In general, most examples of ECH are difficult to compute directly. However the ECH of an ellipsoid with a certain standard contact form can be computed in a page or two. So let us now do that, as this should help with understanding some of the definition of ECH.
Contact type hypersurfaces
Before describing the example, let me review some basics. Let be a symplectic manifold. An oriented hypersurface in is said to be of contact type if there exists a contact form on (in dimensions this means that ) such that . If there is a Liouville vector field defined in a neighborhood of (the Liouville condition means that ) such that is transverse to , then is of contact type, and we can take . If is compact then the converse is true, i.e. if is of contact type then there exists a transverse Liouville vector field. (The proof is a bit trickier but we will not need this.) If is of contact type, then up to scaling by positive functions, the Reeb vector field does not depend on , and is given by the kernel of .
Now let with the standard symplectic form . Then the radial vector field is Liouville vector field. Call a hypersurface star-shaped if it is transverse to the radial vector field . Then by the above discusion, any star-shaped hypersurface is of contact type, and a corresponding contact form is given by the restriction of .
Given positive real numbers , define the ellipsoid
This is a symplectic four-manifold with boundary, where the symplectic form is the restriction of the standard symplectic form on . By the above discussion, its boundary
is a contact type surface in , and
is a corresponding contact form on . Let us now compute . (We are omitting the argument from the notation since this is necessarily .)
The Reeb vector field
We first need to find the Reeb orbits. Even this is difficult for general contact manifolds, but for the present example it is an exercise to show the following:
(1) The Reeb vector field is given by
Here we are using polar coordinates on , so that .
(2) is nondegenerate if and only if is irrational. In this case there are only two embedded Reeb orbits, namely and . Both of these orbits are elliptic. There is a canonical trivialization of over and such that has rotation angle , and has rotation angle (see installment 5 for the definition of rotation angle).
The chain complex
Assume that is irrational. Since the Reeb orbits are are elliptic, an ECH generator corresponds to a pair of nonnegative integers , where we take with multiplicity . We will denote this generator by .
Since is torsion and , the chain complex has a canonical -grading. Since all Reeb orbits are elliptic, it follows from the parity property of the ECH index (see the previous installment) that all generators have even grading. Since the differential decreases the grading by one (I should have said that in the previous installment but it is obvious), it follows that the differential is zero. So to complete the calculation of the ECH in this case, we just need to calculate the grading of each generator.
A calculation is usually easier if you know the answer in advance. In the present case, the calculation of the Seiberg-Witten Floer homology of is a (relatively) easy example in Kronheimer-Mrowka’s book. The result of this calculation together with Taubes’s isomorphism tell us (if we use coefficients) that
So let us try to calculate the ECH index and see if we can recover this result.
Calculation of the grading
What is the grading of the generator ? Since we have normalized the grading of the empty set (corresponding to ), the grading of is where is unique. We can write this as
where the notation is that , , and .
By the above calculation of rotation angles, and the fact that rotation angle is multiplicative under iteration, we have
Putting the above together, we get that
So evidently is always an even nonnegative integer. But why on earth should there be exactly one generator for each even nonnegative integer grading?
What we need to remember about ECH is that it is kind of a symplectic shadow of Seiberg-Witten theory. Things in ECH that are natural from a Seiberg-Witten perspective can appear bizarre when viewed from a purely symplectic perspective. So if Seiberg-Witten theory suggests that we do something which looks crazy, then don’t panic, and it should work out.
Now let’s take a deep breath and stare at the above formula for . Observe that the right hand side can be interpreted as follows: let be the triagle in the plane which is bounded by the coordinate axes together with the line through with slope . Then is just the number of lattice points in the triangle (including the boundary) minus . (To see this, divide the triangle into a rectangle and two smaller triangles along the lines and .) Now take a line with slope and move it up and to the right. Every time the line crosses a lattice point in the first quadrant, this gives a new ECH generator whose grading is two more than the grading of the previous one. QED. (Phew!)