From SFT to ECH, Part 3

References:

  • [NOTES] M. Hutchings, Lecture notes on embedded contact homology
  • [OBG1] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I

Sorry, I overreached a bit in my previous blog post. In the previous post, I made some “claims” (which I have now downgraded to “conjectures”) about the structure of the SFT Hamiltonian. I then explained how these claims/conjectures can be used to deduce that the ECH differential \partial satisfies \partial^2=0, and that ECH (at least as an isomorphism class of {\mathbb Q}-vector spaces) does not depend on the almost complex structure. I was intending to use the present post to deduce these claims/conjectures from some basic axioms about the SFT Hamiltonian which a reasonable abstract perturbation scheme should satisfy. I thought I would be just quickly explaining an old idea, but I realize now that it is a bit more complicated. That is, I have some axioms to state, but they don’t quite imply the claims/conjectures. Some more axioms are needed, and some deep thought is needed to figure out what these should be. (My old notes contain such additional axioms, but I haven’t yet recalled what exactly I was thinking and this will probably require some modification anyway.) The problem lies somewhere between understanding what algebraic/combinatorial structure can be consistent, and understanding how abstract perturbations should work. Anyway, let me now explain the basic idea of this, and describe what remains to be done.

The issue is to understand how the SFT Hamiltonian should count ECH index 1 holomorphic buildings that include index 0 branched covers of trivial (i.e. {\mathbb R}-invariant) cylinders. The idea of the “+ version” of the claim/conjecture is that we can arrange that certain buildings with branched covers of trivial cylinders on the top do not contribute. We would arrange this either by choosing the abstract perturbations appropriately, or by conjugating the Hamiltonian by a “repartitioning operator”, see below. The idea of the “- version” of the claim/conjecture is that we alternately can arrange that certain buildings with branched covers of trivial cylinders on the bottom do not contribute. As I explained in the previous post, either of these claims implies that \partial^2=0, and both of them together imply that an exact symplectic cobordism with no negative ECH index multiple covers induces a chain map on ECH. The nice thing about this last argument is that one does not need to know anything about the details of the cobordism map on SFT, except that it is a chain map with respect to the Hamiltonians and does not include contributions from negative ECH index holomorphic buildings.

1. Setup

More precisely, as before, let (Y,\lambda) be a nondegenerate closed contact three-manifold, let J be a generic almost complex structure on {\mathbb R}\times Y satisfying the usual conditions, and let H be the (currently mythical) SFT Hamiltonian as defined using some abstract perturbation of the compactified moduli spaces of J-holomorphic curves.

The issue that we need to sort out is how H should count certain holomorphic buildings that include branched covers of trivial ({\mathbb R}-invariant) cylinders. More specifically, let \alpha and \beta be ECH generators. If there is an ECH index 1 curve from \alpha to \beta (without trivial cylinders), then this will have positive ends at the SFT generator \alpha_+ and negative ends at the SFT generator \beta_-, and thus will contribute \pm1 to the count of curves n(\alpha_+;\beta_-) that enters into the SFT Hamiltonian. Now if \alpha includes an elliptic orbit with multiplicity greater than one, then there exist SFT generators x\neq \alpha_+ with |x|=\alpha such that there is an index 0 union of covers of trivial cylinders from x to \alpha_+. Likewise, if \beta includes an elliptic orbit with multiplicity greater than one, then there exist SFT generators y\neq \beta_- with |y|=\beta such that there is an index 0 union of covers of trivial cylinders from \beta_- to y. Putting this all together gives a holomorphic building from x to y. Indeed there is a moduli space of such buildings whose dimension equals twice the total number of branch points. Now the equation is, what is the contribution from such buildings to the count n(x;y)? I will argue below that at least some of these (moduli spaces of) buildings must make nonzero contributions to n(x;y). In fact, the equation H\circ H basically requires these contributions to satisfy certain relations which imply that some of them are nonzero.

2. The gluing matrix

Before describing the contributions of the above holomorphic buildings to the SFT Hamiltonian, we first need to introduce the gluing matrix.

Define a partial order on the set of SFT generators as follows: we say that x\ge y if there exists a Fredholm index zero union of (possibly branched) covers of trivial cylinders from x to y. In particular |x|=|y|. See [NOTES, Ex. 3.14(b)] for the proof that this is a partial order.

Now suppose that x\ge y. Suppose that u_+ is a somewhere injective, Fredholm index 1, irreducible (i.e. without trivial cylinders) curve with negative ends at x, and suppose that u_- is a somewhere injective, Fredholm index 1, irreducible curve with positive ends at y. Then, according to [OBG1, Thm. 1.13], it is possible to glue x and y by inserting a Fredholm index 0 union of branched covers of trivial cylinders between them. The signed number of ways to glue equals the product of the signs associated to x and y with a “gluing coefficient” which I will denote here by G(x;y)\in{\mathbb Z}.

There is an explicit combinatorial formula for the gluing coefficients in [OBG1, Sections 1.5-1.6]. We do not need to know this formula here; we just need to know one key property, which is that if \alpha is an ECH generator then G(\alpha_-;\alpha_+)=1. This is the key property which enters into the proof that the ECH differential \partial^2=0, because it says that there is (counted with signs) one way to glue a pair of irreducible curves with ECH index 1. (The proof that \partial^2=0 has some additional complications because one also needs to consider holomorphic currents that include unions of trivial cylinders, but we will not go into this right now.)

And one other property (which is pretty obvious), namely G(x;x)=d(x) (see the previous post for the definition of d(x)).

I would now like to think of the gluing coefficients G(x;y) as defining a linear map G:Q\to Q. This is the “gluing matrix”.

3. Repartitioning operators

Before proceeding, we need a few more definitions.

First, define a “repartitioning operator” to be a linear map S:Q\to Q such that \langle Sx,x\rangle=1 for each SFT generator x, and \langle Sx,y\rangle\neq 0 implies that x\ge y. For example, \kappa^{-1}G and G\kappa^{-1} are repartitioning operators. (See the previous post for the definition of \kappa.) Note that any repartitioning operator is invertible, because it is upper triangular with respect to the partial oder \ge (and respects the decomposition of Q into a sum over orbit sets of finite dimensional vector spaces).

Second, let u be a somewhere injective, Fredholm index 1, irreducible curve from x to y. Let us say that u is “isolated as a current” if it cannot be glued to a union of index 0 branched covers of trivial cylinders above and/or below it. That is, if x'\ge x and y\ge y', then any building consisting of index 0 branched covers from x' to x, followed by u, followed by index 0 branched covers from y to y', is isolated in the compactified moduli space \overline{\mathcal M}. For example, this automatically holds if u has ECH index one, because of the partition conditions in the ECH index inequality.

In fact, I have some ideas for maybe proving that any somewhere injective, Fredholm index 1, irreducible curve is isolated as a current if J is generic. (A special case of this appeared in David Farris’s thesis.) I can blog about this later if anyone is interested.

Anyway, given u, x', y' as above, we expect (under any reasonable perturbation scheme) that there is a well-defined contribution to the curve count n(x';y') from the set of all such buildings. Let us denote this contribution by n(x';u;y')\in{\mathbb Q}.

I now conjecture that the abstract perturbations can be chosen so that the SFT Hamiltonian has the following property:

(P7) There are repartitioning operators S_+ and S_- such that:

(a) If u is a somewhere injective, Fredholm index 1, irreducible curve from x to y which is isolated as a current, and if x'\ge x and y\ge y', then n(x';u';y') = \langle S_+ x',x\rangle \epsilon(u) \langle S_-y,y'\rangle where \epsilon(u) denotes the sign of u.

(b) S_+\kappa S_- = G.

Why do I expect this property? Part (a) is just the simplest way I can think of that these buildings would be counted. It is vaguely plausible that as one perturbs the moduli spaces, the first step would be to perturb the moduli spaces of index 0 branched covers of trivial cylinders (which of course would make them disappear), and the way in which one does this would determine the repartitioning operators.

If we believe part (a), then part (b) is more or less forced by the gluing theorem in [OBG1] so that broken curves with index 0 branched covers of trivial cylinders in the middle will be counted properly in the proof that H^2=0. To spell this out more explicitly, suppose that we have somewhere injective, Fredholm index 1, irreducible curves u_+ from w to x and u_- from y to z that are isolated as currents, where x\ge y. Then we know from [OBG1] that the number of ends of the index 2 moduli space from w to z that converge to buildings consisting of u_+ and u_- with index 0 branched covers of trivial cylinders between them equals G(x;y)\epsilon(u_+)\epsilon(u_-). This should be the local contribution to \langle H^2\kappa^{-1}w,z\rangle. Let us use a subscript “loc” to indicate this local contribution. We can also express this local contribution as a sum over SFT generators x' with x\ge x'\ge y. We get

G(x;y)\epsilon(u_+)\epsilon(u_-) = \langle H^2\kappa^{-1}w,z\rangle_{loc}

= \sum_{x\ge x'\ge y}\langle H\kappa^{-1}w,x'\rangle_{loc} \langle Hx',z\rangle_{loc}

= \sum_{x\ge x'\ge y} n(w;u_+;x') d(x') n(x';u_-;z)

= \sum_{x\ge x'\ge y} \epsilon(u_+)\langle S_-x,x'\rangle d(x') \epsilon(u_-)\langle S_+x',y\rangle

(by property (a))

= \epsilon(u_+)\epsilon(u_-)\langle S_+\kappa S_-x,y\rangle.

Two remarks. First, since the gluing matrix is appearing in the SFT Hamiltonian, in order to describe the Hamiltonian explicitly like this (starting from a polyfold perturbation etc.), one will probably have to do work equivalent to the obstruction bundle computations in [OBG1]. Second, property (P7) will force some of the curve counts n(x;y) coming from buildings with ECH index one to be in {\mathbb Q}\setminus {\mathbb Z}.

4. Towards the claims/conjectures, and more questions.

I was hoping that we could use property (P7) to prove the claims/conjectures from the previous post, but this is not quite sufficient. The idea is that we can arrange that S_+=1 (either by choosing the abstract perturbations suitably or by conjugating the Hamiltonian by a repartitioning operator), and then this should imply the “+ version” of the claim. Likewise the “- version” of the claim/conjecture should hold when S_-=1, which we should likewise be able to arrange.

To prove the claims/conjectures, we need to understand how the SFT Hamiltonian should count certain holomorphic buildings of ECH index 1. Such a holomorphic building consists of an embedded, irreducible, Fredholm index 1 curve u, possibly together with some branched covers of trivial cylinders.

Property (P7) tells us how the SFT Hamiltonian should count such buildings with branched covers of trivial cylinders above and/or below u. However, such buildings may also include branched covers of trivial cylinders “on the side”, namely in the same level as u. Property (P7) does not tell us how the Hamiltonian should count such buildings, and hence is not sufficient to prove the claims/conjectures.

So what we really want to understand (and we only need certain cases of this to prove the claims/conjectures) is:

  • How does the SFT Hamiltonian count arbitrary ECH index 1 holomoprhic buildings?

Assuming we know what the embedded, ECH index 1 irreducible curves are, then we can describe all of these buildings combinatorially. So this is an example of a more general question:

  • If transversality fails, but you still know what all the holomorphic curves are, how do you count them?

Clearly there is no shortage of problems to work on here.

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