Following, e.g., Wikipedia's definitions, the (small) quantum cohomology ring of $X$ is defined over a "Novikov ring" consisting of formal power series of the form $$ \sum_{\beta \in H_2(X;\mathbb{Z})} a_\beta e^\beta,$$ where the $a_\beta$ are in some fixed ring (which is probably usually $\mathbb{Q}$ or $\mathbb{C}$).

On the other hand, in the papers of Fukaya and company for instance, there seems to be a different "Novikov ring", consisting of power series of the form $$\sum_{i=1}^\infty a_i T^{\lambda_i},$$ where $a_i \in \mathbb{Q}$, $\lambda_i \in \mathbb{R}$, and $\lim_{i \to \infty} \lambda_i = \infty$.

What's up with this? Why are there apparently two different Novikov rings? How do I reconcile this? I'm guessing that the $\lambda_i$ in the second definition should correspond to something like $\int_\beta \omega$ where $\beta$ is as in the first definition (and where $\omega$ is the symplectic form), but beyond this I'm not sure...