Rational SFT using only q variables IV

It is now time to explain how to use the q-variable version of rational SFT to define symplectic capacities and symplectic embedding obstructions. Before getting into details, I should tell you some good news and bad news. The bad news is that the rational SFT capacities are not very powerful. The good news is that rational SFT contains stronger embedding obstructions than the rational SFT capacities, and you can access these stronger obstructions if you can compute cobordism maps. This is a very interesting area for further exploration.

Spectral invariants

Let be a nondegenerate closed contact manifold as usual. If is a nonzero class in the q-variable rational SFT, define to be the infimum over such that is in the image of the map induced by the inclusion of chain complexes.

If is a weakly exact symplectic cobordism from to , recall that there is an induced map

which is the direct limit of maps

as . It follows as in ECH that if , if , and if , then

This is the basic inequality which leads to obstructions to symplectically embedding one Liouville domain into another.

As with the ECH spectrum, if is degenerate, one can define as the limit of where is a sequence of nondegenerate contact forms converging in the topology to . The above inequality still holds for a weakly exact cobordism between possibly degenerate contact manifolds.

Definition of capacities

Let be a Liouville domain. What I mean by this is that is exact, and there is a contact form on such that . Note that the unit in the symmetric algebra on the polynomial algebra on the variables, which I denote by , is a cycle in the chain complex computing . The homology class is nonzero, because the cobordism map maps it to .

If is a nonnegative integer, we now define to be the minimum of , where ranges over classes in such that whenever is a composition of of the maps associated to the components of (possibly repeated). If no such class exists, we define . The same or similar arguments to the definition of ECH capacities show the following:

• does not depend on the choice of .
• If is another Liouville domain of the same dimension which symplectically embeds into , then for all .
• We have the disjoint union property

Examples

I did some quick calculations to get an idea of how good these capacities are. I can explain the details of these calculations later, but for now here are the results (which you should regard as preliminary since I didn’t check every detail):

• The capacities of the four-dimensional ball , starting at , are . That is, for .
• The capacities of the four-dimensinal polydisk for are given by .
• If then .

The bad news

To test how powerful these capacities are, let us consider the question of when the disjoint union of four-dimensional balls of capacity be symplectically embedded into the four-ball of capacity . (Recall that ECH capacities are sharp for this embedding problem.) McDuff-Polterovich showed that if such an embedding exists, then when , when , when , and when , and these bounds are optimal.

We can compute the rational SFT capacities of using the disjoint property. We find that , while , so we recover the inequality when . We also compute that while , so we recover the inequality when . Unfortunately, when or we do not even recover the volume constraint .

The ultimate reason for this is that the first inequality comes from curves of degree in , the second inequality comes from curves of degree , the third inequality comes from curves of degree , and the fourth inequality comes from curves of degree . Since the latter two curves have positive genus, rational SFT does not see them. One might hope that rational SFT capacities see some non-embedded rational curves that ECH does not see, e.g. for embedding a polydisk into an ellipsoid where the ECH capacities do not give sharp obstructions, but so far I have not found an example where rational SFT capacities say anything more than ECH capacities.

When , rational SFT capacities say even less about packing a disjoint union of balls into a ball. All they tell us is that if symplectically embeds into , then , which was known to Gromov.

The good news

The good news is that rational SFT gives stronger obstructions to symplectic embeddings than rational SFT capacities, if you know something about the cobordism map.

In particular, rational SFT does recover the optimal obstructions to symplectically embedding the disjoint union of seven or eight four-balls of equal size into a four-ball! To see why, suppose there exists a symplectic embedding of into where . Let be the symplectic cobordism obtained by removing the image of the embedding from . We can perturb this to obtain a cobordism between nearly-round ellipsoids with nondegenerate contact forms. Each ellipsoid has two embedded elliptic Reeb orbits which we denote by and , where denotes the shorter orbit. The ECH generator differential vanishes because all generators have even grading.

We know from general properties of ECH cobordism maps that when (resp. ), the map induced by the cobordism has a nonzero coefficient from the ECH generator (resp. ) to the ECH generator consisting of in one of the balls and in the other six balls (resp. in one of the balls and in the other seven balls). Since the cobordism map decreases symplectic action, we conclude that when and when , which are the optimal inequalities.

Now since the above coefficient of the ECH cobordism map is nonzero, there exists a (more precisely ) holomorphic curve between the corresponding ECH generators. In general one could get a broken and/or multiply covered curve, but (I think) one can rule that out in the present case because otherwise one would obtain a stronger embedding obstruction which cannot be true. Moreover, one can use the ECH partition conditions to see that each end of this holomorphic curve is at a singly covered Reeb orbit, so that it has ends when and ends when . One can then use the relative adjunction formula to show that the holomorphic curve has when and when . This implies that it is rational. So the cobordism map on rational SFT sees it!

More precisely, in rational SFT, the differential vanishes here because all holomorphic curves have even index, so we can identify elements of the chain complex with homology classes. If denotes when and when , and if denotes the cobordism map on rational SFT, then includes a monomial with of the variables when and of the variables when . It follows that when and when , so the inequality at the beginning of this post gives the optimal symplectic embedding obstruction.

In fact, as far as I know it it possible that all of the ECH obstructions to embedding a disjoint union of four-dimensional ellipsoids into an ellipsoid are seen by rational SFT cobordism maps.

The puzzle

In the above example of embedding seven or eight four-dimensional balls into a ball, it is a nontrivial exercise (which I haven’t really tried yet) to compute the relevant part of the rational SFT cobordism map without “cheating” and using information from ECH.

One can get some information about the cobordism map by using the fact that it commutes with the U maps, compare Lemma 3.2 in “The asymptotics of ECH capacities”. However in the present case one has to work harder because the U map is no longer injective on generators.

In dimension , if one can somehow compute the rational SFT cobordism map coming from an embedding of a disjoint union of balls into a ball, then one might get stronger obstructions to ball packing than the Gromov obstruction recalled above.

Likewise, for embedding one ellipsoid into another in higher dimensions, I suspect that rational SFT capacities say nothing more than Ekeland-Hofer capacities, but if one can compute the cobordism map one might be able to obtain stronger obstructions.

Obstructing symplectic embeddings into CP^2 and S^2xS^2

Hello from the CRM in Montreal, where I am at a very nice conference this week. In my talk I attempted (among other things) to present the following theorem. Unfortunately my explanation was not very organized and I said a couple of things slightly wrong. (Sorry audience!) So let me attempt to explain all this more clearly here. Below, denotes the ECH capacity.

Theorem. Let be a disjoint union of finitely many star-shaped domains in . If can be symplectically embedded into , then for all . Likewise, if can be symplectically embedded into , then for all .

That is, as far as ECH capacities are concerned, embedding into is no easier than embedding into a ball; and embedding into is no easier than embedding into a polydisk. (One can prove things for more general Liouville domains , for example using the completed ECH capacities introduced here last August, but assuming that is a union of star-shaped domains makes life simpler.)

First, recall that if is a strong symplectic cobordism from to , then for each real number and each class there is an induced map

.

Here is a homomorphism which measures the failure of and the contact forms to obey Stokes’s theorem. (I explained this here last May.) This cobordism map commutes with the maps on the top and bottom and preserves the contact invariant. If is closed and , then is the Gromov/Seiberg-Witten invariant mod for any . One can recover Gromov invariants with as follows: suppose is a contact type hypersurface splitting into two unions of components and so that there is a contact form on making into a cobordism from the empty set to and into a cobordism from to the empty set. Suppose for simplicity that maps to zero in . Let denote the restriction of to . Let denote any composition of of the maps (possibly repeated) associated to the components of . Then

for any . The idea here is that counts holomorphic curves in in the class with point constraints, and to prove the formula we can stretch the neck along and put the constraining points into the neck. (The real proof of course uses Seiberg-Witten.)

Now to prove the first part of the theorem, let be a disjoint union of star-shaped domains and suppose that symplectically embeds into . Recall that the ECH capacities of the ball are given by whenever

So it is enough to show that if then . For this purpose let denote the complement of the image of in , and let where denotes the homology class of a line. Then and mod 2 (by the wall crossing formula for Seiberg-Witten invariants). Also, . So for any , the above composition property implies that is a class in such that . Note that sends the contact invariant to and all classes of positive grading to zero. It follows from the previous two sentences, similarly to the proof of Proposition 4.5 in my paper “Quantitative ECH”, that . Since was arbitrary, we are done.

The argument for embeddings into is similar. Recall that

where and are nonnegative integers. Now suppose that is a disjoint union of star-shaped domains that embeds into . It is enough to show that if are nonnegative integers with then . For this purpose let . Then , and mod 2 by the wall crossing formula for Seiberg-Witten invariants. The rest of the argument is the same as before.

I’ll continue my series about rational SFT shortly; I have done some calculations and while there were some initial disappointments, things are now getting very interesting.

 

Rational SFT using only q variables III

Let be a nondegenerate closed contact manifold of dimension . In the previous two posts, I defined (modulo the usual transversality issues) a modified version of rational SFT, which we are temporarily denoting by . I now want to define a map on this theory, by analogy with the map on ECH.

Definition of the U map

Recall that denotes the (completed) algebra generated by the and variables. We choose a generic almost complex structure on and perturbation of the holomorphic curve equation, and then denotes the Hamiltonian counting connected index one rational curves.

Now pick a generic point . We can then consider connected rational curves of index in with a marked point mapping to . Counting these curves with the usual signs and combinatorial factors, assuming transversality as usual, defines an element . Looking at ends of moduli spaces of index rational curves with a marked point mapping to shows that the Poisson bracket . This is enough to show that induces a map on the usual rational SFT (which has been studied in the SFT literature).

Now recall from the first post in this series that denotes the symmetric algebra on the polynomial algebra on the variables, and we have a multiplication . This is related to the Poisson bracket as follows: if are monomials in and then

where denotes the grading parity of . It follows from the above identity and that multiplication by defines a differential on , whose homology we are calling . It also follows from the above identity and that is a chain map from to itself. The induced map on is the map, which we denote by .

The map respects the symplectic action filtration, by the usual Stokes’s theorem argument.

When is connected, does not depend on the choice of base point . To see this, let be another base point and let be a generic path from to . We then consider pairs where and is an index rational curve with a marked point mapping to . Counting these with appropriate signs and combinatorial factors defines an element such that . It follows from this equation and the above identity that and induce the same map on .

If is disconnected, then there is a different map for each component of . If are the components of , then it follows from the definition that , and the map associated to the component of is the tensor product of the map on with the identity on the other factors. Just like in ECH. [Correction 5/1/13: It is not quite right that . Rather, there is a map , induced by the obvious inclusion of chain complexes, and this map is injective because it has a right inverse induced by a chain map which "disconnects" Reeb orbits in different components of . Under these maps, the map on associated to the component agrees with the tensor product of the map on with the identity on the other factors.]

Similar arguments show that the map (or maps when is disconnected) do not depend on (and the perturbation of the holomorphic curve equation). Rather than explain this, let us explain a more general phenomenon which implies it, namely how the map behaves under cobordisms.

The U map and cobordism maps

Let be an exact symplectic cobordism from to . In the previous post we defined a cobordism map which respects the symplectic action filtrations. I claim if denotes the map on determined by one of the components of , and if denotes the map on determined by one of the components of , and if these components of and are contained in the same component of , then

To see this, recall from the previous post that counting connected index zero rational curves in the completed cobordism defines an element , and (see the previous post for explanation of this notation) induces the cobordism map .

Now pick a base point . Then counting index connected rational curves in the completed cobordism with a marked point mapping to defines an element . Define

This counts possibly disconnected index rational curves in the completed cobordism with a marked point mapping to . Considering ends of moduli spaces of possibly disconnected index rational curves in the completed cobordism with a marked point mapping to shows that

Hence induces a map .

Now this induced map is equal to if is the map on corresponding to a component of in the same component of as . It is also equal to if is the map on corresponding to a component of in the same component of as . One defines a chain homotopy by taking a path which starts at and goes off to infinity on one of the ends and counting index rational curves in the completed cobordism with  a marked point mapping to a point on .

If is a weakly exact cobordism from to , then the cobordism map commutes with the maps in the same way, by the same argument.

Now we are ready to define capacities! I will do this in the next post.

Rational SFT using only q variables II

In the previous post, I outlined how to define a modified version of rational SFT using only q variables. To continue the discussion, I now want to explain how to define cobordism maps and a U map on this version of rational SFT.  This algebra may be big and unwieldy, but that doesn’t really matter, because at the end of the day we are just going to extract real numbers from it! These are the new symplectic capacities which I am building up to defining.

Review from last time

Let be a closed nondegenerate contact manifold of dimension . The usual rational SFT defines a supercommutative algebra generated by variables and for the good Reeb orbits , with a Poisson bracket . Counting index rational -holomorphic curves in (after suitable abstract perturbations of the -holomorphic curve equation which will hopefully soon be available) defines a “Hamiltonian” in . This satisfies , and so defines a differential on , whose homology is the usual rational SFT.

The alternate version I introduced last time uses an algebra , which (as Janko Latschev points out) can be succintly described as the symmetric algebra on the polynomial algebra on the variables. There is a multiplication , and defines a differential on , whose homology is the “q variable only” version of rational SFT. For lack of a better notation, let us temporarily denote this by .

For every real number , there is also a filtered version , which is the homology of the subcomplex generated by tensor products of monomials in the variables in which the total symplectic action of the Reeb orbits is less than .

Cobordism maps

I now want to define cobordism maps on (which in particular will prove that it does not depend on the choice of ). This is formally very much analogous to the definition of ECH cobordism maps (with the technical difference that ECH cobordism maps use Seiberg-Witten theory instead of polyfolds to perturb the holomorphic curve equation). As in the ECH case, there are three basic types of cobordisms to consider: exact symplectic cobordisms, “weakly exact” symplectic cobordisms, and strong symplectic cobordisms.

Suppose first that is an exact symplectic cobordism from to . We want to define an induced map

The idea is to define a chain map by counting index zero, genus zero, possibly disconnected holomorphic curves in the the completion of , keeping track of what is allowed to be glued without increasing the genus. The formalism is straightforward based on the ideas introduced in the previous post, but a bit messy to write down. Anyway here are the details.

Algebraic preliminaries

Let and denote the algebras and for , and let and denote the corresponding algebras for . Introduce a new algebra , which is the supercommutative algebra generated by the variables for and the variables for , completed to allow infinite sums of monomials in the variables each multiplied by a polynomial in the variables. Also introduce the symmetric algebra on this, which we denote by . One can think of a tensor product of monomials in as representing the ends of a holomorphic curve in (the completion of) with components, each of genus zero, keeping track of which ends of the holomorphic curve are in the same component.

There is now a multiplication map , defined analogously to the multiplication . To multiply a tensor product of monomials in by a monomial in , let denote the number of variables in . One then sums over all ways of cancelling each of the variables in with a variable in , where the variables that get cancelled are in distinct factors in the tensor product. One obtains the remaining and variables in , together with the variables in , grouped into factors as before.

There is also a multiplication map . One multiplies a tensor product of monomonials in by a tensor product of monomials in as follows. The product is zero unless there exists a bijection between the variables in and the variables in that pairs up and variables corresponding to the same Reeb orbit.  Given such a bijection , let denote the graph whose vertices are the tensor factors in and , and whose edges are determined by . One now sums over all bijections such that the graph does not contain any loops. The contribution from , up to the usual signs and combinatorial factors, consists of the variables in , grouped into tensor factors according to the connected components of .

The above multiplications are associative, i.e. there is a well-defined multiplication

obtained by multiplying in either order. The way I like to think of this multiplication is that a tensor product of monomials in can be regarded (up to orientation data) as a graph with components, each consisting of a single vertex with some and edges attached to it. A monomial in can be regarded as a graph with a single vertex with some and edges attached to it. And a tensor product of monomials in can be regarded as a graph with components, each consisting of a single vertex with some edges attached to it. A triple product , where , , and are graphs as above, multiplied in either order, is (with appropriate signs and combinatorial factors) the sum over all ways of gluing all of the edges in and to all of the edges in and , so that glued edges correspond to the same Reeb orbit, and the graph obtained from the gluing contains no loops, and then collapsing the glued edges.

There are analogous multiplication maps and , which are associative so that there is a well-defined triple product

A different associativity relation which is more important for us right now is that there is a well-defined triple product

The chain map

To define the cobordism chain map, we now complete to by attaching symplectization ends as usual, choose an almost complex structure on satisfying the usual conditions, and (hopefully) perturb the -holomorphic curve equation to obtain transversality. Now let be the sum, over all index (perturbed) connected genus zero holomorphic curves in with positive ends at and negative ends at , of the monomial , times the usual sign and combinatorial factor. We then count possibly disconnected genus zero curves by defining

If I set up the algebra correctly, then consideration of ends of moduli spaces of index one genus zero curves in shows that

where denotes the Hamiltonian for . It now follows from the associativity relations explained above that if then

This means that multiplication by defines a chain map

The induced map on homology is the cobordism map we are looking for. Also, respects the symplectic action filtration by the usual calculation with Stokes’s theorem, so it induces maps on filtered homology

for each real number .

Chain homotopies

To complete the story, we should define chain homotopies to show that the above cobordism maps do not depend on choices and are functorial with respect to composition of cobordisms. Assuming relevant transversality as always, the chain homotopies count possibly disconnected index rational curves, consisting of one index component together with any number of index components, in the same way that counts possibly disconnected index rational curves. Part of the transversality that one needs to arrange here is that perturbed holomorphic curves containing the same index component more than once do not exist; this is a tricky foundational issue. Anyway, assuming these foundational issues can be worked out, I don’t think there is anything new in the formalism (correct me if I am wrong).

The weakly exact case

If is a weakly exact symplectic cobordism from to , then one can use the same formalism to define cobordism maps

Here denotes the summand of which is the homology of the subcomplex spanned by tensor products of monomials in the variables in which the total homology class of the corresponding Reeb orbits is . This is very much analogous to the story with ECH.

Strong symplectic cobordisms

A strong symplectic cobordism from to induces a map on a Novikov-type completion of . This is analogous to the (somewhat long) ECH story, and we can define the most fundamental capacities without this, so I won’t go into further detail for the moment.

The U map

There is also a map on analogous to the ECH story. Since this is a crucial ingredient in the definition of capacities, I should explain this in detail. Next post.

Rational SFT using only q variables

I have talked a lot about ECH capacities in this blog and elsewhere. These give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. This obstruction is sharp for embedding one ellipsoid into another, as shown by McDuff. However is not so good for say embedding a polydisk into a ball, as shown by simple examples.

A separate point which I would like to reiterate here is that one can use formalism similar to the definition of ECH capacities to define capacities using other versions of contact homology. For example, the analogue in linearized contact homology of the “full ECH capacities” conjecturally agrees with the Ekeland-Hofer capacities, at least for convex domains in .

I would now like to explain how to define an analogue of ECH capacities for rational SFT. Work of Hind-Kerman and Hind-Lisi (obtaining some powerful results about embeddings by studying rational curves) suggests that one might obtain some useful new embedding obstructions this way. To do so, we first need to modify the definition of rational SFT to make it more suitable for defining capacities.

The usual definition of rational SFT

Let us first briefly review the usual definition of rational SFT due to Eliashberg, Givental, and Hofer. Let be a closed contact manifold of dimension and let be a nondegenerate contact form on . For each (possibly multiply covered) “good” Reeb orbit , one introduces two formal variables and . (A Reeb orbit is “bad” if it is the double cover of a Reeb orbit whose Conley-Zehnder index has opposite sign, otherwise it is “good”.) One defines an algebra which is a suitable Novikov completion of the supercommutative algebra over generated by the variables and . (In the supercommutativity, the parity of the grading of a variable or is the parity of the Conley-Zehnder index of , plus the parity of .) The algebra has a Poisson bracket defined via the symplectic form . Here denotes the covering multiplicity of the Reeb orbit (which is if and only if is embedded). What this means is that the Poisson bracket of two monomials is the sum of all ways of annihilating a factor in one monomial with a factor in the other monomial, counted with appropriate signs and combinatorial factors.

To define the differential on , one chooses a generic almost complex structure on satisfying the usual conditions to define any kind of contact homology. Moreover, one needs to make some abstract perturbation of the -holomorphic curve equation to arrange transversality of moduli spaces of (abstractly perturbed) -holomorphic curves. Hopefully polyfold theory will soon make this possible; let us assume that this has been done and pretend that all relevant moduli spaces of holomorphic curves are cut out transversely, so that we can focus on trying to understand the formalism.

One now defines the Hamiltonian to be a sum over all index holomorphic curves modulo translation. A holomorphic curve with positive ends at Reeb orbits and negative ends at Reeb orbits contributes a term of , times an appropriate sign and combinatorial factor, to the Hamiltonian. Looking at ends of index 2 moduli spaces of genus zero curves shows that

Here the Poisson bracket formalism cleverly keeps track of the fact that if a connected genus zero curve splits into two pieces, then it does so along a single circle. This is because as mentioned above, when you take the Poisson bracket of two monomials, only one variable can annihilate a corresponding variable.

One now defines a differential on the algebra by . The Jacobi identity for the Poisson bracket (with some funny signs coming from the supercommutativity) together with the identity imply that . Then rational SFT is the homology of this differential. Also, the Poisson bracket makes it into an algebra, which is very nice.

The problem

Unfortunately this is not very well suited to defining capacities. To define capacities we first want to define a filtered theory, considering only generators of symplectic action less than . However for the differential to decrease symplectic action, we should define the symplectic action of to be minus the symplectic action of ; but then there would be infinitely many generators of action less than , involving arbitrarily long Reeb orbits, so this would not work so well.

Instead, we need a version of rational SFT involving only the variables. While doing so, we need to somehow keep track of the fact that we are never allowed to glue together the same pair of genus zero curves more than once to obtain a higher genus curve. Here is how I propose to do this.

Rational SFT with only q variables

Define a new algebra over as follows. A generator of is an ordered -tuple , for , where each is a nonconstant monomial in the variables. We mod out by the usual supercommutativity for changing the order of the variables in one of the monomials . Also, we can re-order the elements in the -tuple and multiply by whenever we commute two odd degree elements.

One can alternately think of a generator of as a monomial in the variables, in which the factors have been partitioned into equivalence classes. The idea is that this is a list of Reeb orbits, where Reeb orbits in the same equivalence class are boundary components of the same connected genus zero curve above the Reeb orbits, and so are not allowed to be glued to the same component of a holomorphic curve below the Reeb orbits.

We next define a product as follows. Let be a monomial in , and let be a generator of . The idea is to think of as a connected genus zero curve with positive ends and negative ends, and sum over all ways of attaching the positive ends to without increasing the genus. More precisely,  is the sum over all ways of annihilating each of the factors in with a factor in , such that the factors in that are annihilated are in distinct equivalence classes. The resulting new generator of has equivalence classes. One equivalence class consists of the variables , together with the remaining variables in the equivalence classes from which variables were annihilated. The remaining equivalence classes are the equivalence classes in from which no variable was annihilated. (This is easier to explain by drawing a picture…) One multiplies by the same sign and combinatorial factor as in the definition of the Poisson bracket on .

If are odd degree monomials in , and if is a generator of , then we have

The idea is that both sides of the equation sum over all ways of gluing a positive end of or to a negative end of or , and then gluing all remaining positive ends to distinct equivalence classes in .

We now define a differential on by . The above identity together with imply that . The homology of is the “q-variable only” version of rational SFT, which can be used to define capacities. In particular it has a symplectic action filtration, in which the symplectic action of a monomial in the variables is the sum of the symplectic actions of the Reeb orbits corresponding to the variables.

Did that make any sense at all? If so, I’ll explain in a subsequent post how to use the above to define some new symplectic capacities (which may or may not be interesting, we’ll see).

ECH lecture notes posted

The ECH lecture notes are now posted at arXiv:1303.5789. Here is a brief guide to what in these lecture notes is new, i.e. has not appeared in previous papers. (Some of this material was in blog postings last July.)

Section 1 is an introduction to the basic structure of ECH and its application to symplectic embedding problems. There is nothing really new here; however I’m hoping it will be useful for people who haven’t seen this material before.

Section 2, after reviewing some basics about holomorphic curves, computes Taubes’s Gromov invariant in the -invariant case and uses the result to motivate the choice of generators of the ECH chain complex. This is new (in the sense that it hasn’t appeared in print; of course this was known from the beginning of the development of ECH), and I hope it will demystify the definition of ECH a bit.

Section 3 gives more details about the definition of ECH and computes the ECH of an ellipsoid. There is nothing new here, although again I hope it will be useful for readers who haven’t seen it before and don’t want to dig through the original papers.

Section 4 computes the U map on the ECH of a (nearly round) ellipsoid, outlines the computation of the ECH of T^3, and then explains the computation of the ECH capacities of toric domains. The U map computation hasn’t appeared before. Of course the computation of the ECH of was discussed extensively in my paper with Michael Sullivan. Actually there is a connection with tropical geometry which is discussed in general in Brett Parker’s thesis and which he gave a talk about long ago, and a similar approach appears in Taubes’s work on holomorphic curves in , but I think there should be another expository article explaining this beautiful picture for the case. The computation of ECH capacities of toric domains appeared before in my paper on Quantitative ECH, but the explanation here is maybe a little simpler.

Section 5 gives an introduction to the foundations of ECH. Since the foundations are quite complicated, hopefully this introduction will make it not so hard to get some idea of what is involved. Also, section 5.2 discusses the “topological complexity” (roughly speaking the genus) of the holomorphic curves counted by ECH; while this isn’t new, I have hopefully spelled it out more simply than before.

Section 6 compares ECH with SFT, responding to various frequently asked questions.

Finally there are various exercises and some of them have answers in the appendix.

Also, there are a few new things about the terminology. For example, I now use the term “holomorphic currents” to describe unions of somewhere injective holomorphic curves with positive integer multiplicities. This is appropriate for ECH because when multiple covers arise we do not care about a covering map; we only care about the degree of the cover. It would also make sense to refer to the generators of the ECH chain complex as “Reeb currents”, but I didn’t do that.

Update on the short Reeb orbit conjecture

In an earlier post, I stated a conjecture that every contact three-manifold has a Reeb orbit with an upper bound on its symplectic action in terms of the volume of the contact manifold. There hasn’t been much progress on this conjecture, so now I would like to state a slightly stronger conjecture (the constant is improved) and discuss some examples.

To state the conjecture, let be a closed contact three-manifold. Define the volume . If is a Reeb orbit, define its symplectic action by .

Conjecture 1 (“Short Reeb orbit conjecture”). If is a closed contact three-manifold, then there exists a Reeb orbit with .

Example 1: -invariant contact forms. Let be an -bundle over a surface with Euler class , and let be a connection 1-form on with nowhere vanishing curvature , regarded as a real 2-form on . Let denote the projection and let be a positive function. Then is a contact form on . I claim that the conjecture holds for this contact form, with equality if and only if and is constant.

To see this, observe first that . In particular, if denotes the minimum value of , then . On the other hand, if satisfies , then is a Reeb orbit with . The claim follows immediately.

A special case of this example is where is a compact star-shaped hypersurface in which is -invariant, where acts on as multiplication by scalars of absolute value .

Example 2: geodesics in . Let be any Riemannian metric on and let be the unit cotangent bundle of with its canonical contact form , so that Reeb orbits correspond to closed geodesics, and symplectic action corresponds to length of geodesics. Then . The conjecture then implies that there is a closed geodesic whose length satisfies .

For example, if is the standard round metric, then a great circle has and so the desired inequality is , which holds with a factor of to spare. However there is a famous example in systolic geometry where the minimum of is greater than that of . The example is obtained by taking two equilateral triangles in the plane, say of side length 1, and gluing their edges together, to obtain with a metric which is flat except for three cone singularities. Smooth these. The area is then approximately , while the length of the shortest geodesic (which goes from a vertex of the triangle to the center of the opposite side and back) is approximately .  The desired inequality is now , which still holds, but with less than a factor of to spare. And this is possibly a little worrisome, because of:

Example 3: even geodesics in . As I learned from discussions with Vinicius Gripp, the conjecture actually says a little more about geodesics in . Continuing with the notation from Example 2, observe that . We can pull back the contact form to the double cover of , which is , to obtain a contact form with twice the volume. Also, there are two different kinds of geodesics on : let us call a geodesic “even” if it corresponds to a homotopically trivial loop in , and “odd” if it corresponds to a homotopically nontrivial loop in . Then Reeb orbits in correspond to even geodesics (including even degree covers of odd geodesics). So Conjecture 1 above implies the following:

Conjecture 2. For any Riemannian metric on , there exists an even geodesic of length with .

This conjecture is now sharp for the standard metric on . The reason is that a great circle is an odd geodesic, so the shortest even geodesic is a double cover of this which has length .

In general, you can see whether a geodesic is even or odd by removing a point from which is not on the geodesic to get a plane curve and seeing whether the rotation number of the plane curve is even or odd. Also, the parity of the rotation number of the plane curve is opposite the parity of the number of double points of the curve.

Now what about the example with two triangles glued together? Here the shortest geodesic is again odd. Its double cover is even, but it has length , so the desired inequality here is , which fails. Does this mean that we have a counterexample to Conjecture 1?

No. There is another, slightly longer geodesic which is even. This geodesic consists of six segments going between the centers of the sides of the triangles and has length . It is an even geodesic because it has an odd number (three) of double points. Here the desired inequality is , which holds.

Conclusion. We don’t yet have any counterexample to Conjecture 1, although I think one should try to make one using open books and see what happens. As for Conjecture 2, there is lots of literature on systolic geometry (concerning the existence of a closed geodesics which are short with respect to the Riemannian volume), although I don’t know if this specific conjecture about “even” geodesics has been considered.