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 (Y,\lambda) be a closed nondegenerate contact manifold of dimension 2n-1. The usual rational SFT defines a supercommutative algebra {\mathcal A} generated by variables p_\gamma and q_\gamma for the good Reeb orbits \gamma, with a Poisson bracket \{\cdot,\cdot\}. Counting index 1 rational J-holomorphic curves in {\mathbb R}\times Y (after suitable abstract perturbations of the J-holomorphic curve equation which will hopefully soon be available) defines a “Hamiltonian” H in {\mathcal A}. This satisfies \{H,H\}=0, and so d=\{H,\cdot\} defines a differential on {\mathcal A}, whose homology is the usual rational SFT.

The alternate version I introduced last time uses an algebra {\mathcal A}', which (as Janko Latschev points out) can be succintly described as the symmetric algebra on the polynomial algebra on the q variables. There is a multiplication {\mathcal A}\times {\mathcal A}'\to{\mathcal A}', and d'x=H\cdot x defines a differential on {\mathcal A}', whose homology is the “q variable only” version of rational SFT. For lack of a better notation, let us temporarily denote this by HQ(Y,\lambda).

For every real number L, there is also a filtered version HQ^L(Y,\lambda), which is the homology of the subcomplex generated by tensor products of monomials in the q variables in which the total symplectic action of the Reeb orbits is less than L.

Cobordism maps

I now want to define cobordism maps on HQ^L (which in particular will prove that it does not depend on the choice of J). 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 (X,\omega) is an exact symplectic cobordism from (Y_+,\lambda_+) to (Y_-,\lambda_-). We want to define an induced map

\Phi(X,\omega): HQ(Y_+,\lambda_+) \to HQ(Y_-,\lambda_-).

The idea is to define a chain map by counting index zero, genus zero, possibly disconnected holomorphic curves in the the completion of X, 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 {\mathcal A}_+ and {\mathcal A}'_+ denote the algebras {\mathcal A} and {\mathcal A}' for (Y_+,\lambda_-), and let {\mathcal A}_- and {\mathcal A}'_- denote the corresponding algebras for (Y_-,\lambda_-). Introduce a new algebra {\mathcal A}_{-,+}, which is the supercommutative algebra generated by the p variables for \lambda_+ and the q variables for \lambda_-, completed to allow infinite sums of monomials in the p variables each multiplied by a polynomial in the q variables. Also introduce the symmetric algebra on this, which we denote by {\mathcal A}'_{-,+}=Sym(\mathcal{A}_{-,+}). One can think of a tensor product of k monomials in {\mathcal A}'_{-,+} as representing the ends of a holomorphic curve in (the completion of) X with k 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 {\mathcal A}'_{-,+}\times {\mathcal A}_+ \to {\mathcal A}'_{-,+}, defined analogously to the multiplication {\mathcal A}_+\times {\mathcal A}'_+ \to {\mathcal A}'_+. To multiply a tensor product x of k monomials in {\mathcal A}'_{-,+} by a monomial y in {\mathcal A}_+, let l denote the number of q variables in y. One then sums over all ways of cancelling each of the q variables in y with a p variable in x, where the p variables that get cancelled are in l distinct factors in the tensor product. One obtains the remaining q and p variables in x, together with the p variables in y, grouped into k-l+1 factors as before.

There is also a multiplication map {\mathcal A}'_{-,+}\times {\mathcal A}'_+\to{\mathcal A}'_-. One multiplies a tensor product x of k monomonials in {A}'_{-,+} by a tensor product z of l monomials in {A}'_+ as follows. The product is zero unless there exists a bijection between the q variables in z and the p variables in x that pairs up p and q variables corresponding to the same Reeb orbit.  Given such a bijection \phi, let G_\phi denote the graph whose vertices are the k+l tensor factors in x and z, and whose edges are determined by \phi. One now sums over all bijections \phi such that the graph G_\phi does not contain any loops. The contribution from \phi, up to the usual signs and combinatorial factors, consists of the q variables in x, grouped into tensor factors according to the connected components of G_\phi.

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

{\mathcal A}'_{-,+}\times {\mathcal A}_+ \times {\mathcal A}'_+ \to {\mathcal A}'_-

obtained by multiplying in either order. The way I like to think of this multiplication is that a tensor product of k monomials in {\mathcal A}'_{-,+} can be regarded (up to orientation data) as a graph with k components, each consisting of a single vertex with some p and q edges attached to it. A monomial in {\mathcal A}_+ can be regarded as a graph with a single vertex with some p and q edges attached to it. And a tensor product of l monomials in {\mathcal A}'_+ can be regarded as a graph with l components, each consisting of a single vertex with some q edges attached to it. A triple product x\cdot y \cdot z, where x\in{\mathcal A}'_{-,+}, y \in {\mathcal A}_+, and z\in {\mathcal A}'_+ 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 q edges in y and z to all of the p edges in x and y, 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 {\mathcal A}_-\times{\mathcal A}'_{-,+}\to \mathcal{A}'_{-,+} and {\mathcal A}'_-\times{\mathcal A}'_{-,+}\to {\mathcal A}'_+, which are associative so that there is a well-defined triple product

{\mathcal A}'_-\times{\mathcal A}_-\times{\mathcal A}'_{-,+}\to {\mathcal A}'_+.

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

{\mathcal A}_- \times {\mathcal A}'_{-,+} \times {\mathcal A}'_+ \to {\mathcal A}'_-.

The chain map

To define the cobordism chain map, we now complete X to \overline{X} by attaching symplectization ends as usual, choose an almost complex structure J on \overline{X} satisfying the usual conditions, and (hopefully) perturb the J-holomorphic curve equation to obtain transversality. Now let \phi_1\in{\mathcal A}_{-,+} be the sum, over all index 0 (perturbed) connected genus zero holomorphic curves in \overline{X} with positive ends at \alpha_1,\ldots,\alpha_k and negative ends at \beta_1,\ldots,\beta_l, of the monomial q_{\beta_1}\cdots q_{\beta_l}p_{\alpha_1}\cdots p_{\alpha_k}, times the usual sign and combinatorial factor. We then count possibly disconnected genus zero curves by defining

\phi = \exp(\phi_1) = \sum_{m=0}^\infty \frac{1}{m!} \underbrace{\phi_1\otimes \cdots \otimes \phi_1}_{m}\in{\mathcal A}'_{-,+}.

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

\phi\cdot H_+ = H_-\cdot\phi \in {\mathcal A}'_{-,+}

where H_\pm\in {\mathcal A}_\pm denotes the Hamiltonian for (Y_\pm,\lambda_\pm). It now follows from the associativity relations explained above that if x\in {\mathcal A}'_+ then

H_-\cdot(\phi\cdot x) = \phi\cdot(H_+\cdot x).

This means that multiplication by \phi defines a chain map

\phi : ({\mathcal A}'_+,H_+) \to ({\mathcal A}'_-,H_-).

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

\Phi^L(X,\omega) : HQ^L(Y_+,\lambda_+) \to HQ^L(Y_-,\lambda_-)

for each real number L.

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 -1 rational curves, consisting of one index -1 component together with any number of index 0 components, in the same way that \phi counts possibly disconnected index 0 rational curves. Part of the transversality that one needs to arrange here is that perturbed holomorphic curves containing the same index -1 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 (X,\omega) is a weakly exact symplectic cobordism from (Y_+,\lambda_+) to (Y_-,\lambda_-), then one can use the same formalism to define cobordism maps

\Phi^L(X,\omega): HQ^L(Y_+,\lambda_+,0) \to HQ^L(Y_-,\lambda_-,0).

Here HQ^L(Y,\lambda,0) denotes the summand of HQ^L(Y,\lambda) which is the homology of the subcomplex spanned by tensor products of monomials in the q variables in which the total homology class of the corresponding Reeb orbits is 0\in H_1(Y). This is very much analogous to the story with ECH.

Strong symplectic cobordisms

A strong symplectic cobordism from (Y_+,\lambda_+) to (Y_-,\lambda_-) induces a map on a Novikov-type completion of HQ. 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 U map on HQ analogous to the ECH story. Since this is a crucial ingredient in the definition of capacities, I should explain this in detail. Next post.

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