Instantons in Deformed Super Yang-Mills Theories

We study the instanton effective action in deformed four-dimensional N=2 and N=4 super Yang-Mills (SYM) theories. These deformed gauge theories are defined on the D-brane world-volume in the presence of constant, self-dual Ramond-Ramond (R-R) 3-form field strength background $\mathcal{F}$ which is scaled as $(2 \pi \alpha')^{1/2} \mathcal{F} = fixed$ in the zero-slope limit $\alpha' \to 0$. The instanton effective action is obtained by solving equations of motion of the deformed N=2 SYM action. We show that this effective action correctly reproduces the string theory result derived from D3/D(-1)-branes system in the lowest order of the backgrounds and gauge coupling constant.