D3-brane action in a supergravity background: The fermionic story

Using the $\kappa$-symmetric action for a D3-brane, we study the interaction between its world-volume fermions and a bosonic type IIB supergravity background preserving 4-dimensional Lorentz invariance. We find that the renormalizable terms in the action include only coupling between the fermions and the 3-form flux in the combination ${(*}_6{G}_{\mathrm{}\left(3\right)\mathrm{}}-{\mathrm{iG}}_{\mathrm{}\left(3\right)\mathrm{}}),$ which is zero for a class of supersymmetric and nonsupersymmetric solutions. We also find the magnetic and electric dipole moments for the fermions, which are proportional to the derivative of the dilaton axion. We show that different gauges to fix the $\kappa$ symmetry give the same interaction terms, and prove that these terms are also $SL(2,\mathbf{R})$ self-dual. We interpret our results in terms of $\mathcal{N}=1\mathrm{}$ supersymmetric gauge theory on the D-brane.