TITLE: Different definitions of Novikov ring?
QUESTION [7 upvotes]: 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...

REPLY [5 votes]: We must think of $T$, the generator of the Novikov ring, as a coordinate on the moduli space of symplectic structures $\mathcal M_{sym}(X)$. We can define $<\cdots>_g^\psi:=\sum_\beta<\cdots>_{g,\beta}^\psi e^{-\omega(\beta)}$ , but we don't know if the right hand side converges and so we put $T^{\omega(\beta)}=e^{-(-\log T\omega)(\beta)}$ and we can think of this as large volume limit, i.e. neighborhood of singular point in $\mathcal M_{sym}(X)$ where $\omega\to \infty$
For a $J$-holomorphic curve $u$ we have $E(u)=ω([u])=∫_Σu^∗ω≥0$, and we must take $λ_i≥0$. Moreover if $ω$ belongs to an integral cohomology class, then $ω(β)∈\mathbb Z_{≥0}$ and we can replace the Novkov ring with $\mathbb Q[[T]]$
See Denis Auroux's
paper https://arxiv.org/pdf/0902.1595.pdf .