Higher Derivative Corrections to the Abelian Born-Infeld Action using   Superspace Methods
  In this thesis we study two-dimensional supersymmetric non-linear sigma-models with boundaries. We derive the most general family of boundary conditions in the non-supersymmetric case. Next we show that no further conditions arise when passing to the N = 1 model and we present a manifest N = 1 off-shell formulation. Subsequently, we determine under which conditions a second supersymmetry exists. Finally we recast some of our results in N = 2 superspace. Leaning on these results we then calculate the beta-functions through three loops for an open string sigma-model in the presence of U(1) background. Requiring them to vanish is then reinterpreted as the string equations of motion for the background. Upon integration this yields the low energy effective action. Doing the calculation in N = 2 boundary superspace significantly simplifies the calculation. The one loop contribution gives the effective action to all orders in alpha' in the limit of a constant fieldstrength. The result is the well known Born-Infeld action. The absence of a two loop contribution to the beta-function shows the absence of two derivative terms in the action. Finally the three loop contribution gives the four derivative terms in the effective action to all orders in alpha'. Modulo a field redefinition we find complete agreement with the proposal made in the literature. By doing the calculation in N = 2 superspace, we get a nice geometric characterization of UV finiteness of the non-linear sigma-model: UV finiteness is guaranteed provided that the background is a deformed stable holomorphic bundle. 
Stijn Nevens