Boundary conditions, supersymmetry and A-field coupling for an open string in a B-field background

We discuss the non-linear sigma model representing a NSR open string in a curved background with non-zero Bμν-field. With this coupling the theory is not automatically supersymmetric, due to boundary contributions. When B=0 supersymmetry is ensured by the conditions that follow as the boundary contribution to the field equations. We show that inclusion of a particular boundary term restores this state of affairs also in the presence of a B-field. The boundary conditions derived from the field equations in this case agree with those that have been proposed for constant B-field. A coupling to a boundary Aμ-field will modify both the boundary conditions and affect the supersymmetry. It is shown that there is an A-coupling with non-standard fermionic part that respects both the supersymmetry and the shift symmetry (in the B and A fields), modulo the (modified) boundary conditions.