Let $X$ be a Calabi-Yau threefold and let us fix a homology class $\beta\in H_2(X,\mathbb Z)$, just for simplicity. The generating series of Gromov-Witten invariants of $X$ in class $\beta$, $$\mathsf Z=\sum_{g}\textrm{GW}_{g,\beta}(X)\color{red}{u}^{2g-2},$$ has the variable $\color{red}{u}$, taking care of the varying genera, weighted by $\color{red}{2g-2}$. Some time ago, I remember having read this variable is called the string coupling constant, sometimes denoted $g_s$.

Question. Why is this variable weighted by $2g-2$, instead of, say, $g$? Is there a reason coming from Physics, maybe from String Theory?

I feel the reason why I cannot answer my question is that I do not really understand the meaning of $u$ (or $g_s$), or its role inside $\mathsf Z$. It seems not to be just a variable, or indeterminate, which one uses to write down $\mathsf Z$. It seems, for instance, that it makes sense to talk about "small values" of the coupling constant (which confuses me even more, since it is called a constant). I hope I made clear my confusion. And, my apologies for the non-research-level of the question.

Thanks you all.