Asymptotic Geometry – Relationship between Gromov-Witten and the Gromov invariant of Taubes

Fix a symplectic and compact four collector ($ X $, $ omega $).

Recall the Gromov invariant of Taubes is a certain full value function on $ H ^ 2 (X; mathbb {Z}) $ defined by the weighted number of pseudoholomorphic curves in $ X $. In particular, this number combines curves of all kinds.

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

Is there a relationship, conjectural or otherwise, between these two invariants stronger than "both have holomorphic curves"? Of course, this would require looking at the Gromov-Witten invariants for all genres $ g $. Personally, I do not understand the best way to define them for the genre $ g> $ 0. I understand that Zinger has a construction for $ g = $ 1 it's a bit more refined than looking at all the modules space.