Fix a field ${\mathbin{\mathbb{F}}}$, in which we will do ‘counting’ of $J$-holomorphic curves. If nontrivial $J$-holomorphic ${\mathbin{\mathbb{CP}}}^{1}$’s can exist in the symplectic manifold $(M,\omega)$ we are interested in, the virtual counts can be rational, so ${\mathbin{\mathbb{F}}}$ must have characteristic zero, and ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{Q}}},{\mathbin{\mathbb{R}}}$ or ${\mathbin{\mathbb{C}}}$ are the obvious possibilities. If $M$ has no $J$-holomorphic ${\mathbin{\mathbb{CP}}}^{1}$’s (for example, if $\omega$ is exact, or if $\pi_{2}(M)=0$) then ${\mathbin{\mathbb{F}}}$ can be arbitrary, so we can take ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{Z}}}_{2}$, for instance, which means we do not have to worry about orientations on moduli spaces of $J$-holomorphic curves.

The Novikov ring $\Lambda_{\rm nov}$ is the field of formal power series $\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ for $a_{i}\in{\mathbin{\mathbb{F}}}$ and $\lambda_{i}\in{\mathbin{\mathbb{R}}}$ with $\lambda_{i}\rightarrow+\infty$ as $i\rightarrow\infty$, for $P$ a formal variable. Write $\Lambda_{\rm nov}^{\geqslant 0}$ for the subring of $\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ in $\Lambda_{\rm nov}$ with all $\lambda_{i}\geqslant 0$, and $\Lambda_{\rm nov}^{+}$ for the ideal of $\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ in $\Lambda_{\rm nov}$ with all $\lambda_{i}>0$.