Nekrasov & Okounkov proved (https://arxiv.org/pdf/hep-th/0306238.pdf) that the Seiberg-Witten prepotential can be given by \begin{equation} \mathcal{F}(\mathbf{a},\Lambda) = \lim_{\hbar\rightarrow 0}\hbar^2\log Z(\mathbf{a},\hbar,\Lambda). \end{equation} Where $Z(\mathbf{a},\hbar,\Lambda) = Z(\mathbf{a},+\hbar,-\hbar,\Lambda)$ is the Nekrasov's Partition function with $\epsilon_1 = -\epsilon_2 = \hbar$. Then there exists a Seiberg-Witten differential $dS = (1/2\pi i)zdw/w$ on the Seiberg-Witten curve $\Lambda^N(w + 1/w) = P_N(z)$ such that $A$ and $B$-periods are related by: \begin{equation} a_i := \oint_{A_i}dS, \qquad a^D_i := 2\pi i\oint_{B_i}dS = \frac{\partial \mathcal{F}(\mathbf{a},\Lambda)}{\partial a_i}. \end{equation} My question is what if we don't consider the $\hbar\rightarrow 0$ limit and simply define $\mathcal{F}(\mathbf{a},\hbar,\Lambda) = \hbar^2\log Z(\mathbf{a},\hbar,\Lambda)$, then can we find a $\hbar$-dependent version of Seiberg-Witten differential $dS_\hbar$ on the same Seiberg-Witten curve $\Lambda^N(w + 1/w) = P_N(z)$ such that \begin{equation} a_i(\hbar) = \oint_{A_i}dS_\hbar, \qquad a^D_i(\hbar) = 2\pi i \oint_{B_i}dS_\hbar = \frac{\partial \mathcal{F}(\mathbf{a},\hbar,\Lambda)}{\partial a_i(\hbar)}? \end{equation} In particular I'm interested about the asymptotic expansion of $dS_\hbar$ as $\hbar \rightarrow 0$: \begin{equation} dS_\hbar \approx dS + \hbar dS^{(1)} + \hbar^2dS^{(2)} + ... \end{equation} or something similar. Is there any good references for this result or any suggestion where I can read more about it?


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