ECH as a (symplectic) field theory

Consider the following question:

Given a closed symplectic 4-manifold $(X,\omega)$ with a separating contact type hypersurface $Y$, how can one compute Taubes’s Gromov invariant of $X$ by cutting along $Y$ and pairing some kind of relative ECH of the two pieces?

If I’m not mistaken, I can answer this using ideas from the previous post. Here is a sketch.

Taubes’s Gromov invariant. First let’s briefly review Taubes’s Gromov invariant. This invariant is parametrized by classes $A\in H_2(X)$. Define an even integer

$I(A) = \langle c_1(TX),A\rangle + A\cdot A.$

This is the closed four-manifold version of the ECH index.  Now choose a generic $\omega$-compatible almost complex structure $J$ on $X$. The Gromov invariant $Gr(X,A)$ is, roughly speaking, a count of $J$-holomorphic curves which are required to pass through $I(A)/2$ generic points in $X$. These holomorphic curves are not required to be connected. They turn out to be embedded, except that they may contain multiple covers of tori with zero self-intersection (which have to be counted in a subtle way), and multiple covers of spheres with negative self-intersection are not permitted. Taubes showed that $Gr(X,A)$ agrees with the Seiberg-Witten invariant of $X$ for the spin-c structure $\frak{s}_\omega + A$.

A “Gromov series”. To get a simpler formalism, let us package the Gromov invariants for the different classes $A$ into a single “series”. For this purpose define a kind of Novikov ring $\Lambda$ in two variables $t,u$ as follows. An element of $\Lambda$ is a (possibly infinite) formal sum $\sum_{r\in{\mathbb R}}a_rt^r$ where $a_r\in{\mathbb Z}[u]$. We require that for all $R\in{\mathbb R}$, the number of $r\in{\mathbb R}$ with ${r}<{R}$ and $a_r\neq 0$ is finite. Multiplication is defined by $t^rt^{r'}=t^{r+r'}$, and the product of two elements of $\Lambda$ is well defined by the finiteness condition.

The idea of the formalism that we will set up is that whenever we count a holomophic curve of area/action $r$, we multiply by $t^r$; and whenever we require a holomorphic curve to pass through $k$ points, we multiply by $u^k$.

To start, we can define a “Gromov series” by

$Gr(X) = \sum_{A\in H_2(X)}Gr(X,A)u^{I(A)/2}t^{\omega(A)}\in\Lambda.$

We have $Gr(X)\in\Lambda$ by Gromov compactness. The Gromov series determines the full Gromov invariant provided that there is no nonzero class $A\in H_2(X)$ with $\omega(A)=I(A)=0$; this can always be achieved by adding a small closed $2$-form to $\omega$. For each nonnegative integer $k$, define $Gr^k(X)$ to be the $u^k$ part of $Gr(X)$.

A “completion” of ECH. Next let $(Y,\lambda)$ be a closed contact 3-manifold; we will define a “Novikov completion” of the ECH of $(Y,\lambda)$. This is the homology of a chain complex over $\Lambda$, which we denote by $\overline{ECC}(Y,\lambda)$. An element of $\overline{ECC}(Y,\lambda)$ is a (possibly infinite) formal sum $\sum_{\alpha,r}a_{\alpha,r}t^r\alpha$ where the sum is over $r\in{\mathbb R}$ and ECH generators $\alpha$, and the coefficients $a_\alpha\in{\mathbb Z}[u]$. We impose the finiteness condition that for each $R\in{\mathbb R}$, there are only finitely many pairs $(\alpha,r)$ such that $\mathcal{A}(\alpha)+r and $a_{\alpha,r}\neq 0$. Here $\mathcal{A}(\alpha)$ denotes the symplectic action of $\alpha$.  Now pick an almost complex structure on ${\mathbb R}\times Y$ as needed to define the usual ECH differential $\partial$. The  differential on $\overline{ECC}(Y,\lambda)$, which we denote by $\overline{\partial}$, is defined in terms of $\partial$ by

$\overline{\partial}\alpha=\sum_{\beta}\langle\partial\alpha,\beta\rangle t^{\mathcal{A}(\alpha)-\mathcal{A}(\beta)}\beta.$

This gives a well-defined differential on $\overline{ECC}(Y,\lambda)$ by the finiteness condition and the fact that $\partial$ decreases symplectic action. Note also that there is an inclusion of chain complexes ${ECC}(Y,\lambda)\to \overline{ECC}(Y,\lambda)$ sending an ECH generator $\alpha$ to $t^{-\mathcal{A}(\alpha)}\alpha$.  We denote the homology of this complex by $\overline{ECH}(Y,\lambda)$.

Recall that if $y\in Y$ is a generic point, then there is a chain map $U_y$ on $ECC(Y,\lambda)$ counting $I=2$ curves passing thrugh $U_y$. We now define a map $\overline{U_y}$ on $\overline{ECC}(Y,\lambda)$ by

$\overline{U_y}\alpha = \sum_{\beta}\langle U_y\alpha,\beta\rangle ut^{\mathcal{A}(\alpha)-\mathcal{A}(\beta)}\beta.$

This induces a map on $\overline{ECH}(Y,\lambda)$ which depends only on the component of $Y$ containing $y$. When $Y$ is connected or when there is no confusion about which component this is, we will denote this map by $\overline{U}$.

Cobordism maps. Now let $(Y_+,\lambda_+)$ and $(Y_-,\lambda_-)$ be closed contact three-manifolds (not necessarily connected), and let $(X,\omega)$ be a connected strong symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$ (see the previous post for this terminology) with a homology orientation. I want to define a cobordism map

$\Phi_X:\overline{ECH}(Y_+,\lambda_+)\to \overline{ECH}(Y_-,\lambda).$

Heuristically, to define this we choose an appropriate almost complex structure $J$ on the “symplectization completion” of $X$ and suitably count holomorphic curves $C$ of ECH index $2k$ passing through $k$ generic points, multiplied by $u^kt^{\int_C\omega}$. But as discussed in previous posts, we cannot (with current technology) count such curves directly and instead need to use Seiberg-Witten theory.

I have to think a bit about how to say the definition correctly, but here is an approximation. Recall that in the previous post (see that post for the notation I am about to use), for each $A\in H_2(X,\partial X)$ and for each $L\in{\mathbb R}$ we defined a map

$\Phi_{X,A}^L:ECH^L(Y_+,\lambda_+,\Gamma_+) \to ECH^{L+\rho(A)}(Y_-,\lambda_-,\Gamma_-)$

This is induced by a (noncanonical) chain map which is nonzero only in the presence of a (possibly broken) holomorphic curve.  We know that if we replace $L$ by $L'$, then $\Phi_{X,A}^L$ is compatible with the inclusion-induced maps on homology (and the direct limit agrees under Taubes’s isomorphism with the map on Seiberg-Witten Floer cohomology corresponding to the spin-c structure $\frak{s}_\omega + A$).  Now (here is the approximation part) suppose that this is also true at the chain level, i.e. that we can choose the aforementioned chain maps inducing $\Phi_{X,A}^L$ to be compatible with the inclusions of chain complexes. The limit of these chain maps is then a chain map

$\phi_{X,A}:ECC(Y_+,\lambda_+,\Gamma_+)\to ECC(Y_-,\lambda_-,\Gamma_-)$

which is nonzero only in the presence of a (possibly broken) holomorphic curve. Now define a chain map

$\phi_X^0:\overline{ECC}(Y_+,\lambda_+)\to \overline{ECC}(Y_-,\lambda_-)$

by

$\phi_X^0\alpha_+=\sum_{A\in H_2(X,\partial X)} \sum_{\alpha_-}\langle\phi_{X,A}\alpha_+,\alpha_-\rangle t^{\mathcal{A}(\alpha_+)-\mathcal{A}(\alpha_-)+\rho(A)}\alpha_-.$

Here $\alpha_+$ is an ECH generator for $(Y_+,\lambda_+)$, and we are summing over ECH generators $\alpha_-$ for $(Y_-,\lambda_-)$.  This is well-defined on $\overline{ECC}(Y_+,\lambda_+)$ by Gromov compactness, because if $\langle\phi_{X,A}\alpha_+,\alpha_-\rangle\neq 0$, then there is a (possibly broken) holomorphic curve from $\alpha_+$ to $\alpha_-$ in the class $A$, and the integral of $\omega$ over this is $\mathcal{A}(\alpha_+) - \mathcal{A}(\alpha_-) + \rho(A)$.  We then obtain an induced map on homology

$\Phi_X^0:\overline{ECH}(Y_+,\lambda_+)\to \overline{ECH}(Y_-,\lambda_-).$

Next, for each nonnegative integer $k$, define

$\Phi_X^k=u^k\overline{U}^k\Phi_X^0=u^k\Phi_X^0\overline{U}^k.$

Here the map $\overline{U}$ can be defined using any component of $Y_+$ or $Y_-$. Finally,

$\Phi_X = \sum_{k\ge 0}\Phi_X^0.$

If $Y_+$ and $Y_-$ are empty, so that $X$ is closed, then the above definition of $\Phi_X^k$ only makes sense when $k=0$ and agrees with $Gr^0(X)$. By extension we define $\Phi_X=Gr(X)$.

Composition law. Suppose $X$ is the composition of a connected strong cobordism $X_+$ from $(Y_+,\lambda_+)$ to $(Y_0,\lambda_0)$ with a connected strong cobordism $X_-$ from $(Y_0,\lambda_0)$ to $(Y_-,\lambda_-)$. I claim (and this should follow directly from Seiberg-Witten theory if I said it right) that for any nonnegative integers $k_+$ and $k_-$,

$\Phi_{X_-}^{k_-}\circ \Phi_{X_+}^{k_+} = \Phi_X^{k_-+k_+}.$

To say this in a fancier way,

$\Phi_X = u^{-1}\int \Phi_{X_-}\circ\Phi_{X_+}du.$

If $Y_+$ and $Y_-$ are empty, then this determines the Gromov invariant of $X$ from invariants associated to $X_+$ and $X_-$.

Comparison with Seiberg-Witten and Heegaard Floer. Recall that in Seiberg-Witten and Heegaard Floer theory, invariants of closed 4-manifolds are defined in a more complicated way. For example in Heegaard Floer, if $X$ is a smooth cobordism between connected three-manifolds $Y_-$ and $Y_+$ which is a composition of two cobordisms each with $b_2^+>0$, then (ignoring spin-c structures) one obtains a “mixed” invariant

$\Phi_X^{mix}:HF^-(Y_-)\to HF^+(Y_+).$

If $X$ is closed and $b_2^+(X)>1$ then one defines the invariant of $X$ by removing two balls to obtain a cobordism from $S^3$ to $S^3$ and applying the above construction. One has to use the mixed invariant because if one simply mapped from $HF^-$ to $HF^-$ or from $HF^+$ to $HF^+$ then the cobordism map would be zero! How can we reconcile this with what we are doing in ECH?

Here is my best guess (although this could be completely wrong).  Let’s work with field coefficients so that we do not have to worry about torsion and we can identify homology with cohomology.

The first and most important point to make is that if we take a closed symplectic manifold $X$ (say with $b_2^+>1$) and if we remove two balls from it, then we do not get a symplectic cobordism from $S^3$ to $S^3$. Rather, we get a symplectic cobordism from the empty set to $S^3\sqcup S^3$.

Now suppose more generally that we have a symplectic cobordism $X$ from the empty set to $(Y_1,\lambda_1)\sqcup (Y_2,\lambda_2)$ where $Y_1$ and $Y_2$ are connected. Then in my formalism we get an element of the completed ECH of $Y_1\sqcup Y_2$, and this space (maybe after some completion) corresponds to $HF^-(-Y_1)\otimes HF^-(-Y_2)$. (Note that $HF^-$ of $Y_1\sqcup Y_2$ is not directly defined because the latter is disconnected. This is one of the few structural advantages of ECH over the other two theories: it handles disconnected manifolds with no trouble.) By duality, the latter space agrees with $HF^-(-Y_1)\otimes HF_+(Y_2)=Hom(HF_-(-Y_1),HF_+(Y_2))$. My conjecture is that (after identifying homology with cohomology since we are using field coefficients), this is exactly the mixed invariant, where we regard $X$ now as a cobordism from $-Y_1$ to $Y_2$.