Nonholomorphic corrections from three-branes inFtheory

We construct solutions of type IIB supergravity corresponding to seven-branes, an $O7\mathrm{}$ plane, and three-branes. By considering a probe moving in this background, with constant coupling and an ${\mathrm{AdS}}_5$ component in its geometry, we are able to reproduce the exact low energy effective action for $\mathcal{N}=2\mathrm{}$ super Yang-Mills theory with gauge group SU(2) and $N_f=4\mathrm{}$ massless flavors. After turning on a mass for the flavors we find corrections to the ${\mathrm{AdS}}_5$ geometry. In addition, the coupling shows a power law dependence on the energy scale of the theory. The origin of the power law behavior of the coupling is traced back to instanton corrections. Instanton corrections to the four derivative terms in the low energy effective action are correctly obtained from a probe analysis. We study how these instanton corrections are reflected in the background geometry by calculating the quark-antiquark potential. Finally we consider a solution corresponding to an asymptotically free field theory. Again, the leading form of the four derivative terms in the low energy effective action is in complete agreement with field theory expectations.