Instanton calculus in deformed N=4 super Yang-Mills theories

We study the instanton effective action of four-dimensional deformed N = 4 supersymmetric Yang-Mills theory in the presence of constant, self-dual Ramond-Ramond (R-R) 3-form background in type IIB superstring theory. We compare the instanton effective action with the low-energy effective action on D(-1)-branes in the D3-D(-1) system in the same background. We find that discrepancy appears at the second order of the R-R background, which was also observed in deformed N = 2 theory. This discrepancy is resolved if the action of the deformed gauge theory is improved by introducing a term with coordinate-dependent gauge coupling. We obtain the same deformed instanton effective action from super Yang-Mills theory in ten-dimensional Omega-background by dimensional reduction. We also discuss another type of R-R 3-form background which corresponds to massive deformations of the instanton effective action.