A global stability analysis of the boundary layer in the leading edge of a swept wing is performed in the incompressible flow regime. It is demonstrated that the global eigenfunctions display the features characterizing the local instability of the attachment line, as in swept Hiemenz flow, and those of local cross-flow instabilities further downstream along the wing. A continuous connection along the chordwise direction is established between the two local eigenfunctions. An adjoint-based receptivity analysis reveals that the global eigenfunction is most responsive to forcing applied in the immediate vicinity of the attachment line. Furthermore, a sensitivity analysis identifies the wavemaker at a location that is also very close to the attachment line where the corresponding local instability analysis holds: the local cross-flow instability further along the wing is merely fed by its attachment-line counterpart. As a consequence, global mode calculations for the entire leading-edge region only need to include attachment-line structures. The result additionally implies that effective open-loop control strategies should focus on base-flow modifications in the region where the local attachment-line instability prevails.