# Relationship between Gromov-Witten and Taubes' Gromov invariant

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.

• 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). – Chris Gerig Apr 12 '19 at 0:34