Four Dimensional Supersymmetric Theories in Presence of a Boundary

In this paper, we study $\mathcal{N} =1$ supersymmetric theories in four dimensions in presence of a boundary. We demonstrate that it is possible to preserve half the supersymmetry of the original theory by suitably modifying it in presence of a boundary. This is done by adding new boundary terms to the original action, such that the supersymmetric variation of the new terms exactly cancels the boundary terms generated by the supersymmetric transformation of the original bulk action. We also analyze the boundary projections of such supercharges used in such a theory. We study super-Yang-Mills theories in presence of a boundary using these results. Finally, we study the Born-Infeld action in presence of a boundary. We analyse the boundary effects for the Born-Infeld action coupled to a background dilaton and an axion field. We also analyse the boundary effects for an non-abelian Born-Infeld action. We explicitly construct the actions for these systems in presence of a boundary. This action preserves half of the original supersymmetry.