Abelian and Non-abelian D-brane Effective Actions

In this Ph.D. thesis we review and elaborate on a method to find the D-brane effective action, based on BPS equations. Firstly, both for the Yang-Mills action and the Born-Infeld action it is shown that these configurations are indeed BPS, i.e. solutions to these equations saturate a Bogomolny bound and leave some supersymmetry unbroken. Next, we use the BPS equations as a tool to construct the D-brane effective action and require that (a deformation of) these equations should still imply the equations of motion in more general cases. In the abelian case we managed to calculate all order in $\alpha'$ four-derivative corrections to the effective action and the BPS equations while in the non-abelian case we obtained the effective action up to order $\alpha'^4$. Furthermore, we discuss a check based on the spectrum of strings stretching between intersecting branes. Finally, this Ph.D. thesis also discusses the construction of a boundary superspace which is the first step to use the method of Weyl invariance in N=2 superspace in order to again construct the D-brane effective action. A more detailed summary of each chapter can be found in the introduction.