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.