On D3-brane potentials in compactifications with fluxes and wrapped D-branes

We study the potential governing D3-brane motion in a warped throat region of a string compactification with internal fluxes and wrapped D-branes. If the Kähler moduli of the compact space are stabilized by nonperturbative effects, a D3-brane experiences a force due to its interaction with D-branes wrapping certain four-cycles. We compute this interaction, as a correction to the warped four-cycle volume, using explicit throat backgrounds in supergravity. This amounts to a closed-string channel computation of the loop corrections to the nonperturbative superpotential that stabilizes the volume. We demonstrate that for warped conical spaces the superpotential correction is given by the embedding equation specifying the wrapped four-cycle, in agreement with the general form proposed by Ganor. We verify that the corrected gauge coupling on wrapped D7-branes is holomorphic. Finally, our results have applications to cosmological inflation models in which the inflaton corresponds to a D3-brane moving in a warped throat.