\epsilon-Corrected Seiberg-Witten Prepotential Obtained From Half Genus Expansion in beta-Deformed Matrix Model

We consider the half-genus expansion of the resolvent function in the $\beta$-deformed matrix model with three-Penner potential under the AGT conjecture and the $0d-4d$ dictionary. The partition function of the model, after the specification of the paths, becomes the DF conformal block for fixed $c$ and provides the Nekrasov partition function expanded both in $g_s = \sqrt{-\epsilon_1 \epsilon_2}$ and in $\epsilon = \epsilon_1+\epsilon_2$. Exploiting the explicit expressions for the lower terms of the free energy extracted from the above expansion, we derive the first few $\epsilon$ corrections to the Seiberg-Witten prepotential in terms of the parameters of SU(2), $N_{f} =4$, ${\cal N}= 2$ supersymmetric gauge theory.