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 $g\_s$.



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



In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?
Here is a purely mathematical reason why we prefer to put $u^{2g-2}$ in our generating function instead of, say, $u^g$. 


The generating function you write, 
$$\sum\_{g,\beta} GW\_{g,\beta}(X) u^{2g-2}Q^{\beta},$$
is the generating function for the *connected* Gromov-Witten invariants of $X$ (I've thrown in another variable that tracks the class $\beta$). Connected here means that the invariants are obtained from the moduli space of maps with connected domains. This generating function is usually called $F$ because we reserve the letter $Z$ for the *disconnected* generating function:


$$Z = \exp \left(\sum\_{g,\beta} GW\_{g,\beta}(X) u^{2g-2} Q^{\beta}\right)$$


The coefficients of this generating function are the *disconnected* Gromov-Witten invariants -- i.e. the invariants obtained from the moduli space of maps with (possibly) disconnected domains. 


In order for the relationship between the connected and disconnected invariants to be $Z=\exp(F)$, we need that the quantity tracked by the variable to be *additive* under disjoint union and so we prefer the Euler characteristic $2g-2$ to the genus $g$ ($g$ is not additive under disjoint union). For disconnected invariants, Euler characteristic is much more natural than genus (which would have to be defined via Euler characteristic anyway). Note that much of the interesting features in GW theory are formulated via the disconnected invariants (e.g. the MNOP conjecture). 


