From SFT to ECH, Part 2

References:

  • [NOTES] M. Hutchings, Lecture notes on embedded contact homology
  • [FABERT] O. Fabert, Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Continuing the previous post, I now want to explain how to use the fact that the SFT Hamiltonian H satisfies H\circ H = 0 to deduce that the ECH differential \partial satisfies \partial^2=0. We saw in the previous post how to extract \partial from H, but are now going to have to do this in a different way which is more amenable to showing that \partial^2=0. To do so, we will need to assume some conjectural additional properties of the SFT Hamiltonian. In the next post I will try (not completely successfully) to explain why these should be true for reasonable perturbation schemes.

1. Filtration by the ECH index

Fix a nondegenerate closed contact three-manifold (Y,\lambda) and a generic almost complex structure J on {\mathbb R}\times Y, and imagine that we have made some abstract perturbations of the moduli spaces of holomorphic curves allowing us to define a Hamiltonian satisfying properties (P1)-(P3) from the previous post.

Recall that the Hamiltonian H is a formal differential operator which defines a map H:Q\to Q, where Q is the space of supercommutative polynomials in the good Reeb orbits over {\mathbb Q}. If we apply H to an SFT generator x (a monomial in Q), then each term in Hx can be regarded as counting the union of two things:

  • a Fredholm index 1 (abstractly perturbed) holomorphic curve with positive ends at some of the Reeb orbits in x (those that are differentiated),
  • a union of trivial cylinders over the remaining Reeb orbits in x (those that are not differentiated).

This union has a well-defined ECH index. We can now define, for each integer k, a map H_k:Q\to Q, where H_kx is the sum of those terms in Hx such that the corresponding union of a Fredholm index 1 curve and trivial cylinders has ECH index k. Thuse we have a decomposition

H = \sum_k H_k : Q \to Q.

(Note that H_k is not a differential operator. In particular, it is not the sum of the terms in H corresponding to index 1 curves of ECH index 1. The reason is that taking the union of a curve with a trivial cylinder may increase its ECH index.)

We have H_k=0 when k<0, because by [NOTES, Prop. 3.7] there are no (possibly broken) holomorphic currents of negative ECH index. Furthermore, results in [FABERT] strongly suggest that

(P4) H_0=0.

To prove this, one just needs to check that branched covers of {\mathbb R} cross a hyperbolic orbit with one branch point cannot contribute to the Hamiltonian. The results in [FABERT] should imply this for any reasonable abstract perturbation scheme.

Thus, we now have a decomposition of the Hamiltonian (as a map Q\to Q) of the form

H = H_1 + H_2 + \cdots.

Furthermore, the additivity property of the ECH index allows us to split the equation H\circ H=0 into the equations

H_1^2=0, H_1H_2+H_2H_1=0, H_1H_3 + H_2^2 + H_3H_1=0, …

This looks promising: it gives us a spectral sequence where the first term is the homology of H_1, the second term is the homology of H_2 acting on the homology of H_1, and so on. One could imagine a spectral sequence starting from ECH and converging to SFT. However this is not so simple because the ECH and SFT generators are quite different, as we have seen. Here H_1 acts on SFT generators, and in fact basic examples (such as a birth/death bifurcation) show that the homology of H_1 is not invariant under deformation of the contact form.

On the other hand, the previous post does show that the ECH differential \partial can be extracted from some of the terms in H_1. We now want to deduce that \partial^2=0 from H_1^2=0. The difficulty here is that contributions to \langle\partial\alpha,\beta\rangle from holomorphic curves without trivial cylinders count curves going from \alpha_+ to \beta_-. There are problems composing these to try to prove that \partial^2=0, because usually \beta_+\neq \beta_-. There are also many other partitions that could arise, adding additional terms to the equation.

2. Where we are going.

We can resolve the above issues as follows. First we need one more bit of notation. If x is an SFT generator, define a positive integer d(x) to be the product, over each Reeb orbit \gamma of multiplicity d that appears n times in x, of n!d^n. (This is the number of ways of pregluing a holomorphic curve with positive ends at x to a holomorphic curve with negative ends at x.) Now define an operator \kappa:Q\to Q by \kappa(x) = d(x)x. We can now state:

Conjecture (+ version). Assuming additional properties of H stated below, the abstract perturbations can be chosen so that if \alpha and \beta are ECH generators, then:

(a) If \langle H_1 x,\beta_+\rangle \neq 0 and |x|=\alpha, then x=\alpha_+.

(b) \langle\partial\alpha,\beta\rangle = \langle \kappa H_1\kappa^{-1}\alpha_+,\beta_+\rangle.

Assuming this conjecture, which might appear completely outrageous right now, we can deduce that \partial^2=0 as follows. Let \alpha and \gamma be ECH generators. Then

0 = \langle H_1^2\kappa^{-1}\alpha_+,\kappa^{-1}\gamma_+\rangle

= \sum_x \langle H_1\kappa^{-1}\alpha_+,\kappa^{-1}x\rangle \langle H_1\kappa^{-1}x,\kappa^{-1}\gamma_+\rangle

(where the sum is over SFT generators x; here we are expanding in the basis \{\kappa^{-1}x\} of Q)

= \sum_\beta \sum_{|x|=\beta} \langle H_1\kappa^{-1}\alpha_+,\kappa^{-1}x\rangle \langle H_1\kappa^{-1}x,\kappa^{-1}\gamma_+\rangle

(where the sum is over ECH generators \beta; in the previous sum, |x| can never be an orbit set which is not an ECH generator because of the partition conditions, see [NOTES, Section 3.9])

= \sum_\beta \langle H_1\kappa^{-1}\alpha_+,\kappa^{-1}\beta_+\rangle \langle H_1\kappa^{-1}\beta_+,\kappa^{-1}\gamma_+\rangle

(by part (a) of the conjecture)

=\sum_\beta \langle \kappa H_1\kappa^{-1}\alpha_+,\beta_+\rangle \langle \kappa H_1\kappa^{-1}\beta_+,\gamma_+\rangle

= \sum_\beta \langle\partial\alpha,\beta\rangle \langle\partial\beta,\gamma\rangle

(by part (b) of the conjecture)

= \langle \partial^2\alpha,\gamma\rangle.

Note that this argument proves that \partial^2=0 over {\mathbb Z}.

3. Cobordism chain maps

We can use a similar argument to define (at least over {\mathbb Q}) chain maps on ECH induced by an (exact) symplectic cobordism when the (completed) cobordism admits an almost complex structure without negative ECH index multiple covers. This is a very special situation, but it does happen sometimes (at least up to large symplectic action), and defining cobordism chain maps in this case allows one to prove that ECH (at least as an isomorphism class of {\mathbb Q}-vector spaces) does not depend on the choice of almost complex structure used to define it.

More specifically, let (Y_+,\lambda_+) and (Y_-,\lambda_-) be closed nondegenerate contact three-manifolds. Let Q_\pm denote the {\mathbb Q}-vector space spanned by SFT generators (modulo anticommutation as usual) for (Y_\pm,\lambda_\pm).

Let (X,\omega) be an exact symplectic cobordism from (Y_+,\lambda_+) (the convex end) to (Y_-,\lambda_-) (the concave end). Choose generic almost complex structures J_+ on {\mathbb R}\times Y_+ and J_- on {\mathbb R}\times Y_- as needed to define ECH. Let \overline{X} denote the completed cobordism and choose a “cobordism admissible” almost complex structure on \overline{X} extending J_\pm, see [NOTES, Section 5.5].

After suitably abstractly perturbing the moduli spaces of J-holomorphic curves in \overline{X} (compatibly with the abstract perturbations of moduli spaces of J_\pm-holomorphic curves in {\mathbb R}\times Y_\pm, which we should do first), we expect to obtain a map

\Phi: Q_+ \to Q_-

with the following properties:

(P5) \Phi satisfies the obvious analogues of properties (P1) and (P2).

(P6) \Phi\circ H_+ = H_-\circ \Phi, where H_\pm is the Hamiltonian for (Y_\pm,\lambda_\pm).

As in the earlier part of this post, we can decompose \Phi=\sum_k\Phi_k where \Phi_k is the contribution from (possibly broken) holomorphic currents of ECH index k for each integer k. Then, for each integer k, we have

\sum_{i+j=k}\Phi_i (H_+)_j = \sum_{i+j=k} (H_-)_i \Phi_j.

Suppose now that \overline{X} contains no negative ECH index holomorphic currents. (The ECH index inequality implies that if J is generic then any holomorphic curve in \overline{X} without multiply covered components has nonnegative ECH index. However multiple covers may have negative ECH index.) Then by (P5) we have \Phi_k=0 when k<0. It then follows from the above equation with k=1 that

\Phi_0 (H_+)_1 = (H_-)_1 \Phi_0.

We would now like to use \Phi_0 to define a map from the ECH chain complex of (Y_+,\lambda_+) to that of (Y_-,\lambda_-), and we would like to use the above equation to show that it is a chain map.

We first need an alternate, symmetric version of the above conjecture.

Conjecture (- version). Let (Y,\lambda) be a nondegenerate closed contact three-manifold and let J be a generic almost complex structure on {\mathbb R}\times Y. Assuming additional properties of H stated below, the abstract perturbations can be chosen so that if \alpha and \beta are ECH generators, then:

(a) If \langle H_1 \alpha_-,y\rangle \neq 0 and |y|=\beta, then y=\beta_-.

(b) \langle\partial\alpha,\beta\rangle = \langle \kappa H_1\kappa^{-1}\alpha_-,\beta_-\rangle.

We now want to define H_+ using abstract perturbations satisfying the “- version” of the claim for (Y_+,\lambda_+,J_+), and define H_- using abstract perturbations satisfying the “+ version” of the claim for (Y_-,\lambda_-,J_-). We then define a map \phi on the ECH chain complexes as follows: If \alpha is an ECH generator for (Y_+,\lambda_+), and if \beta is an ECH generator for (Y_-,\lambda_-), then

\langle \phi\alpha,\beta\rangle = \langle\kappa\Phi_0\kappa^{-1}\alpha_-,\beta_+\rangle.

We can now prove that \phi is a chain map on the ECH chain complexes, i.e. \phi\partial_+ = \partial_-\phi, by the same argument as before. This time we let \alpha be an ECH generator for (Y_+,\lambda_+) and \beta an ECH generator for (Y_-,\lambda_-), and start with the equation

0 = \langle (\Phi_0) (H_+)_1 - (H_-)_1\Phi_0)\kappa^{-1}\alpha_-,\kappa^{-1}\beta_+\rangle.

4. Limitations of this approach

If there are negative ECH multiple covers in \overline{X}, then the above approach does not work to define a chain map on the ECH chain complexes (at least not without some additional, missing ingredient), because in this case we no longer know that \Phi_0(H_+)_1 = (H_-)_1\Phi_0; the equation has additional terms such as \Phi_{-1} (H_+)_2 etc. which might be nonzero. We need some way of eliminating terms like this, and I don’t know what that is (but I hold out hope that there may be some magic identities that will allow us to do this).

Chain homotopies are also problematic. They will be fine if in a one-parameter family we don’t see multiple covers of ECH index less than -1, but this is unlikely to hold even in the best cases. (Chain homotopies are also a highly nontrivial issue in SFT because in a one-parameter family there can be holomorphic buildings with repeated Fredholm index -1 curves.)

Finally, ECH cobordism maps should be defined over {\mathbb Z}, but I’m not sure if we can see that using this approach. (This might be obvious, but I have a headache.)

5. Still to come

I would now like to deduce the above conjectures from more plausible properties of the SFT Hamiltonian. I will try to do this in the next post (but will not completely succeed).

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