I am trying to read this paper. At page $34$, the authors define a symplectic form $\omega$ on some $\mathbb{C}$-vector space $V\oplus W^*$ (I don't want to go too much into the details here, because I don't think it is necessary for my question). About half a page later (beginning of section $2.1.3$), we find the following sentence:
The symplectic form $\omega$ is scaled by the action of $\mathbb{C}^\times$ scaling the second summand in $V\oplus W^*$. We denote by $\hbar$ its weight. When there are other $\mathbb{C}^\times$ around, we denote this one by $\mathbb{C}^\times_\hbar$.
I don't understand what this means. Does anybody have a clue? Any help would be greatly appreciated.