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 .
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.
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 .
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.