  We consider the one-dimensional (1D) topological superconductor that may form
in a planar superconductor-metal-superconductor Josephson junction in which the
metal is is subjected to spin orbit coupling and to an in-plane magnetic field.
This 1D topological superconductor has been the subject of recent theoretical
and experimental attention. We examine the effect of perpendicular magnetic
field and a supercurrent driven across the junction on the position and
structure of the Majorana zero modes that are associated with the topological
superconductor. In particular, we show that under certain conditions the
Josephson vortices fractionalize to half-vortices, each carrying half of the
superconducting flux quantum and a single Majorana zero mode. Furthemore, we
show that the system allows for a current-controlled braiding of Majorana zero
modes.
