Refined topological amplitudes in N = 1 flux compactification

We study the implication of refined topological string amplitudes in the supersymmetric N = 1 flux compactification. They generate higher derivative couplings among the vector multiplets and graviphoton with generically non-holomorphic moduli dependence. For a particular term, we can compute them by assuming the geometric engineering. We claim that the Dijkgraaf-Vafa large N matrix model with the β-ensemble measure directly computes the higher derivative corrections to the supersymmetric effective action of the supersymmetric N = 1 gauge theory.