On the quantization constraints for a D3-brane in the geometry of NS5 branes

A D3-brane in the background of NS5-branes is studied semi-classically. The conditions for preserved supersymmetry are derived, leading to a differential equation for the shape of the D3-brane. The solutions of this equation are analyzed. For a D3 brane intersecting the NS5-branes, the angle of approach is known to be restricted to discrete values. Four different ways to obtain this quantization are described. In particular, it is shown that, assuming the D3-brane avoids intersecting a single NS5 brane, the above discrete values correspond to the different possible positions of the D3-branes among the NS5-branes.