Someone at the Munich workshop suggested that it would be helpful to have a comparison of ECH versus SFT (Symplectic Field Theory, due to Eliashberg-Givental-Hofer). I think this would make a nice conclusion to an introduction to ECH, but there are more things which belong in an introduction which I have not yet posted notes about. However I have given up on trying to present things in any kind of reasonable order. So here we go with the comparison.
ECH is only defined for three-dimensional contact manifolds (and in some cases stable Hamiltonian structures) and certain four-dimensional symplectic cobordisms between them.
SFT is defined in all dimensions.
It is an interesting question whether there is a higher-dimensional analogue of ECH. Actually the first question is, what does this question mean exactly? Maybe after going through the rest of the comparion one might have a better idea of what to look for, although I have no particular reason to expect that such an object exists.
ECH depends only on the three-manifold (at least if you regard it as absolutely graded by homotopy classes of oriented two-plane fields; Dan Cristofaro-Gardiner has shown, in a paper to appear soon, that Taubes’s isomorphism identifies this absolute grading with the absolute grading of Seiberg-Witten Floer cohomology.)
SFT depends heavily on the contact structure; for example, the basic versions are trivial for overtwisted contact structures. On the other hand, ECH does contain the contact invariant (the homology class of the empty set of Reeb orbits) which can distinguish some contact structures. Note that the ECH contact invariant is analogous to the unit in the contact homology algebra.
The full version of SFT counts all Fredholm index 1 holomorphic curves (after suitable perturbation to make the moduli spaces transverse). Other versions of SFT just count genus 0 Fredholm index 1 curves (rational SFT), or genus 0 Fredholm index 1 curves with one positive end (the contact homology algebra).
ECH counts holomorphic curves with ECH index 1, without explicitly specifying their genus (although this is more or less determined indirectly by the theory, in a way which I plan to explain later). These also have Fredholm index 1, although the way we are selecting a subset of the Fredholm index 1 curves to count (by setting the ECH index equal to 1) is very different from the way this is done in SFT (by setting the genus to 0, etc.).
SFT is relatively graded by the Fredholm index. Absolute gradings in SFT are a subtle matter which I am not qualified to comment on.
ECH is relatively graded by the ECH index, and has an absolute grading by homotopy classes of oriented 2-plane fields.
SFT has some algebra structure (for example the contact homology algebra is an algebra).
ECH doesn’t. (There is a natural way to “multiply” ECH generators, but the differential and grading are not well behaved with respect to this “multiplication”.)
Multiply covered Reeb orbits.
In an ECH generator, we only care about the total multiplicity of each Reeb orbit.
In an SFT generator, one keeps tracks of individual covering multiplicities of Reeb orbits.
For example, if is an elliptic Reeb orbit, and if denotes the -fold multiple cover of , then and are distinct SFT generators which correspond to the same ECH generator . Likewise, the SFT generators , and all correspond to the ECH generator .
Disallowed Reeb orbits.
In ECH, hyperbolic orbits cannot have multiplicity greater than 1.
In SFT, “bad” Reeb orbits are thrown out; in the three-dimensional case, a bad Reeb orbit is an even cover of a negative hyperbolic orbit. The reasons for discarding bad orbits in SFT are similar to the reasons for disallowing multiply covered hyperbolic orbits in ECH; see installments 3 and 10 of the lecture notes.
Keeping track of topological complexity.
In SFT, there is a formal variable which keeps track of the topological complexity of holomorphic curves; whenever you count a curve with genus and positive ends, you multiply by .
In ECH, there is a quantity called which is similar to but a bit more subtle. I hope to explain this at some point here, but if you want to see it right away, it is explained in section 6 of “The ECH index inequality revisited”.
SFT defines invariants of Legendrian knots.
ECH doesn’t. At least not yet. I don’t have any good idea how to do this, but I still hope it might be possible somehow.
ECH defines capacities.
Other kinds of contact homology or SFT can also be used to define capacities. This has not been explored very much, and I hope to write more about it here later.
ECH has a U map counting holomorphic curves passing through a base point, and also an action of of the three-manifold, counting holomorphic curves intersecting a 1-cycle.
There are analogous structures on SFT (which can be more interesting for higher dimensional contact manifolds with lots of homology).
Field theory structure.
SFT can recover Gromov-Witten invariants of closed symplectic manifolds by cutting them into pieces, e.g. along contact-type hypersurfaces.
ECH can similarly recover Taubes’s Gromov invariant of closed symplectic four-manifolds; I outlined this in a blog posting in May and am currently writing up the details.
There is an exact triangle describing how linearized contact homology behaves under Legendrian surgery, due to Bourgeois-Ekholm-Eliashberg.
There is also an exact triangle describing the behavior of ECH under Legendrian surgery, although so far we can only define this by using Taubes’s isomorphism with Seiberg-Witten Floer cohomology. This played a key role in the proof of the Arnold chord conjecture in three dimensions.
Technical difficulties with multiply covered holomorphic curves.
Both SFT and ECH have serious technical difficulties arising from multiply covered holomorphic curves that are not cut out transversely. I think the difficulties are worse in SFT, at least in higher dimensions.
In SFT, it is expected that the polyfold theory of Hofer-Wysocki-Zehnder will resolve these difficulties.
In ECH, we could manage these difficulties to prove that using holomorphic curves (in the 200-page gluing paper with Taubes), but defining cobordism maps is harder and currently requires Seiberg-Witten theory. See installment 11 of these notes for a detailed discussion of the issues.