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Fix a compact, symplectic four-manifold ($X$, $\omega$).

Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; \mathbb{Z})$ defined by weighted counts of pseudoholomorphic curves in $X$. In particular, this count combines curves of any genus.

On the other hand, the Gromov-Witten invariants for a fixed genus $g$ and homology class $A \in H_2(X; \mathbb{Z})$ are (very roughly) integers derived from the "fundamental class" of the moduli space $\mathcal{M}_g^A(X)$ of pseudoholomorphic maps from a surface of genus $g$ into $X$ representing the class $A$.

Is there any sort of relationship, conjectural or otherwise, between these two invariants that is stronger than "they both count holomorphic curves"? Of course, this would require looking at Gromov-Witten invariants for any genus $g$. I personally don't understand the best way to define these for genus $g > 0$. I do understand that Zinger has a construction for $g = 1$ that is a bit more refined than looking at the entire moduli space.

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  • $\begingroup$ With respect to Ionel-Parker's paper (which is the answer), note that GT allows a single curve to be disconnected with "whatever" genus whereas GW considers a connected curve of fixed genus, so you should expect that their generating series are the same (roughly speaking). $\endgroup$ – Chris Gerig Apr 12 at 0:34
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Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper: https://arxiv.org/abs/alg-geom/9702008

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