Different definitions of Novikov ring? 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...
 A: If you read around, you'll find plenty of other variants of the Novikov ring... The underlying point is that to apply the Gromov compactness theorem (or its algebraic counterpart) you need a bound on the energy (= symplectic area) of ridid  holomorphic spheres. Since there's no a priori bound in general, you instead count curves according to their area, or according to some refinement thereof, such as homology class.
Your idea is correct: if we fix a base field $k$, the homomorphism $H_2(X)/tors. \to \mathbb{R}$ given by pairing with $[\omega]$ induces a homomorphism from the completed group $k$-algebra on $H_2(X)/tors.$ to the "universal Novikov field" (that is, the field of series $\sum_{\lambda \in \mathbb{R}}{a(\lambda)T^\lambda}$ where the support of $a \colon \mathbb{R}\to k$ contains only finitely many reals less than any given $C$). After making a small perturbation to $\omega$, changing its cohomology class, we can make this homomorphism injective. If memory serves, the smaller Novikov ring is a PID, and the universal Novikov field is flat over it because it's a torsion-free module over a PID. So you can safely tensor with it after passing to cohomology.
The advantages of the universal Novikov field are that it's universal (meaning biggest, essentially), that it's a field, and that it comes with a natural valuation. The disadvantage is that it's bigger than is strictly necessary.
A: 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 .
