t$ an action class $[A_t]\in H^1(C;\mathbb{R})$. Then $K_t = \ker I \cap \ker [A_t] \subset H_1(C;\mathbb{R})$. The point is that $\ker I$, the group of periods of the action functional $A_t$, shows up in the Floer-theoeretic differential, and if the behaviour of the action functional on $\ker I$ changes then unexpected cancellations could occur. Note that this does indeed jump in the circle example above (I get $K_t=\mathbb{R}$ for $t\neq 0$, $K_t=0$ for $t=0$).

The principle can be proved in particular instances either using the homotopy method (bifurcation analysis), which is very general but analytically hard, or Floer's continuation method (in which case you'll need good energy estimates).

We get bit more flexibility using a local system $l$ on $C$. If our coefficient ring is $\Lambda$, this assigns an automorphism of $\Lambda$ to each homotopy class of paths from $c_1$ to $c_2$, where $c_1,c_2\in crit(A_t)$, compatibly with concatenation of paths. We can build this system into the Floer differential. The local system defines a class $[l]\in H^1(C;\Lambda)$. If $R$ comes with a homomorphism $f\colon \mathbb{R}\to \Lambda$ (think Novikov ring!) then a local system $l_\mathbb{R}$ with coefficients in $\mathbb{R}$ defines one with coefficients in $\Lambda$. If contributions to the differential are weighted by $f\circ A_t$, then for such local systems we should modify the definition of $K_t$, since the contribution of periods to the differential has changed: now the definition should be $$K_t= \ker I \cap \ker ([A_t]+l_{\mathbb{R}}).$$ The continuity principle is as before.

As an example, one can see this way that if $L_t$ is a Lagrangian isotopy with flux $F\in H^1(L_0;\mathbb{R})$ then the local Floer homology $HF(L_0,L_1)_{loc}$ (taken in a suitable small neighbourhood of $L_0$ so as to avoid "instanton corrections") should be isomorphic to $H_\ast(L_0; \Lambda_F)$, where $\Lambda_F$ is the local system of Novikov rings determined by $F$.

Sample references:

Le Hong Van, Kaoru Ono, "Symplectic fixed points, the Calabi invariant and Novikov homology", MR1308493.

Kaoru Ono, "Floer-Novikov cohomology and the flux conjecture", MR2276532.

Yi-Jen Lee, "Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori. I", MR2199540

Mihai Damian, "Constraints on exact Lagrangians in contangent bundles of manifolds fibered over the circle". MR2534477